The Pennsylvania State University
The Graduate School
A Study on Heat Transfer in Dropwise
Condensation
A Dissertation in
Mechanical Engineering
by
Sanjay Adhikari
© 2020 Sanjay Adhikari
Submitted in Partial Fulfillment
of the Requirements
for the Degree of
Doctor of Philosophy
August 2020
ii
The dissertation of Sanjay Adhikari was reviewed and approved by the following:
Alexander S. Rattner
Assistant Professor of Mechanical Engineering
Dissertation Adviser
Chair of Committee
Fan-Bill Cheung
George L. Guillet Professor of Mechanical and Nuclear Engineering
Tak-Sing Wong
Wormely Family Early Career Professor in Engineering
Assistant Professor of Mechanical Engineering
Kristen Fichthorn
Merrell Fenske Professor of Chemical Engineering and Professor of Physics
Daniel Haworth
Associate Head for Graduate Programs, Interim
iii
Abstract
Dropwise condensation (DWC) is a high intensity heat transfer phenomenon, with heat fluxes up to an order
of magnitude higher than in filmwise condensation. Many industrial applications like desalination, thermal
management, power generation and fog water harvesting may benefit from employing DWC. Considerable work
has been done in this field and yet there are some challenges which remain open for research and further
investigation. High resolution numerical studies that capture the complex hydrodynamics of larger droplets as they
coalesce and move on the surface, effectively refreshing it for renucleation, have still not been realized as the drop
size distribution spans six orders of magnitude. A new Volume of Fluid (VOF) based multi-scale approach, which
resolves the grid scale larger droplets O(50 μm) and models the heat transfer for the subgrid scale smaller droplets,
is proposed in this dissertation. To supplement this approach, heat transfer models for individual droplets (Chapter
1) and droplet interaction (Chapter 2) are presented. Modeling work described in chapter 2 also solves many
challenges associated with resolving grid scale droplets in VOF simulations. The proposed multi-scale approach
also requires a transient heat transfer closure model for the subgrid scale droplets. Experimental studies have so far
focused only on the steady state behavior of DWC. The startup period is still not well understood. Transient
phenomena in DWC startup may significantly affect the performance of proposed applications like vapor chambers
subjected to pulsed heat fluxes, or refrigeration cycles operating intermittently. This dissertation provides first
characterization of fast transients during startup (chapter 4). A unique experimental facility and instrumentation are
developed for measurements. Inverse numerical methods are employed to extract data (q" and HTC) from these
measurements. Three distinct phases during DWC startup have been identified, which further highlight the
importance of characterizing this period. The experimental data can be coupled with the numerical work described
in this dissertation to formulate a robust, widely applicable simulation framework.
iv
This dissertation is based upon the work partially funded by the United States National Science Foundation under
the Award Number CBET-1652578. This report was prepared as an account of work sponsored by an agency of the
United States Government. Neither the United States Government nor any agency thereof, nor any of their
employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy,
completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use
would not infringe privately owned rights. References herein to any specific commercial product, process, or
service-water by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its
endorsement, recommendation, or favoring by the United States Government or any agency thereof. The views and
opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or
any agency thereof.
v
Table of Contents
List of Figures ........................................................................................................................................................ viii
List of Tables .......................................................................................................................................................... xiii
Acknowledgements ................................................................................................................................................ xiv
Chapter 1 Introduction and Literature Review ......................................................................................................1
1.1 Dropwise Condensation: Application and Mechanisms .............................................................................2
1.2 Prior Research in Dropwise Condensation .................................................................................................7
1.2.1 Major Advances in the History of Dropwise Condensation ...............................................................7
1.2.2 Heat Transfer Models of Dropwise Condensation .............................................................................7
1.2.3 Computational Approaches to Modeling Dropwise Condensation ..................................................10
1.2.4 Goals of the Present research and Dissertation .................................................................................13
Chapter 2 Detailed Characterization of Individual Droplet Heat Transfer .........................................................17
2.1 Modeling Approach ..................................................................................................................................19
2.2 Simulation Studies ....................................................................................................................................21
2.3 Results and Discussion .............................................................................................................................24
2.3.1 Proposed Correlation ........................................................................................................................24
2.3.2 Validation of Results ........................................................................................................................25
2.3.3 Internal temperature Distribution in Droplets ...................................................................................29
2.3.4 Implications of varying Conduction Resistance Factor in Dropwise Condensation ........................31
2.4 Conclusion ................................................................................................................................................33
Chapter 3 Characterizing Droplet Interactions During Coalescence ..................................................................35
3.1 Modeling Approach ..................................................................................................................................39
3.1.1 Governing equations .........................................................................................................................39
3.1.2 Surface Tension Model .....................................................................................................................42
3.1.3 Phase Change Model ........................................................................................................................43
3.1.4 Dynamic Contact Angle Model ........................................................................................................44
3.1.5 Solution Algorithm ...........................................................................................................................45
3.2 Simulation Studies ....................................................................................................................................47
3.2.1 Studied Domain and Case setup .......................................................................................................47
vi
3.2.2 Grid Resolution, Sensitivity Study and Dynamic Contact Angle Validation ...................................51
3.3 Validation of Results ................................................................................................................................54
3.3.1 Results and Discussion .....................................................................................................................57
3.3.2 Characterizing Coalescence ..............................................................................................................59
3.3.3 Effect of Coalescence on Heat Transfer ...........................................................................................64
3.3.4 Simulation Data and Proposed Correlation for Droplet Coalescence Time Constant ......................69
3.4 Conclusions ..............................................................................................................................................71
Chapter 4 Heat Transfer Measurements of Fast Transient Startup of Dropwise Condensation..........................73
4.1 Experimental Design and Setup ...............................................................................................................76
4.1.1 Apparatus ..........................................................................................................................................77
4.1.2 Test Section ......................................................................................................................................80
4.1.3 Sensor Plate and Amplification Module ...........................................................................................84
4.1.4 Dielectric Thermal Conductivity Measurement ...............................................................................88
4.2 Results and Discussion .............................................................................................................................92
4.2.1 Transient Heat Transfer Measurements ............................................................................................92
4.2.2 Heat Flux and Heat Transfer Coefficient Calculation Methodology ................................................94
4.2.3 Validation Using Constriction Resistance and Theoretical Drop-Size Distribution Model .............98
4.2.4 Discussion on Transient Heat Fluxes and Heat Transfer Coefficients ...........................................104
4.2.5 Conclusions ....................................................................................................................................110
Chapter 5 Preliminary Study on Multi-Scale Simulation of Dropwise Condensation ......................................111
5.1 Modeling Approach ................................................................................................................................114
5.1.1 Grid-Scale Modeling ......................................................................................................................114
5.1.2 Subgrid Scale Modeling .................................................................................................................115
5.2 Illustrative Domain and Case Setup .......................................................................................................117
5.3 Preliminary Results and Discussion .......................................................................................................119
5.4 Conclusions ............................................................................................................................................122
Chapter 6 Conclusions and Recommendations for Future Research ................................................................123
6.1 Numerical Simulations of dropwise condensation .................................................................................124
6.2 Experimental Measurement of Heat Transfer During Dropwise Condensation Startup ........................125
6.3 Future Research Work ............................................................................................................................126
vii
Appendix ................................................................................................................................................................128
A 1 Calibrating the Test Plate RTDs .........................................................................................................128
A 2 Population Balance Theory ................................................................................................................130
Bibliography ...........................................................................................................................................................134
viii
List of Figures
Figure 1.1 a) Schematic of the Droplet with radius r, contact angle θ, conduction resistance Rcond and interfacial
resistance Rint. b) Magnified view of the interfacial resistance ..................................................................................4
Figure 1.2 Comparison between droplet covered surfaces a) From the numerical study of Burnside and Hadi [1];
b) Proposed multi-scale simulation test surface. ..................................................................................................... 13
Figure 2.1 Representative mesh sizes: (a) coarse Δ3 = 2.2 × 102mm; (b) fine Δ2 = 1.2 × 10 − 2mm; (c) finest
Δ1 = 6.0 × 10 − 3mm ........................................................................................................................................... 21
Figure 2.2 𝑓𝐵𝑖, 𝜃 vs. Bi-1 for different contact angles. ............................................................................................ 23
Figure 2.3 Comparison of absolute values of f predicted with proposed correlation and conduction simulations. 25
Figure 2.4 Nu vs. Bi for different contact angles (50° to 80°). ............................................................................... 26
Figure 2.5 Non-dimensional temperature 𝑻−𝑻𝒘𝒂𝒍𝒍
𝑻𝒔𝒂𝒕−𝑻𝒘𝒂𝒍𝒍 distribution inside a droplet with varying Bi. a) Bi = 1000, b)
Bi = 2 and c) Bi = 0.2. ............................................................................................................................................. 29
Figure 2.6 Heat flux along the condensing surface of the droplet with 𝜽 = 𝟗𝟎° for different Bi. 𝜷 is the angle
subtended by any point on the droplet surface and the horizontal (inset). .............................................................. 30
Figure 2.7 Non-dimensional temperature distribution inside droplets with different contact angles for Bi = 1000.
a) 𝛉 = 𝟒𝟎°, b) 𝛉 = 𝟗𝟎° and c) 𝛉 = 𝟏𝟒𝟎°. .............................................................................................................. 31
Figure 2.8 Comparison of droplet growth between OpenFoam VOF simulation, theoretical relation with f obtained
using the correlation and theoretical relation with f = 0.25. .................................................................................... 33
Figure 3.1 Coalescing droplet mid-plane temperature profiles. Both parent droplet radii are 𝟐𝟓 𝛍𝐦. The
maximum velocity observed during coalescence is 𝐔𝐦𝐚𝐱 ~ 𝟑 𝐦 𝐬 − 𝟏. .............................................................. 38
ix
Figure 3.2 Advancing and receding contact angles measured using volume changing technique (liquid injection or
removal from the top of the droplet). a) Increasing volume of the drop. b) Decreasing volume of the drop. ......... 45
Figure 3.3 Phase change volume-of-fluid solver algorithm flow chart. .................................................................. 46
Figure 3.4 Representative domain for the VOF coalescence study. The face showing the droplets is the symmetry
plane; the bottom surface, which is not visible in the figure, is the condensing surface; and the remaining 4 faces
are open atmosphere. ............................................................................................................................................... 47
Figure 3.5 Steady state temperature distribution on the droplet surfaces before merging. ..................................... 49
Figure 3.6 Representative droplet isosurfaces, indicating VOF mesh resolutions for the droplets from the
parametric study. a) For smaller droplet radius 𝒓𝟏 = 𝟏 𝛍𝐦 and radius ratio Rt = 1. b) For 𝒓𝟏 = 𝟏 𝛍𝐦 and Rt = 2.
c) For 𝒓𝟏 = 𝟏 𝛍𝐦 and Rt = 4.................................................................................................................................. 51
Figure 3.7 Heat transfer grid independence study (Q vs 𝒏𝒓 where 𝒏𝒓 = # 𝒄𝒆𝒍𝒍𝒔/𝒓𝟏) for 𝒓𝟏 = 𝟏 𝛍𝐦 and 𝒓𝟏 =
𝟐𝟓 𝛍𝐦. .................................................................................................................................................................... 52
Figure 3.8 Colliding droplet contact patch diameter on surface (D) vs. time (t) for different grid resolutions and
experimental data [121]. .......................................................................................................................................... 53
Figure 3.9 Comparison of the simulation data (heat transfer rate (Q) vs. Biot number (Bi)) with that of Sadhal and
Martin [91], and conduction-only finite element COMSOL simulations. ............................................................... 55
Figure 3.10 Representative mesh sizes: (a) coarse 𝚫𝟑 = 𝟏. 𝟗𝟕 × 𝟏𝟎 − 𝟗𝛍𝐦− 𝟏. 𝟗𝟗 × 𝟏𝟎 − 𝟓𝛍𝐦; (b) fine 𝚫𝟐 =
𝟓. 𝟖𝟒 × 𝟏𝟎 − 𝟏𝟎𝛍𝐦− 𝟓.𝟗𝟎 × 𝟏𝟎 − 𝟔𝛍𝐦; (c) finest 𝚫𝟏 = 𝟏. 𝟒𝟖 × 𝟏𝟎 − 𝟏𝟎 𝛍𝐦− 𝟏. 𝟓𝟎 × 𝟏𝟎 − 𝟔𝛍𝐦. ........ 56
Figure 3.11 Total heat transfer rate (Q) vs. droplet radius (r) for water condensation at atmospheric pressure. The
dashed lines indicate the rate of variation of heat transfer with droplet radius for different size droplets. ............. 60
Figure 3.12 . Comparison of a) condensing droplet heat transfer rate vs. time, b) base area vs. time, and c) contact
line (perimeter) length vs. time for two merging drops of radii 𝟔. 𝟐𝟓 𝛍𝐦 (𝐁𝐢 ≈ 𝟎. 𝟑)........................................... 62
x
Figure 3.13 Decaying exponential function superimposed over Q vs Time plot of two equal merging drops of radii
𝟔. 𝟐𝟓 𝛍𝐦 each. ........................................................................................................................................................ 63
Figure 3.14 Differences between subgrid scale heat transfer due to microscale condensing droplets around two
coalescing r = 25 μm droplets: a) the dynamic coalescence case; and b) the static coalescence case. Top-down
view snapshots of the droplet shapes, represented in black, are presented. The colored area underneath the
droplets shows the evolution of subgrid scale heat flux due to small condensing droplets around the primary
coalescing droplets 𝐪"𝐒𝐆𝐒 ∗ 𝐭 = (𝐪"𝐒𝐆𝐒 −𝐦𝐢𝐧(𝐪"𝐒𝐆𝐒(𝐭)))/(𝐦𝐚𝐱(𝐪"𝐒𝐆𝐒(𝐭)) −𝐦𝐢𝐧(𝐪"𝐒𝐆𝐒(𝐭))). ............... 67
Figure 3.15 Cumulative heat transfer for an interface tracked dynamic coalescence case and a static case. Initially,
for both cases, both droplets have radius 𝒓 = 𝟐𝟓 𝛍𝐦. ............................................................................................ 68
Figure 3.16 Dimensionless time constant (𝝉 ∗) vs. dimensionless smaller droplet radius (𝒓𝟏 ∗) and droplet radius
ratio Rt. The curves are generated using the correlation given in Eqn. 3.22 and the simulation data is represented
by the markers. ........................................................................................................................................................ 70
Figure 4.1 Experimental setup to capture transient heat transfer data during DWC. a) Schematic of the setup, b)
Photograph of the full test stand, and c) photographs of major components: (1) Steam generator; (2) Test section;
(3) Test plate with resistance temperature detectors (RTDs) printed on both sides and heater for drying between
tests; (4) Full cone spray nozzle; (5) Primary pressurized cooling water tank; (6) Auxiliary water collection tank;
(7) Venting and vacuum arrangement for steam; (8) Peripheral heater for the test plate; (9) constant current source
for RTDs and signal amplifier; (10) Data acquisition system (DAQ); (11) View-port for camera. ........................ 78
Figure 4.2 Test Section: 3 – Test plate; 4 – Full cone spray nozzle; 2.1 – Coolant supply line; 2.2 – Coolant
discharge port to the auxiliary tank; 2.3 – Condensate return port; 2.4 – Steam inlet; 2.5 – Sight glass assembly;
2.6 – Port for venting, vacuum, and pressure transducer; 2.7 – Temperature sensor port. (Dimensions in mm) ... 80
Figure 4.3 a) Evolution of steam temperature 𝐓𝐄 with time; b) Evolution of steam pressure 𝐏𝐄 with time. ......... 81
xi
Figure 4.4 a) High speed camera arrangement; b) Camera view during the experimental run. .............................. 83
Figure 4.5 Test plate: 1 – Gold-plated copper track used as an RTD in a 4-wire configuration; 2 – Gold finger
contacts for connection to the RTD processing circuits; 3 – Area visible to the camera (Figure 4.4-b). ................ 84
Figure 4.6 a) Test plate dimensions in 𝐦𝐦. b) Stack-up of layers over the aluminum substrate on the condensing
side of the plate: (1) 100 nm thick conformal coat of parylene-C; (2) 80 nm thick (nominal) ENIG® finish layer
(3) 8 𝛍𝐦 thick copper tracks; (4) 85 𝛍𝐦 thick COBRITHERM (Benmayor Aismalibar. proprietary) dielectric
layer. ........................................................................................................................................................................ 85
Figure 4.7 a) Constant current supply and signal amplification board: 1 – Edge connector, which mates with the
test plate terminals; 2 – Signal amplification block; 3 – Constant current supply block; 4 – Signal cable connector
to the DAQ; b) Schematic of a single RTD, current and amplifier loop: 1 – Constant current supply; 2 – Voltage
signal amplification. ................................................................................................................................................ 87
Figure 4.8 Conduction resistance network schematic between surface temperature measurements on the cold and
the condensing faces of the test plate. ..................................................................................................................... 89
Figure 4.9 COMSOL® Multiphysics mesh to determine thermal conductivity of the dielectric layer: 1 – Coarse
mesh; 2 – Fine mesh; 3 – Finest mesh. .................................................................................................................... 90
Figure 4.10 DWC on the vertical test plate at different times after startup (pressure = 1 atm) ............................... 93
Figure 4.11 Schematic representing 1 dimensional discretization of the test plate, which is divided into separate
material zones with different material properties and discretization lengths: 1 – Dielectric zone (discretization
length 𝚫𝐱𝐝); 2 – Aluminum zone (discretization length 𝚫𝐱𝐀𝐥); 3 – Dielectric zone (discretization length 𝚫𝐱𝐝); 4 –
Boundary; 5 – Aluminum-dielectric interface; 6 – Aluminum-dielectric Interface; 7 – Boundary. ....................... 94
Figure 4.12 Temperature profile through the plate for time t = 0.3 s. ..................................................................... 97
xii
Figure 4.13 Schematic showing the constriction resistance 𝐑𝐬 and the promoter layer resistance 𝐑𝐂 between the
surface temperature measured by the RTDs on the condensation side (𝐓𝐬𝐮𝐫𝐟) and the Steam temperature (𝐓𝐄). 99
Figure 4.14 a) Surface temperature on the Condensing side 𝐓𝐬𝐮𝐫𝐟, surface temperature on the cooling side 𝐓𝐂
and the steam temperature 𝐓𝐄 during the experiment: 1 – First inflection point; 2 – Second inflection point; b)
Magnified presentation of the chart area indicated above it. ................................................................................. 105
Figure 4.15 Solved heat fluxes on the cooled and condensing faces of the test plate. .......................................... 106
Figure 4.16 Evolution of heat transfer coefficients on the condensing face (a, 𝐇𝐓𝐂𝐃𝐖𝐂) and cooled face (b,
𝐇𝐓𝐂𝐂). ................................................................................................................................................................... 107
Figure 4.17 Schematic cross-section representing the test plate RTD calibration setup: (1) Reference temperature
probe; (2) High thermal conductivity aluminum block; (3) intermediate thermal conductivity stone block; (4)
Temperature controlled heated plate; (5) – Thermal insulation; (6) Signals to data acquisition system; (7) Test
plate. ...................................................................................................................................................................... 128
Figure 5.1 Illustrative distribution of grid scale and subgrid scale drop sizes. Grid scale droplets are resolved using
VOF, and an average heat flux model is applied instead of resolving subgrid scale droplets............................... 113
Figure 5.2 Average heat transfer coefficient vs time of a surface (given by Glicksman and Hunt [28]). ............. 116
Figure 5.3 Representative VOF domain for multi-scale simulation test case. ...................................................... 117
Figure 5.4 a) Liquid coverage on the condensing surface after the elapse of 1 minute in the multi-scale simulation
b) 3D rendering of another test simulation with contact angle set to 𝟏𝟎𝟎°. ......................................................... 119
Figure 5.5 Cumulative surface coverage by the grid scale droplets, of the condensing surface vs droplet radii for
different times. ....................................................................................................................................................... 120
Figure 5.6 a) Grid scale heat transfer map; b) Subgrid scale heat transfer map. ................................................... 121
xiii
List of Tables
Table 2.1 Coefficients for the correlation given in Eqn. 2.10. ................................................................................ 24
Table 2.2 Average heat flux through the base of droplets with representative hydrophobic Bi and 𝜃 values,
computed with VOF solver, COMSOL conduction studies, and proposed correlation. ......................................... 28
Table 3.1 Boundary conditions for the VOF simulation domain ............................................................................ 48
Table 3.2 Simulation test matrix for the parametric study of coalescing droplets, with varying smaller radius (r1)
and the ratio between the radii of the merging droplets (Rt). Time constant 𝜏 is obtained from the simulation
results. ...................................................................................................................................................................... 58
Table 3.3 Constants for the droplet coalescence time constant correlation (Eqn. 3.22) .......................................... 69
Table 3.4 𝝉 for representative test cases obtained from best-fit exponential curves to the experimental data and the
proposed correlation (Eqn. 3.22). ............................................................................................................................ 70
Table 4.1 Grid independence calculation summary for the 1D, unsteady conduction equation solver. .................. 97
Table 4.2 Comparison between theoretically calculated heat flux and heat transfer coefficient with mean steady
state quantities from the experiment. ..................................................................................................................... 103
Table 5.1 Boundary conditions for the VOF domain. ........................................................................................... 118
xiv
Acknowledgements
First of all, I would like to thank my advisor Dr. Alexander S. Rattner who has guided me through the
entire period of my PhD. He has been patient and extremely kind. I will forever be grateful and indebted to
him. I would also like to thank all my past and present lab mates, especially Mahdi Nabil and Chris Greer, for
helping me out whenever I needed it.
I appreciate the financial support of this project by the U.S. National Science Foundation and
Department of Mechanical Engineering at The Pennsylvania State University during the past five years.
I am really thankful to my committee members: Dr. Fan-Bill Cheung, Dr. Tak-Sing Wong and Dr.
Kristen Fichthorn.
Finally, I would like to thank my family, Anand, Ashley, Hema and Sachin, for their unwavering love
and support.
1
Chapter 1
Introduction and Literature Review
2
1.1 Dropwise Condensation: Application and Mechanisms
Dropwise condensation has been identified as a promising heat transfer mechanism that can sustain
heat fluxes up to an order of magnitude higher than filmwise condensation [1]–[5]. It could play an important
role in a number of practical applications, including power generation condensers [6], desalination [7], and
thermal management [8]. For example, Brunt and Minken [9], in their 1958 study on seawater evaporative
desalination, found a 50% increase in the output of such a system using dropwise condensation. Recently,
investigations have also explored ambient fresh water harvesting using dropwise condensation (e.g., fog water
collection [10], [11] and water collection in arid/desert conditions [12]).
The dropwise condensation process is initiated from minute (<1 μm diameter) primary droplets at
discrete nucleation sites (often 107 − 109 sites mm−2), which are distributed over the condensing surface [13].
If the radius of a primary droplet is greater than the smallest thermodynamically stable value [14], it will grow
due to condensation and coalescence with neighboring droplets until it is large enough to be removed by body
forces (e.g., gravity). Larger droplets slide during coalescence and departure, clearing portions of the cooled
surface, and allowing the formation of new primary droplets at the nucleation sites.
After the initiation of dropwise condensation, the surface state evolves towards a quasi-steady condition
[2], [14]. A final drop size distribution is eventually obtained when the mean rate of condensation is balanced
by the rate of liquid removal due to departing droplets. Moreover, the number distribution (Area fraction
covered by droplets of each size range) is bimodal [15] with large droplets 𝑂(10−3) m and very small droplets
𝑂(10−6) m. The problem of predicting the steady state heat transfer rate in dropwise condensation can therefore
be divided into two parts: (1) predicting the heat transfer through individual droplets of varying sizes, and (2)
quantifying the distribution of drop sizes over the condenser surface. The effects of sliding, departing and
3
merging droplets is a localized phenomenon on a steady state surface, and a time and area averaged heat transfer
coefficient should also capture such effects. In an individual droplet, the heat of condensation transfers through
the liquid-vapor interface (R"int) and then conducts through the base of the droplet (Rcond) to the cooled surface.
Figure 1.1 illustrates the heat transfer process in a droplet. Curvature of the droplet at the liquid-vapor interface
results in lowering of the saturation temperature at the interface [16]–[18]. Many studies have claimed that the
effect of curvature and interfacial resistance is negligible [19]–[21] as compared to conduction resistance in the
droplet. However, it is shown in Chapter 2 that interfacial resistance effect is not negligible in typical
condensation conditions. Eqn. 1.1 [13] present one of the most widely used analytic models for the average
heat flux q” through a steady state surface and takes into account Rcond, R"int, curvature effect and droplet
distribution.
q" =
1
3r23
×∫
{
ΔT −2σTsathlvρr
K1rk+ K2 (
0.6270.664)
Tsathlv2 ρv
(γ + 1)(γ − 1)
[RTsat2π ]
2
}
r−23dr
r
r
1.1
Here, r is the maximum droplet radius (𝑂)~10−3m [22] for water vapor condensing at Patm, r is the
minimum thermodynamically viable radius (𝑂)~10−9m [22], ΔT is the temperature difference between the
surface and the vapor, σ is the surface tension, Tsat is the saturation temperature, hlv is the enthalpy of
vaporization, ρ is the liquid density, K1 and K2 are constants. K1 and K2 are typically assumed as constants close
to unity [13]. ρv is the vapor density, γ is the ratio of the principal specific heat capacities (cp,v/cv,v), and R is
the specific ideal-gas constant. The second term in the numerator of the integrand is due to curvature of the
droplet. The first term in the denominator of the droplet is due to conduction resistance, and the second term is
due to interfacial resistance. The drop size distribution generally used in Eqn. 1.1 is given in Eqn. 1.2 [14].
4
Figure 1.1 a) Schematic of the Droplet with radius r, contact angle θ, conduction resistance
Rcond and interfacial resistance Rint. b) Magnified view of the interfacial resistance
The drop size distribution on a surface is a critical component of heat transfer models. It has been shown
that heat transfer through a droplet of radius r can be integrated over the drop size distribution to predict overall
heat flux through a surface in steady state [22]–[24]. Several experimental [23], [25], [26], numerical [1], [27]–
[30], and theoretical [2], [14], [17] studies have helped in developing a theory in drop size distribution over a
surface during steady state dropwise condensation. Some key points are described as follows.
Eqn. 1.2 [17] gives the fraction (χ) of the surface covered by the drops having radii in the range of r to
r.
5
χ = χ (r
r)
χ(1) = 0
1.2
The above form is supported by the observation that photograph of a surface undergoing dropwise
condensation appear the same under different magnifications (self-similar) [22], [30]–[34]. Le Fevre and Rose
[14] have proposed the following relation for 𝜒, which is supported by data from the experimental studies [23],
[25], [26].
χ = 1 − (r
r)
13
1.3
Heat transfer through a droplet q"(r) of radius r can be combined with the drop size distribution to
predict average heat flux through a surface [17].
q" = ∫ q"(r) × (−χ′ (
r
r))dr
r
r
r
1.4
1.1.1 Transient Heat Transfer Modeling
q"(r) has recently been completely characterized by Chavan et al. [35] (hydrophobic surfaces) and
Adhikari et al. [36] (hydrophilic and hydrophobic surfaces). The latter study in conjunction with an evolving
transient drop size distribution, and using the approach given by Eqn. 1.4, can be utilized to predict the overall
transient heat flux through a surface. However, Limited transient drop size distribution [28] or heat flux [28],
[37] data are available in literature.
6
Most of the experiments in literature have employed temperature sensors in the substrate, under the
condensing surface, at varying depths[14], [26], [38]–[40]. Surface temperature can be extrapolated from such
measurements using Fourier’s law. However, this approach may be difficult to adapt to transient heat transfer
measurements of DWC startup. During startup, the temperature profile through the substrate will be non-linear,
making instantaneous heat flux estimation difficult. Further, conventional DWC heat transfer test sections have
relatively thick substrates, which makes transient flux measurements for significantly shorter periods infeasible.
Owing to the above-mentioned difficulties, almost all of the experimental and the numerical studies focus on
the steady state description of DWC. Limited data are available in literature for transient processes, which are
often encountered in practice. Parin et al. [41] performed a quasi-transient experimental study of DWC on nano-
structured aluminum surfaces. Their temporal resolution was on ~80 s. Numerical studies [1], [28] indicate that
such coarse resolution might be inadequate to capture the variation of transients during DWC. Results from the
present study, which uses high-intensity spray cooling, achieve start-up times of only a few seconds. The present
dissertation provides the first experimental measurements of transient heat transfer during the startup of DWC
at relatively high heat fluxes.
Prior numerical studies of dropwise condensation have either applied the concepts of fractals [30], [34]
or tracked individual droplets (Lagrangian approach) [1], [28]. These approaches have been able to give
satisfactory drop size distribution and average heat flux during steady state, with modest computational costs.
However, their transient predictions for startup have not yet been experimentally verified. The experimental
work described in this study also provides the first experimental verification for some of the earlier numerical
works. As implemented in the literature, these approaches have a number of limitations. In particular, these
approaches have treated droplets as rigid hemispheres, neglected the histories of droplets (e.g., non-quasi-steady
temperature distributions in droplets that have recently merged), and neglected the complex hydrodynamics
7
associated with droplet merging and sliding. Multi-scale VOF simulations, which utilize the numerical and
experimental findings from the present dissertation could potentially mitigate these challenges.
1.2 Prior Research in Dropwise Condensation
1.2.1 Major Advances in the History of Dropwise Condensation
Dropwise condensation was first investigated rigorously more than 80 years ago by Schmidt et al. [42]
which garnered significant attention from the scientific community as the heat transfer coefficients were
reported to be an order of magnitude higher than in filmwise condensation. Studies by Tanzola and Wiedman
[43], Kajanne [44], Brunt and Minken [9], Poll et al. [45], etc. have found significant efficiency improvements
in various heat transfer systems enabled with dropwise condensation. Due to its industrial importance, most of
the literature deals with water/steam as the working fluid. Some researchers have studied relatively high surface
tension organic fluids (e.g., ethylene glycol, propylene glycol, and nitrobenzene etc.) [4], [46], [47] and metals
(for e.g., mercury) [48]–[51] as well. As in the case of water and steam, the heat transfer coefficients were found
to be significantly greater than those found in filmwise condensation, for the above-mentioned fluids.
1.2.2 Heat Transfer Models of Dropwise Condensation
Micro-cine studies by McCormick and Westwater [52], [53], and Peterson and Westwater [54] looked
into the process of droplet nucleation and established that droplets grow at specific nucleation sites. Umur and
8
Griffith [16], studied and developed theory for droplet growth due to condensation, and validated their work by
comparing it with observed data by McCormick and Baer [20].
Due to limitations of the instrumentation and also lack of awareness about the effect of non-condensable
gases on the overall heat transfer during condensation, early experimental heat transfer data was inconsistent.
Later experiments took great care to eliminate non-condensable gases and used more sensitive instrumentation,
yielding several mutually agreeing heat transfer datasets. Studies by Tanner et al. [55], Graham [25],
Wilmshurst and Rose [56], Stylianou and Rose [5], and Tanaka and Tsuruta [57] have experimentally
characterized the average heat flux of a surface at steady state, resulting in a simple empirical equation for steam
condensation given by Rose[13].
q"[kWm−2] = T0.8 [5
ΔT
kv+ 0.3 (
ΔT
kv)2] 1.5
Here, T is the vapor temperature in Celsius, and kv is the thermal conductivity of the vapor.
Le Fevre and Rose [14], provided a theoretical approach in which heat flux through the base of a droplet
q"(r) of radius r is integrated over the entire drop size distribution over the condensing surface to predict the
overall average heat flux through the surface. Their theory has been corroborated and adjusted over the years
in the experimental studies by Rose [58] and Rose et al. [59]. The steady state distribution functions used in
such an approach have been verified by experimental works of Graham [25], and Tanasawa and Ochiai [26],
Numerical works of Glicksman and Hunt [28], and Burnside and Hadi [1], theoretical works of Rose and
Glicksman [2], and LeFevre [17], and others. Recently Kim and Kim [60] have formulated a theoretical drop
size distribution model for superhydrophobic surfaces (contact angle θ > 120°) and experimentally
corroborated it. The present work proposes to utilize such an approach to integrate q"(r) over the smaller size-
9
scale portion of an evolving drop size distribution to predict average subgrid-scale (SGS) heat flux through a
surface. The SGS heat flux result can then be used for closure in a multi-scale simulation
Stylianou and Rose [5], Griffith and Lee [61], and others [62], [63] looked into the effect of the
condensing surface thermal conductivity on the overall heat transfer and found the effect to be negligible.
However, Tanner et al. [62], Hannemann and Mikic [40], and Wilkins and Bromley [63] reported that the effect
is significant. Later studies by Tsuruta et al. [64], [65] have seemed to confirm the importance of substrate
thermal conductivity. Analysis of experimental data in this project incorporates treatment for the constriction
resistance due to thermal conductivity of the condensing surface, which is found to be a significant effect.
A major challenge in the industrial application of dropwise condensation is developing methods to
prolong it in harsh industrial environment. Researchers used promoters like PTFE (e.g., Wilmshurst and Rose
[56]) or monolayer promoters like dioctadecyl disulphide (e.g., Le Fevre and Rose [14]). The interest in
dropwise condensation subsided in the 1980s as researchers found it difficult to find industrially reliable
promoters. This field of study was revived again in the 1990s with methods identified to promote dropwise
condensation [66]. Newer superhydrophobic and more durable surface treatments [67]–[75], which may last
longer in relevant environments, have led to renewed interest in dropwise condensation.
Recent advances in surface treatment technology have brought low adhesion, nano-textured
superhydrophobic surfaces into existence [76]–[82]. These recent developments have led many researchers to
focus their attention toward understanding the fundamental nature of droplet dynamics and droplet-surface
interactions [83]–[85]. The present work also aims to characterize droplet dynamics with heat transfer in greater
detail.
10
1.2.3 Computational Approaches to Modeling Dropwise Condensation
Due to the experimental challenges, no measurements of the transient start-up of high-heat-flux
dropwise condensation have been reported, to the author’s knowledge. Some computational studies have been
performed to simulate this process. Glicksman and Hunt [28] performed a numerical study in which they tracked
the growth and coalescence of each droplet, treating them as individual liquid hemispheres. This approach can
be termed Lagrangian in the sense that it tracks evolution of individual parcels (droplets) rather than time or
space-averaged quantities (Eulerian approach). They modeled heat transfer assuming each droplet was at a
quasi-steady temperature distribution, and that heat flux did not depend on the history of individual droplets.
Moreover, coalescence of droplets was modeled as an instantaneous process and the intermediate complex
shapes of coalescing droplets, which could affect the time varying heat transfer rate, were neglected.
A similar study was conducted by Mei et al. [30]. As part of another study by Mei et al. [86], they
developed an analytical drop distribution model which was based on the concept of fractals and validated it
with a droplet tracking numerical simulation. They had a much greater number of initial nucleating droplets
which they tracked (around 50,000), compared with Glicksman and Hunt (around 1,000). They compared the
data with the experimental results of Tanasawa and Ochiai [26] and found good agreement. They conducted
numerical experiments with monodispersed (uniform size) and polydispersed initial droplets. They found that
the final steady state drop size distribution was insensitive to the initial distribution. This observation makes
their simulation technique suitable only for steady state studies. Their final predicted drop size distribution did
not match well at the extremities, that is, the smallest and the largest drop dimensions differed from
experimental observations.
11
Using the concept of fractals, Wu et al. [34] developed a drop size distribution model. Their model was
developed by randomly populating square grids and iteratively superimposing them over each other to predict
the drop size distribution. Initially a coarse grid (cell length = r) was used and random cells were chosen to
represent drops according to a fixed χ. Then the grid was refined and the available area was repopulated with
the same value of χ. The grid was refined again and the process is repeated. This method can be continued for
several iterations depending on computing resources. For the finest grid, they predicted heat flux using a mean
heat transfer coefficient hr through the base of a droplet of radius r (model of Rose [22]) as the boundary
condition for the condensing surface. A steady state energy equation was solved to get the overall heat transfer
through the surface. Heat transfer coefficients over the surface were averaged to get the mean heat transfer
coefficient h of the surface.
Because the drop size spectrum is so large (6 orders of magnitude apart [13]), Wu et al. employed a
two-scale approach. In their approach, a much finer grid over was evaluated for a domain whose dimensions
were equal to a single cell on the coarser grid. The results of the finer grid were superimposed over the coarser
grid. This approach is sensitive to the chosen value of ��. To solve the transient heat transfer problem, Wu et al.
also developed a transient model for ��, which assumed quasi-steady behavior for each time step. This method
shares similarities with the multi-scale simulations proposed here, but relies on quasi-steady assumptions. For
example, their method does noes not account for droplet hydrodynamics.
While these prior Lagrangian numerical simulations have provided useful insights into the DWC
process, they share some common limitations. They do not account for the hydrodynamics of large droplets,
treating them instead as point particles. Additionally, quasi-steady temperature and size distributions are used
for closure, which may not apply in the transient startup of DWC.
12
To enable more general predictive capabilities, the present investigation proposes a new multi-scale
transient simulation approach (Chapter 5). Coalescence of droplets is the primary mode of droplet growth for
water droplets ≥ 2 μm [15]. Therefore, it is important to study the hydrodynamics and the heat transfer process
during coalescence of droplets. Fidelity of the transient drop distribution and heat transfer models, to be
developed as part of the present work, can be improved with a proper understanding of the coalescence process
and its effect on the overall heat transfer on the surface. Cooper et al. [87], Sellier and Trelluyer [88], Wang et
al. [89], and Sikarwar et al. [90], etc. have looked into the coalescence process in detail, both numerically and
experimentally. However, these foregoing numerical and experimental studies have focused purely on the
hydrodynamics and do not consider heat transfer. Figure 1.2 shows the comparison between the droplet covered
surface of a droplet tracking numerical study by Burnside and Hadi [1], and another surface from the proposed
multi-scale numerical simulation approach (preliminary results). It can be clearly seen on the condenser surface
of the test case that several droplets are in the process of sliding and coalescing. These mechanisms allow larger
droplets to sweep the surface around them clean and initiate fresh nucleation sites, thereby affecting the overall
heat transfer on the surface.
13
Figure 1.2 Comparison between droplet covered surfaces a) From the numerical study of
Burnside and Hadi [1]; b) One of the proposed multi-scale simulation test surfaces.
1.2.4 Goals of the Present research and Dissertation
The aim of this PhD dissertation research is to address the following fundamental questions about the
phenomena involved in dropwise condensation through both experimental and computational approaches.
I. Develop a Framework for Multi-Scale Simulations of Dropwise Condensation
One of the primary objectives of this work is to develop a framework for multi-scale simulation of
dropwise condensation. Such multi-scale simulations should capture both microscale heat transfer and large-
scale dynamics of dropwise condensation simultaneously with acceptable computational costs. Smaller droplets
are not very mobile, but account for a major portion of surface heat transfer. Therefore, a relatively simple
subgrid scale modeling approach that accurately captures the size distribution and heat transfer of small
14
droplets, but neglects hydrodynamics, could be successful. Larger droplets account for a small portion of the
overall heat transfer directly (less than 10% of the total), but clear surface area during coalescence or sliding,
allowing small droplets to nucleate. Without accurately predicting such hydrodynamics, the rate of small droplet
nucleation and resulting intense heat fluxes would be incorrect. Thus, a successful simulation approach should
capture the hydrodynamics of such larger droplets. To achieve these simulation goals, the following steps are
undertaken:
a) Detailed Analysis of Heat Transfer in Individual Droplets
Models for dropwise condensation have generally assumed a statistical distribution of droplet sizes and
integrated heat transfer over the droplet size spectrum, considering droplet curvature effects on saturation
temperature, conduction thermal resistance, and interfacial resistance. Most earlier studies have assumed a
constant heat transfer factor (f = O(1)) to account for the relative contributions from conduction and interfacial
effects to total thermal resistance. However, f varies with droplet Biot number (Bi = ℎ𝑖𝑟
𝑘), where ℎ𝑖 is the
interfacial heat transfer coefficient of the droplet, r is the base radius, and k is the thermal conductivity and
contact angle (θ). Formulations for f with broad ranges of applicability are not currently available. In this
component of this dissertation, finite element simulations are performed to determine f and corresponding
numerical uncertainties for 0.0001 ≤ Bi ≤ 1000 and 10° ≤ θ ≤ 170°. This spans the active droplet size range
considered in most droplet condensation studies (e.g., for water condensing at Patm on a surface 10 K below the
ambient temperature, active droplets have 0.0005 < Bi < 300). A single explicit correlation is developed for f
and validated with available published data. A Volume of Fluid (VOF) two-phase model for capturing the
interfacial resistance is also developed to enable fully resolved simulations of condensing droplet heat transfer.
15
This study will be used to validate the drop distribution and the average heat flux data from the proposed
experiments.
b) Characterizing Droplet Interactions During Coalescence
Prior models of dropwise condensation have generally described coalescence as an instantaneous event.
However, coalescence and recovery of a quasi-steady droplet temperature profile requires a finite time during
which the droplet heat transfer rate varies. A detailed description of droplet coalescence may be needed to
accurately model the overall dropwise condensation process. Information about condensing droplet heat transfer
during this process has not yet been reported. This portion of the dissertation employs simulations to show that
the transient heat transfer during coalescence (with water as the working fluid) can be fully characterized by a
decaying exponential function with a time constant 𝜏, radius of the smaller droplet 𝑟1, and Ratio (Rt) of the radii
of two coalescing droplets. Volume-of-Fluid (VOF) simulations are performed to determine the relationship
between 𝜏, 𝑟1 , and Rt (1 μm ≤ 𝑟1 ≤ 25 μm; 1 ≤ Rt ≤ 4). For water at Patm this spans the range of droplet sizes
through which most of the heat transfer occurs on a surface (~80%). An explicit correlation for 𝜏 is developed,
which can facilitate the analysis of transient heat transfer in dropwise condensation. The approach developed
in this study can be applied to resolve grid-scale hydrodynamics of larger droplets.
c) Multi-Scale Simulation of Dropwise Condensation
Previous computational studies of dropwise condensation have applied droplet tracking Lagrangian or
fractal-based approaches, in which the drop distribution pattern is assumed to be similar for all magnification
scales. In such studies, droplets are treated as quasi-steady rigid spheres and the droplet history effects are
neglected. Moreover, in these models, the process of coalescence has been assumed instantaneous. Therefore,
such studies may incorrectly predict the overall time-averaged heat transfer rate or have limited generality.
16
Direct interface resolving approaches (e.g., VOF flow simulations) could directly predict condensing droplet
dynamics and heat transfer, but resolving the full droplet size spectrum during condensation would be
prohibitively expensive. In this study a multi-scale modeling approach is proposed that directly resolves large-
scale droplet dynamics with interface capturing flow simulations (e.g., VOF) and couples this with a subgrid
scale model for the evolving heat flux through smaller subgrid scale droplets between grid-scale droplets.
II. Experimental measurement of Transient Heat Transfer for Dropwise Condensation
The experimental thrust of this Ph.D. dissertation research seeks to collect the first experimental data
for early-transient heat transfer in high-flux dropwise condensation. While a number of experimental
investigations have been conducted to measure steady state dropwise condensation heat transfer rates, the
transient process has only been theoretically described through analytic and computational models. Although
some studies on breath figures [15], [91], [92] have considered DWC startup, they were performed at low heat
fluxes and significant non-condensable mass transfer resistances.
The transient startup process could last for a significant portion of the operating cycle of devices that
operate in a cyclic fashion, such as refrigerators or air-conditioners. To enable measurements during the early
transient startup process, the experimental setup will has a thin test section with low conduction resistance and
time scale, be exposed to a pure vapor (water is chosen as the working fluid due to its industrial and scientific
importance) environment on one side, and be cooled with a rapid supply of high-flow-rate coolant (spray
cooling) on the other side. Transient heat transfer data and high-speed visual data of the evolving drops will be
collected. These experiments are designed to enable assessment and closure of the proposed multi-scale
simulation approach.
17
Chapter 2
Detailed Characterization of Individual
Droplet Heat Transfer1
1 Chapter 2 is adapted from the 2017 paper by Adhikari et al. [36], published in the International Journal of Heat and Mass
Transfer
18
In dropwise condensation, heat from the condensation process transfers through the liquid-vapor
interface (interfacial resistance 𝑅int′′ ) and then conducts through the droplet to the cooled wall (Rcond) (Figure
1.1). Internal circulation and convection within droplets is generally assumed negligible [93]. Interfacial heat
transfer resistance is usually significant in dropwise condensation because the conduction length from the
interface to the wall approaches zero at the contact line. Therefore, interfacial thermal resistance is dominant in
the near-wall region. The relative overall contributions of interfacial and conduction resistances in droplet
condensation heat transfer depend on the Biot number (𝐵𝑖 =ℎ𝑖𝑟
𝑘𝑙) and contact angle 𝜃 [94]. Average heat flux
through the base of the droplet is given by the following relation:
𝑞" =
Δ𝑇
𝑓(Bi, 𝜃)𝑟𝑘𝑙
+12ℎ𝑖
2.1
Here, 𝑞" is the average heat flux through the base of the droplet, Δ𝑇 is the temperature difference
between ambient vapor temperature and the base, 𝑟 is the radius of curvature for the droplet, and 𝑘𝑙 is the liquid-
phase thermal conductivity. 𝑓(Bi, 𝜃) is a scaling factor for the conduction contribution to overall droplet thermal
resistance. 𝑓 can be considered as an inverse fin efficiency for the droplet on the cooled wall, and should
approach 0 for an infinitely conductive droplet (limBi→0 𝑓 = 0). Most prior studies assumed a constant factor f
= O(1) [1], [23], [28]. This assumption may be suitable for steady-state analyses where the distribution of drop
sizes is constant. However, an effective average f value would still be specific to the operating conditions. For
cases where the droplet size distribution evolves over time (e.g., dropwise condensation startup), the variation
of f as the droplet size distribution develops must be considered. Several studies have been performed to
determine formulae for f(Bi, θ). Sadhal and Martin [95] and Kim and Kim [60] have proposed analytical models
19
for hydrophilic surfaces (θ < 90°). Recently, Miljkovic et al. [96] and Chavan et al. [35] have reported numerical
and experimental results for hydrophobic surfaces (θ > 90°). Generally valid models for f(Bi, θ) would facilitate
dropwise condensation analyses, but are not currently available.
In this chapter, finite element heat transfer analyses are performed for 2958 cases to develop a single
broadly applicable droplet condensation heat transfer formulation for 0.0001 ≤ Bi ≤ 1000 and 10° ≤ θ ≤ 170°.
Discretization errors are evaluated to determine uncertainty bounds for Richardson-extrapolated f factors.
Results are validated against prior findings from Sadhal and Martin [95] and Chavan et al. [35]. Summarized
data and simulation results from this study are publicly available at [97].
2.1 Modeling Approach
In this study, finite element analyses are performed for heat transfer in sessile droplets using COMSOL
MultiPhysics® (v. 5.2 [98]). In typical dropwise condensation conditions, more than 90% of the heat is
transferred through droplets less than 200 𝜇𝑚 in diameter [2]. Droplets in this size range are sufficiently small,
such that Marangoni circulation and other convection effects can be neglected [57]. Therefore, conduction is
the predominant mode of heat transfer. As droplets grow, the internal temperature distribution varies. However,
the conduction-front propagation velocity (𝛼𝑙
𝑟) is generally much greater than the droplet growth rate (
𝑑𝑟
𝑑𝑡);
therefore, the heat transfer process can be modeled as quasi-steady. For example, for a water droplet with 1 𝜇𝑚
diameter and a 90° contact angle on a surface at 99.15° C (𝑇𝑠𝑎𝑡 = 100.15° 𝐶) the conduction front velocity is
𝛼𝑙
𝑟~1.6 × 10−1 m s−1. Applying a standard droplet condensation heat transfer model (e.g., from [23] with
constant f = 0.25), the predicted growth velocity would only be 𝑑𝑟
𝑑𝑡= 8.4 × 10−5 m s−1. For similar conditions
20
a 1 mm droplet will have 𝛼𝑙
𝑟= 1.6 × 10−4 m s−1 and
𝑑𝑟
𝑑𝑡= 6 × 10−7 m s−1. Based on these arguments, the in-
droplet condensation heat transfer process reduces to conduction in a solid medium, with the following
equation:
∇2𝑇 = 0 2.2
For small droplets, as is the focus here, surface tension forces dominate gravity. Therefore, the droplets
can be modeled as truncated spheres [35], and evaluated with 2-D axisymmetric domains (Figure 2.1). A fixed
temperature boundary condition is applied on the base of the axisymmetric droplet domain (1 K below the
saturation temperature), representing the cooled condenser surface temperature. A convection boundary
condition is applied to the liquid-vapor interface to account for interfacial resistance, with the following heat
transfer coefficient [99]:
ℎ𝑖 = (2𝜎
2 − 𝜎) ∗ (
ℎ𝑙𝑣2
𝑇𝑣𝑣𝑙) ∗ (
𝑀
2𝜋𝑅𝑇𝑣)1/2
2.3
Here, ℎ𝑖 is the interfacial heat transfer coefficient, 𝜎 is the accommodation constant, ℎ𝑙𝑣 is the enthalpy
of vaporization, 𝑇𝑣 is the ambient temperature, 𝑀 is the molecular mass of the working fluid, and R is the
specific gas constant. The saturation temperature of the vapor at the ambient pressure (𝑇𝑠𝑎𝑡) is substituted for
𝑇𝑣 in Eqn. 2.3.
The domain was meshed with nearly uniform size triangular elements (Figure 2.1). The governing
equation was discretized with second order accurate elements and solved to a relative residual of 10−14. The
21
total droplet heat transfer rates obtained from the FEA were used to determine 𝑓(𝐵𝑖, 𝜃), using the relation given
in Eqn. 2.1, for varying Biot number and contact angle values.
Figure 2.1 Representative mesh sizes: (a) coarse 𝚫𝟑 = 𝟐. 𝟐 × 𝟏𝟎
𝟐𝐦𝐦; (b) fine 𝚫𝟐 =𝟏. 𝟐 × 𝟏𝟎−𝟐𝐦𝐦; (c) finest 𝚫𝟏 = 𝟔. 𝟎 × 𝟏𝟎
−𝟑𝐦𝐦
2.2 Simulation Studies
A parametric study for droplet condensation heat transfer was performed. The contact angle (𝜃) was
varied to span both the hydrophilic and hydrophobic regimes (10° – 170°) in 10° increments.Droplet
conductivity (𝑘𝑙) was varied with fixed radius (r = 1 mm) and interfacial resistance (ℎ𝑖=10000 W m-2 K-1) to
evaluate 174 Bi values from 0.0001 to 1000 for each contact angle. For water at atmospheric pressure, this Bi
range spans droplet radii from 310 nm to 3.1 mm. In total, 2958 cases were studied.
Grid sensitivity studies were performed for each case with at least three meshes to extrapolate
converged values of 𝑓(Bi, 𝜃) and corresponding uncertainties following the method of Celik et al. [100]. The
technique is briefly described below.
22
Meshes (Figure 2.1) with average element dimensions: Δ3 = 2.2 × 10−2 mm, Δ2 = 1.2 × 10
−2 mm
and Δ1 = 6.0 × 10−3 mm are employed. Subscript 3 refers to the coarsest mesh and subscript 1 refers to the
finest mesh. The steps are as follows:
𝜀32 = 𝑓3 − 𝑓2; 𝜀21 = 𝑓2 − 𝑓1; 𝑠 = 𝑠𝑔𝑛 (
𝜀32𝜀21) 2.4
𝑅32 =
Δ3Δ2; 𝑅21 =
Δ2Δ1
2.5
𝑞(𝑝) = ln(𝑅21𝑝− 𝑠
𝑅32𝑝− 𝑠
) ; 𝑝(𝑞) =| ln (|
𝜀32𝜀21|) + 𝑞|
ln (𝑅21) 2.6
𝑓𝑒𝑥𝑡 = (𝑅21
𝑝𝑓1 − 𝑓2)/(𝑅21
𝑝− 1) 2.7
𝑒𝑎21 = |
𝑓1 − 𝑓2𝑓1
| ; GCI =1.25𝑒𝑎
21
𝑅21𝑝− 1
2.8
𝑢𝑛𝑐 = 𝑓𝑒𝑥𝑡 × GCI 2.9
The 𝜀 values are the changes in f (the heat transfer scaling factor) between meshes. R values are the
mesh refinement ratios. p is the empirical convergence rate, defined implicitly in terms of parameter q. 𝑓𝑒𝑥𝑡 is
the extrapolated value of f for Δ → 0 (infinitely fine mesh). 𝑒𝑎21 is the relative error between the two finest
meshes. GCI is the grid convergence index, a relative uncertainty estimate for 𝑓𝑒𝑥𝑡. The absolute uncertainty in
𝑓𝑒𝑥𝑡 (unc) is determined from Eqn. 2.9.
All studied cases were found to be monotonically converging (𝑠 = 1) and therefore the grid
convergence index (GCI) uncertainty estimate for 𝑓𝑒𝑥𝑡 was reported (<3% uncertainty in all cases). Summarized
data and simulation results from this study are publicly available at [97].
23
Figure 2.2 shows the variation of 𝑓 with (Bi-1) for different contact angles (50° to 130°). Linear trends
of f with Bi-1 were found to continue for Bi = 10-1 – 10-4, and are not included in Figure 2.2 so that sharper
variations for large Bi are apparent. Similar trends of increasing f with Bi-1 for 𝜃 < 50° and decreasing f with
Bi-1 𝜃 > 130°, were found. Corresponding very low and high contact angle curves are therefore omitted from
Figure 2.2 for presentation clarity. Negative f values were found for hydrophobic droplets with low Bi values.
Such droplets have relatively low internal conduction resistances and greater surface area for condensation than
for the baseline θ = 90° droplet assumed in Eqn. 2.1. Thus, the negative f value yields a lower overall thermal
resistance than for interfacial resistance alone on a θ = 90° droplet.
Figure 2.2 𝒇(𝑩𝒊, 𝜽) vs. Bi-1 for different contact angles.
24
2.3 Results and Discussion
2.3.1 Proposed Correlation
Simulation data (0.0001 ≤ Bi ≤ 1000, 𝜃 = 50° − 130°) for 𝑓(Bi, 𝜃) were fit to the following analytic
correlation using non-linear regression. The range of f is spread over 4 orders of magnitude. Therefore, data
points were weighted by 𝑓−2 in the fitting procedure to offset skew toward high magnitude values. The
correlation is given in Eqn. 2.10 and the coefficients are provided in Table 2.1. In Eqn. 2.10, the contact angle
𝜃 is in degrees.
𝑓(𝐵𝑖, 𝜃) = (𝐴 + 𝐵 ∗ 𝐵𝑖−1) + (𝐶 + 𝐷 ∗ 𝐵𝑖−1) ∗ 𝜃
+ [𝐸 ∗ log10(𝐵𝑖−1) + 𝐹 ∗ {log10(𝐵𝑖
−1)}2 + 𝐺 ∗ {log10(𝐵𝑖−1)}3]
2.10
Coefficient Value
(Uncertainty, 95% confidence interval)
A 0.2160 (±0.0015)
B 0.7278 (±0.0015)
C 0.001465 (±0.000013)
D -0.008086 (±0.000017)
E 0.1012 (±0.0007)
F -0.01378 (±0.00036)
G -0.007361 (±0.00013)
Table 2.1 Coefficients for the correlation given in Eqn. 2.10.
The absolute average deviation (AAD) for the data range (0.0001 ≤ Bi ≤ 1000, 50° ≤ θ ≤ 130°) is 7.4%.
Figure 2.3 compares the absolute values of f obtained from the correlation with conduction simulation results.
The correlation predicts 91% of cases with | f | < 1 within 25% of simulation values. All cases with | f | > 1 are
predicted within 25% of simulation values.
25
Because of the wide range of f values, it was difficult to fit a simple analytic correlation over 10° ≤ θ ≤
170°. Therefore, a Matlab® [101] function for 𝑓(Bi, 𝜃) is provided [97] that directly interpolates simulation
results to provide precise values overall the full parametric space (0.0001 ≤ Bi ≤ 1000, 10° ≤ θ ≤ 170°).
Figure 2.3 Comparison of absolute values of f predicted with proposed correlation and
conduction simulations.
2.3.2 Validation of Results
For hydrophilic conditions the proposed correlation for f (Eqn. 2.10) was validated with data from
Sadhall and Martin [95]. Droplet Nusselt number Nu =𝑞
Δ𝑇𝑟𝑘 was compared. 𝑞 is the total heat transferred
through the base of a droplet. The AAD from the study, for the whole range of Bi and 𝜃, is 3.4%. Figure 2.4
26
compares the variation of Nu with Bi. Numerical uncertainties are not presented with the simulation data points
in the figure because they are too small to be visible (< 10−4% ).
Figure 2.4 Nu vs. Bi for different contact angles (50° to 80°).
The only available data for f in hydrophobic droplets was reported by Chavan et al. [35]. They have
provided three correlations for 10-1 < Bi < 105. This span of Bi does not cover the typical full size range of
active droplets condensing on a surface. For example, for water condensing at 𝑃𝑎𝑡𝑚 on a surface 10 K below
the ambient temperature, active droplets have 0.0005 < Bi < 300 (0.0001 ≤ Bi ≤ 1000 evaluated in the present
27
study). Good agreement between the proposed correlation (Eqn. 2.10) and the correlations of Chavan et al. [35]
are found in the overlapping data region (0.1 < 𝐵𝑖 < 1000, 90° < 𝜃 < 130°), with AAD = 10.93 %. The
comparison is provided in table format [97].
As there are no other published data to validate hydrophobic droplet condensation heat transfer, five
representative hydrophobic cases were also evaluated in interThermalPhaseChangeFoam [102], a volume-of-
fluid (VOF) based CFD solver for phase-change heat transfer processes. VOF is an interface capturing
technique, which tracks the volume fraction of the liquid phase in each mesh cell with a scalar 𝜙 ∈ [0, 1]. Eqn.
2.11 is the advection equation for 𝜙, where 𝑢𝑖 is the velocity vector and �� is the volumetric phase fraction
source due to condensation phase change. A single set of governing equations is solved for both the liquid and
the vapor phases. Fluid material properties are weighted by Eqn. 2.12.
𝜕𝜙
𝜕𝑡+
𝜕
𝜕𝑥𝑖(𝑢𝑖𝜙) = �� 2.11
𝛽 = 𝜙𝛽𝐿 + (1 − 𝜙)𝛽𝑉 𝛽 ∈ [𝜌, 𝜇, 𝑘] 2.12
To reduce the VOF computation time for the sessile, quasi-steady droplets considered here, the
momentum and phase fraction advection (Eqn. 2.11) equations are not evaluated (velocity field 𝑢 = 0, phase-
fraction field 𝜙 constrant). The steady state VOF thermal transport equation (Eqn. 2.13) is solved for a water
droplet (𝑟 = 1 𝑚𝑚) with different contact angles (90° to 130°). ��𝑝𝑐 is the volumetric condensation phase-
change heat source term applied on the interface. Interfacial resistance is applied as a distributed source term
on the diffuse interface region (~4 mesh cells thick). Volume generation is disabled to model the quasi-steady
conduction problem.
28
𝜕
𝜕𝑥𝑖𝜕𝑥𝑖(𝑘𝑇) = ��𝑝𝑐 2.13
The VOF domain is axisymmetric and 4 mm× 4 mm in height and width. Meshes with average
element dimensions: Δ3 = 2.5 × 10−2 mm, Δ2 = 1.25 × 10
−2 mm and Δ1 = 6.25 × 10−3 mm are employed.
The base of the droplet is maintained at a temperature 2 K below the saturation temperature (𝑇𝑠𝑎𝑡 = 373.15° 𝐾).
The heat transfer in the vapor region is not included for comparison, and only the heat transfer through the base
of the droplet is employed for comparison with the COMSOL® (droplet-only) data. The results from
interThermalPhaseChangeFoam are also Richardson-extrapolated following the procedure from Eqns. 2.4 -
2.9.
Table 2.2 compares the average heat flux 𝑞", obtained using interThermalPhaseChangeFoam (VOF),
COMSOL conduction simulations, and the proposed correlation (Eqn. 2.10). Maximum error is less than 1.4
%. This close agreement between results from two different software packages with different simulation
methodologies further confirms the accuracy of reported hydrophobic droplet heat transfer data.
Biot No.
(Bi)
Contact
Angle (𝜽)
q” (W m-2)
VOF
q” (W m-2)
COMSOL
q” (W m-2)
Correlation
1.98 × 10−3 90° 8.87 × 105 ± 4300 8.75 × 105 ± 3.15 8.75 × 105
5.88 × 102 100° 5.43 × 103 ± 0.50 5.40 × 103 ± 700 5.57 × 103
2.40 110° 3.75E × 105 ± 6.0 3.72 × 105 ± 8600 3.82 × 105
0.10 120° 1.50 × 106 ± 7.0 1.47 × 106 ± 27000 1.43 × 106
3.68 × 10−1 130° 1.32 × 106 ± 3200 1.30 × 106 ± 38000 1.28 × 106
Table 2.2 Average heat flux through the base of droplets with representative hydrophobic Bi and 𝜃 values,
computed with VOF solver, COMSOL conduction studies, and proposed correlation.
29
2.3.3 Internal temperature Distribution in Droplets
Temperature distribution trends inside droplets were found to depend more strongly on Bi than θ. As
Bi reduces, the conduction resistance also reduces relative to the interfacial resistance. Thus, as Bi reduces, the
internal droplet temperature distribution becomes more uniform. For higher Bi, internal conduction resistance
is significant and condensation heat transfer primarily occurs in a thin region near the contact line. Figure 2.5
presents the temperature distribution inside a droplet with varying Bi (𝜃 = 90°, droplet geometry and conditions
described in Section 2.2). It is evident from Figure 2.5 that for Bi < 0.2, the droplet is effectively isothermal.
This effect can also be observed in Figure 2.6, which presents the local heat flux at a given angle ranging from
𝛽 = 0° at the contact line to 𝛽 = 90° at the top of the droplet. For Bi = 0.2, condensation heat flux only varies
by 19% over the surface of the droplet. In contrast, for Bi = 1000 (high Bi), 90% of the heat transfer occurs in
the 20° of the droplet surface closest to the contact line.
Figure 2.5 Non-dimensional temperature (
𝑻−𝑻𝒘𝒂𝒍𝒍
𝑻𝒔𝒂𝒕−𝑻𝒘𝒂𝒍𝒍) distribution inside a droplet with varying
Bi. a) Bi = 1000, b) Bi = 2 and c) Bi = 0.2.
30
Figure 2.6 Heat flux along the condensing surface of the droplet with 𝜽 = 𝟗𝟎° for different Bi.
𝜷 is the angle subtended by any point on the droplet surface and the horizontal (inset).
Trends of temperature distribution inside droplets were not found to depend strongly on the contact
angle. As can be observed in Figure 2.7, qualitatively similar temperature distributions are found for θ = 40° -
140° for a droplet at Bi = 1000.
31
Figure 2.7 Non-dimensional temperature distribution inside droplets with different contact
angles for Bi = 1000. a) 𝛉 = 𝟒𝟎°, b) 𝛉 = 𝟗𝟎° and c) 𝛉 = 𝟏𝟒𝟎°.
2.3.4 Implications of varying Conduction Resistance Factor in Dropwise Condensation
Many prior dropwise condensation studies have assumed a constant value of conduction factor f.
However, this approximation can yield incorrect heat transfer predictions for cases with time-varying droplet
size distributions. As an example, consider the growth of a single water droplet (𝜃 = 90°, 𝑟 = 100 μm), on a
surface maintained 10 K below the saturation temperature (𝑇𝑠𝑎𝑡 = 373.15° 𝐾). The quasi-steady heat transfer
through a growing droplet is given as [103]:
𝑞 = 𝜌ℎ𝑙𝑣2𝜋𝑟2𝑑𝑟
𝑑𝑡 2.14
From Eqn. 2.14 and Eqn. 2.1, the growth rate of a droplet is:
𝑑𝑟
𝑑𝑡 =
Δ𝑇
2𝜌ℎ𝑙𝑣×
1
𝑓𝑟𝑘+
12ℎ𝑖
2.15
32
Eqn. 2.15 is integrated for 1.5 s to predict the growth of the droplet. This analysis is performed
assuming both a constant value of f = 0.25 [28] and the proposed correlation for f (Eqn. 2.10), which varies
with the instantaneous droplet radius.
This droplet growth case is also evaluated using the interface-capturing volume-of-fluid (VOF) based
solver, interThermalPhaseChangeFoam to provide a high-resolution reference case. The domain is
axisymmetric and 400 μm × 400 μm in height and width. A fine mesh with an average element dimension Δ =
0.625 μm is employed. The momentum and phase fraction advection equations were solved here, unlike in the
quasi-steady cases described in the previous section.
Assuming a constant value of f yields a 23% relative error in the droplet growth with respect to the
VOF simulation (Δrf = 0.25 = 71 μm, ΔrVOF = 92 μm). Based on the trend at t = 1.5 s (Figure 2.8), this deviation
would continue to grow if the analysis were continued to a later time. However, the proposed correlation for f
results in only a 3% error. This analysis also indicates the accuracy of the proposed correlation, and its
applicability to unsteady droplet condensation cases.
33
Figure 2.8 Comparison of droplet growth between OpenFoam VOF simulation, theoretical relation
with f obtained using the correlation and theoretical relation with f = 0.25.
2.4 Conclusion
In this study, finite element analyses were performed for condensation heat transfer in sessile droplets
accounting for internal conduction and interfacial resistance. Parametric studies were performed over a wide
range of Biot numbers (0.0001 ≤ Bi ≤ 1000) and contact angles (10° ≤ 𝜃 ≤ 170°) to determine conduction
thermal resistance factors (f) and corresponding uncertainties. Summarized data and simulation results from
34
this study are publicly available at [97]. A new correlation was proposed for 𝑓(𝐵𝑖, 𝜃) that is valid over the
studied range of conditions. This model was validated against an earlier hydrophilic model, only applicable for
part of this range. Independent VOF simulations were performed to validate hydrophobic predictions.
An analysis was performed to highlight the importance of accounting for the variation of f with different
size droplets. As shown in Figure 2.8, assuming a fixed value of f can lead to large errors in droplet growth
predictions. The proposed, broadly applicable, correlation for f can thus facilitate future dropwise condensation
analyses and research.
It is important to note that the condensing surface was smooth for these studies and 𝜃 > 120° has not
yet been observed for smooth surfaces. For 𝜃 > 120° surface roughness needs to be accounted for, i.e., Biot
number for the substrate also becomes important for rough surfaces.
35
Chapter 3
Characterizing Droplet Interactions During
Coalescence2
2 Chapter 3 is adapted from the 2018 paper by Adhikari et al. [146], published in the International Journal of Heat and Mass
Transfer
36
Beysens and Knobler [15] described three regimes of dropwise condensation with a 100% relative
humidity nitrogen-steam mixture flowing over a cooled surface (vapor temperature 𝑇𝑉 = 23°C; surface
temperature 𝑇𝑆 = 21°C). Initially, in the first regime, for 𝑡 ≤ 1 s, a surface is covered by a homogenous pattern
of small droplets with both 𝑟 and 𝑎 smaller than 2 μm (here 𝑡 is time, 𝑟 is the radius of a droplet and 𝑎 is the
distance between two droplets). The droplets primarily grow by direct condensation [104]. In the second regime,
for 1 ≤ 𝑡 ≤ 300 s and 2 < 𝑟 ≤ 300 μm, the droplets primarily grow due to coalescence. In the third regime,
𝑡 > 300 s, the droplet size distribution on the surface reaches a statistically steady state condition, with
equilibrium rates of droplet removal, production, and growth. Their finding suggests that coalescence is the
dominant mode of growth for ambient pressure water droplets with r > 2 μm. Following the size distribution
model of Graham and Griffith [23] such droplets would account for 60-70% of the surface coverage during
dropwise condensation, indicating the importance of understanding the coalescence process.
A number of recent studies have investigated the details of droplet coalescence during dropwise
condensation. Chu et al. [105] have studied the effect of surface wettability and contact angle hysteresis on
droplet coalescence. Sprittles et al. [106], [107] have developed a finite element framework, which considers
formation and then gradual disappearance of an internal interface, to study the coalescence of two merging
droplets. Their numerical framework was also used by Enright et al. [108] to model coalescence induced
jumping of droplets on ultra-low adhesion surfaces. Enright et al. [108] also validated their findings
experimentally. Chen et al. [109] have also conducted experiments on superhydrophobic surfaces with
hierarchical microscale/nanoscale roughness to study coalescence induced jumping of droplets.
While many such investigations have advanced understanding of droplet coalescence hydrodynamics
[87], [88], [113]–[117], [105]–[112], limited information is available on heat transfer during coalescence.
Numerical simulations have been conducted of the overall dropwise condensation process, tracking growth of
37
individual (ideal hemispherical) droplets in a Lagrangian sense [1], [30]. In these studies, the heat transfer rate
of individual droplets has been modeled in a quasi-steady function (independent of droplet history), with heat
transfer rates depending on current droplet radius, fluid properties, and operating conditions (surface
temperature and saturation temperature) [112]. Such studies have modeled coalescence as an instantaneous
event [1], [28], [30]. Typically, in these models, when two parent droplets overlap at the end of a simulation
time-step, they are replaced with a single larger hemispherical child droplet (volume conserved) with its center
at the volume-weighted mean of the parent droplet centers. At the next simulation time-step, the heat transfer
rate of the larger droplet corresponds to the quasi-steady value for its size. However, the actual droplet
coalescence proceeds over a finite time interval, and the resulting child droplet requires a period of time to
recover the quasi-steady temperature profile and heat transfer rate. Additionally, these models do not predict
the unsteady hydrodynamics of coalescence. Actual droplets can have complex time varying shapes during
coalescence, and the heat transfer rate depends on the shape of the droplet base [118]. Moreover, coalescing
droplets oscillate and sweep the area around them, repeatedly re-initializing nucleation on the surrounding
surface, and potentially increasing the overall heat transfer rate.
It is not yet known what role coalescing droplet transient heat transfer and hydrodynamics have on the
overall dropwise condensation process. Figure 3.1 shows a representative time series of the coalescence process
with two droplets of equal radii (25 μm), obtained using the interface tracking (Volume of Fluid – VOF)
simulation approach method presented later in this paper. This study seeks to provide initial quantitative data
on droplet coalescence heat transfer to inform improved models of dropwise condensation.
38
Figure 3.1 Coalescing droplet mid-plane temperature profiles. Both parent droplet radii are 𝟐𝟓 𝛍𝐦.
The maximum velocity observed during coalescence is 𝐔𝐦𝐚𝐱 ~ 𝟑 𝐦 𝐬−𝟏.
Considering the simplified coalescence assumption of prior modeling studies of dropwise
condensation, the present study seeks to:
1. Perform interface resolving simulations of droplet coalescence to identify heat transfer trends and
timescales
2. Determine the extent to which the instantaneous coalescence assumption under-predicts
condensing droplet heat transfer rates
3. Formulate an analytic model for time-dependent heat transfer after droplet coalescence, which
could be incorporated into Lagrangian simulations of dropwise condensation to improve heat
transfer predictions
4. Estimate the degree of heat transfer enhancement due to repeated clearing of droplets from the
surrounding wall area by oscillating coalescing droplets
39
To achieve these goals, this study performs Volume-of-Fluid (VOF) simulations of coalescence of pairs
of water droplets in saturated steam environment at 100 kPa on a previously characterized surface (hydrophobic
silane). The VOF approach is validated using hydrodynamic data from the literature [95] and a set of
conduction-only Finite Element (FEM) heat transfer simulations (based on the approach given by Adhikari et
al.[36]). A parametric study is performed for varying droplet radii (𝑟 ∈ [1, 100] μm – most of the heat ~80%
is transferred through this range of droplet sizes [2]) and varying ratios of the merging droplet radii (Rt ∈ [1, 4]).
It is shown that transient condensation heat transfer during coalescence can represented by an exponential curve
with time constant 𝜏. A simple correlation is proposed for 𝜏 for the range of conditions in this parametric study.
A representative coalescence case is simulated with a subgrid scale model to estimate the heat transfer
enhancement due to an oscillating droplet repeatedly clearing surrounding microscopic droplets.
3.1 Modeling Approach
3.1.1 Governing equations
This study employs the Volume of Fluid (VOF) interface tracking formulation, as implemented in
OpenFOAM® [119]. VOF is an interface capturing technique in which the liquid volume fraction in each
computational cell is represented by a scalar field 𝜙 ∈ [0,1] (𝜙 = 0 all vapor in cell, 𝜙 = 1 all liquid in cell),
and a single set of governing mass, momentum, and thermal energy transport equations is solved for both
phases. The phase fraction field is advected by the solved velocity field (Eqn. 3.1). Fluid material properties at
each point are determined as a 𝜙-weighted average of liquid and vapor values (Eqn. 3.2, 3.3).
40
𝜕𝜙
𝜕𝑡+
𝜕
𝜕𝑥𝑖(𝑢𝑖𝜙) = ��𝑝𝑐 3.1
𝛽 = 𝜙𝛽𝐿 + (1 − 𝜙)𝛽𝑉 𝛽 ∈ [𝜌, 𝜇, 𝑘] 3.2
𝑐𝑝 =
𝜙𝑐𝑝,𝐿𝜌𝐿 + (1 − 𝜙)𝑐𝑝,𝑉𝜌𝑉𝜙𝜌𝐿 + (1 − 𝜙)𝜌𝑉
3.3
Here 𝑢𝑖 is the velocity vector, ��𝑝𝑐 is the volumetric phase fraction source due to condensation phase
change, 𝛽 represents the fluid property in a cell and the subscripts 𝐿 and 𝑉 represent the liquid phase and the
vapor phase, respectively. 𝜌 is the density, 𝜇 is the viscosity, 𝑘 is the thermal conductivity, and 𝑐𝑝 is the mass
weighted specific heat capacity in a cell [102], [120], [121]. The fluid properties are assumed constant for the
vapor and liquid phases.
The present simulations focus on the period between initiation of coalescence between two parent
droplets and the recovery of a quasi-steady temperature profile in the child droplet. During this period, the
increase in liquid volume in a droplet due to condensation is assumed to be small. Additionally, the momentum
transfer from the condensing vapor to the droplets is relatively low. Therefore, the process can be modeled
assuming incompressible flow (Eqn. 3.4, ��𝑝𝑐 = 0) and by using the incompressible Navier-Stokes equation
(Eqn. 3.5).
𝜕𝑢𝑖𝜕𝑥𝑖
= 0 3.4
𝜕(𝜌𝑢𝑖)
𝜕𝑡+
𝜕
𝜕𝑥𝑗(𝜌𝑢𝑖𝑢𝑗) −
𝜕
𝜕𝑥𝑗[𝜇 (
𝜕𝑢𝑖𝜕𝑥𝑗
+𝜕𝑢𝑗
𝜕𝑥𝑖)] = −
𝜕𝑝
𝜕𝑥𝑖+ 𝜌𝑔𝑖 + 𝑓𝑠,𝑖 3.5
41
Here, 𝑔𝑖 is the gravity vector and 𝑓𝑠,𝑖 is the surface tension force (applied on a volumetric basis,
described in Section 3.1.2).
The justification for the incompressible flow assumption is as follows. The characteristic velocities (𝑉𝑐)
due to droplet coalescence are much greater than the droplet growth rate (𝑑𝑟
𝑑𝑡). 𝑉𝑐 is defined as the distance
moved by the larger of the two droplets divided by the time it takes for the oscillations to die out. The droplet
front velocity due to condensation phase change is 𝑑𝑟
𝑑𝑡 (Eqns. 3.6, Eqn. 3.7) [122]. Based on simulation results
described later, for a water droplet with 1 μm diameter on a surface at 99.15° C (𝑇𝑠𝑎𝑡 = 100.15° C)
𝑉𝑐 ~ 8.3 m s−1. The predicted droplet growth rate due to condensation is:
𝑑𝑟
𝑑𝑡 ~ 8.4 × 10−5 m s−1. For similar
conditions, a 1 mm droplet will have 𝑉𝑐 ≈ 2.4 × 10−3 m s−1 and
𝑑𝑟
𝑑𝑡= 6 × 10−7 m s−1.
𝑑𝑟
𝑑𝑡 =
Δ𝑇
2𝜌ℎ𝐿𝑉×
1
𝑓𝑟𝑘+
12ℎ𝑖
3.6
ℎ𝑖 = (2��
2 − ��) ∗ (
ℎ𝐿𝑉2
𝑇𝑉𝑣𝐿) ∗ (
𝑀
2𝜋𝑅𝑇𝑉)1/2
3.7
Here, Δ𝑇 is the temperature difference between the cooled surface and the vapor, k is the thermal
conductivity of the liquid, and 𝑓 = 𝑓(Bi, θ) is the coefficient for conduction resistance [36], ℎ𝑖 is the interfacial
heat transfer coefficient, �� (0.03 [122]) is the accommodation constant, ℎ𝐿𝑉 is the enthalpy of vaporization, 𝑇𝑉
is the ambient temperature, 𝑀 is the molecular mass of the working fluid, and R is the specific gas constant.
The saturation temperature of the vapor at the ambient pressure (𝑇𝑠𝑎𝑡) is substituted for 𝑇𝑉 in Eqn. 3.7.
42
Similarly, the relative significance of the momentum imparted by steam vapor condensing into the
interface can be assessed by comparing the vapor stagnation pressure (1
2𝜌𝑉 [
𝑑𝑟
𝑑𝑡(1 + 𝜌𝐿/𝜌𝑉)]
2) to the capillary
pressure (2𝜎 𝑟⁄ ). For these cases, the vapor stagnation pressure (r = 1 μm, Δp = 5 × 10-3 Pa; r = 1 mm, Δp = 3
× 10-7 Pa) is negligible compared with the capillary pressure (r = 1 μm, Δp = 1.2 × 105 Pa; r = 1 mm, Δp = 1.2
× 102 Pa). Based on these arguments, the growth of the interface due to phase change and condensation of vapor
into the interface do not affect the hydrodynamics of the problem, justifying the incompressible flow
assumption.
Eqn. 3.8 is the thermal energy equation, where 𝑘 is the thermal conductivity and ��𝑝𝑐 is the volumetric
phase change heat source. Further details are provided in Section 3.1.3.
𝜕(𝜌𝐶𝑝𝑇)
𝜕𝑡+
𝜕
𝜕𝑥𝑖(𝜌𝐶𝑝𝑢𝑖𝑇) =
𝜕
𝜕𝑥𝑖[(𝐶𝑝𝑘)
𝜕𝑇
𝜕𝑥𝑖] − ��𝑝𝑐 3.8
3.1.2 Surface Tension Model
For the droplets size range of the present study Marangoni effects can be neglected [57], [123].
Therefore, a constant surface tension value is assumed (𝜎 = 0.056 N m−1). Surface tension forces are
calculated in each cell using the continuum surface tension force formulation (Eqn. 3.9) developed by Brackbill
et al. [124]. Resulting momentum fluxes through cell faces are applied as an explicit source term in the pressure
correction process.
43
𝑓𝑠,𝑖 = 𝜎𝜅∇𝜙; 𝜅 = −(∇ ∙ ��); �� =
∇𝜙
|∇𝜙| 3.9
Here, 𝜎 is the surface tension between the liquid and the vapor phases, 𝜅 is the curvature at the interface,
and �� is the unit normal into the vapor phase. In the OpenFOAM VOF implementation, the interface is spread
over a few cells; therefore, ∇𝜙 is effectively smoothed delta function which applies the force over the diffuse
interface.
3.1.3 Phase Change Model
The phase change model is based on the approach of Rattner and Garimella [120]. The volumetric heat
source ��𝑝𝑐 is applied on interface cells (Eqn. 3.8).
��𝑝𝑐 =
ℎ𝑖𝐴Δ𝑇
𝑉𝑐𝑒𝑙𝑙; 𝐴 = |∇ϕ| ∙ Vcell 3.10
Here, ℎ𝑖 is the interfacial heat transfer coefficient, 𝐴 is an approximation of the area of the interface
area contained in a cell, Δ𝑇 is the difference between the saturation temperature and the local temperature (Δ𝑇 =
𝑇𝑠𝑎𝑡 − 𝑇), and 𝑉𝑐𝑒𝑙𝑙 is the volume of the cell. ��𝑝𝑐 is solved implicitly as part of the energy equation (Eqn. 3.8).
The foregoing heat transfer model has been validated by Adhikari et al. [36].
44
3.1.4 Dynamic Contact Angle Model
A static contact angle model is insufficient to predict the process of coalescence because the three-
phase contact line moves during the process. In general, the advancing contact angle 𝜃𝐴, is greater than the
static contact angle θ0, which is greater than the receding contact angle 𝜃𝑅 (Figure 3.2). To model the
coalescence accurately, the dynamic contact angle model of Yokoi et al. (Eqn. 3.11) [125] is used:
𝜃𝐷 =
{
min [θ0 + (
Ca
ka)
13, θma] 𝑖𝑓 Ca ≥ 0
max [θ0 + (Ca
kr)
13, θmr] 𝑖𝑓 Ca < 0
3.11
Here, 𝜃0 is the static contact angle, 𝜃𝐷 is the dynamic contact angle, Ca =μUcl
𝜎 is the local capillary
number, Ucl is the velocity of the moving contact line which is positive in the direction of the droplet outward
normal vector (��) on the condensing surface. ka and kr are liquid-surface pair dependent constants, 𝜃𝑚𝑎 is the
limit for advancing contact angle, and 𝜃𝑚𝑟 is the limit for receding contact angle. In the present study parameters
for water-silicon (with hydrophobic silane grafted on it) are used, as listed by Yokoi et al. [125]. The surface
parameters are as follows: 𝜃𝑚𝑎 = 114°, 𝜃𝑚𝑟 = 52°, 𝜃 = 90°; k𝑎 = 9 × 10−9; and kr = 9 × 10
−8. The model,
as implemented here, is validated with experimental data from [125] in Section 4.2.2.
45
Figure 3.2 Advancing and receding contact angles measured using volume changing technique
(liquid injection or removal from the top of the droplet). a) Increasing volume of the drop. b)
Decreasing volume of the drop.
3.1.5 Solution Algorithm
The OpenFOAM® finite volume formulation is applied to discretize the governing equations, and the
solution algorithm is summarized as a flow chart in Figure 3.3. Expanded discussion of the phase change
simulation approach is provided in the study by Nabil and Rattner [102].
46
Figure 3.3 Phase change volume-of-fluid solver algorithm flow chart.
47
3.2 Simulation Studies
3.2.1 Studied Domain and Case setup
Coalescence of two droplets is symmetric along one center plane. Therefore, the domain only includes
half of the droplets and their surrounding region. Figure 3.4 shows the representative domain with labeled
boundaries.
Figure 3.4 Representative domain for the VOF coalescence study. The face showing the droplets is
the symmetry plane; the bottom surface, which is not visible in the figure, is the condensing
surface; and the remaining 4 faces are open atmosphere.
48
Here, 𝐿 = 6 × 𝑟1 + 2 × 𝑅𝑡 × 𝑟1 and 𝑊 = 𝐻 = 2 × 𝑟1 + 𝑅𝑡 × 𝑟1, 𝑟1 is the radius of the smaller droplet,
and Rt ≥ 1 is the ratio of the two merging droplets. Table 3.1 shows the boundary conditions at the respective
boundaries.
Boundary Boundary condition
Symmetry
∇T ∙ �� = 0
∇𝑃 ∙ �� = 0
𝑈 ∙ �� = 0; ∇𝑈 ∙ �� = 0
∇𝜙 ∙ �� = 0
Atmosphere
∇T ∙ �� = 0
𝑃 = 𝑃𝑎𝑡𝑚
∇𝑈 ∙ �� = 0
∇𝜙 ∙ �� = 0
Wall (condensing surface)
𝑇 = (𝑇𝑠𝑎𝑡 − 1 K)
∇𝑃 ∙ �� = 0
𝑈 = 0
�� ∙ ��𝑤 = cos(𝜃𝐷)
Table 3.1 Boundary conditions for the VOF simulation domain
Here ��𝑤 is the unit normal from the wall pointing into the domain, �� is the unit normal from the liquid
phase to the vapor phase, and �� is the unit normal to the patch. For wall ��𝑤 = ��. The dynamic contact angle
model is applied as a boundary condition for the phase fraction term 𝜙. While the no-slip boundary condition
is applied at the wall, in this finite volume formulation, the mean velocity in cells adjacent to the wall can be
49
non-zero. This introduces enough numerical slip to eliminate the stress singularity at the three-phase contact
line [126].
For the studied case of steam condensing at atmospheric pressure, more than 95% of the heat transfer
occurs through droplets with r < 100 μm [14], [23] – which is the radius of the largest droplet in this parametric
study. Surface tension dominates gravity for such small droplets; therefore, the droplets can be initialized as
truncated spheres [35].
Figure 3.5 Steady state temperature distribution on the droplet surfaces before merging.
For the studied range of droplet sizes (1 μm < r < 100 μm) conduction is the primary mode of heat
transfer [57]. Conduction is a diffusion process [127], and therefore, the thermal front takes a finite amount of
time to propagate from the base of the droplet to the liquid vapor interface. An order of magnitude estimate of
the conduction front propagation velocity is (𝛼𝑙
𝑟). This velocity is generally much greater than the droplet
growth rate (Section 3.1.1). For example, for a water droplet with 1 μm diameter on a surface at 99.15° C
(𝑇𝑠𝑎𝑡 = 100.15° 𝐶) the conduction front velocity is 𝛼𝑙
𝑟~1.6 × 10−1 m s−1, whereas the droplet growth rate is
𝑑𝑟
𝑑𝑡 ~ 8.4 × 10−5 m s−1. Based on the arguments, a quasi-steady state internal temperature distribution can be
50
assumed in droplets just before coalescence begins. Therefore, the droplets are initialized with a steady-state
temperature distribution (Figure 3.5), obtained by solving the steady-state energy equation (Eqn. 3.12).
𝜕
𝜕𝑥𝑖𝜕𝑥𝑖(𝑘𝑇) = ��𝑝𝑐 3.12
51
3.2.2 Grid Resolution, Sensitivity Study and Dynamic Contact Angle Validation
The mesh resolution must be sufficient to resolve heat transfer and hydrodynamic effects. Figure 3.6
shows the representative droplet isosurfaces with VOF meshing, for the parametric study. The mesh is made
of cube cells, and extends similarly in the third dimension.
Figure 3.6 Representative droplet isosurfaces, indicating VOF mesh resolutions for the
droplets from the parametric study. a) For smaller droplet radius 𝒓𝟏 = 𝟏 𝛍𝐦 and radius ratio
Rt = 1. b) For 𝒓𝟏 = 𝟏 𝛍𝐦 and Rt = 2. c) For 𝒓𝟏 = 𝟏 𝛍𝐦 and Rt = 4.
52
Grid resolution in the domain is defined using 𝑛𝑟 = (𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑐𝑒𝑙𝑙𝑠)/𝑟1. Conduction grid
independence studies are carried out for the largest and the smallest radii droplets (𝑟1 = 1 μm and 𝑟1 = 25 μm).
𝑛𝑟 is varied from 5 to 21, and the heat transfer rate (𝑄) through the base of the droplets is evaluated. For each
mesh in the grid sensitivity study, 𝑛𝑟 is increased by a factor of 1.6 (total number of cells increases by 4×). 𝑄
is plotted against 𝑛𝑟. In both the cases, it is observed that 𝑄 becomes asymptotic (Richardson extrapolated
values of Q are given in the supplemental data available publicly [128]) when 𝑛𝑟 ∈ [13,21] (Figure 3.7). The
uncertainties for Q in both the cases are less than 0.4%. Therefore, 𝑛𝑟 = 15 can be assumed the minimum
resolution for heat transfer in the parametric study (Figure 3.6-c). As an example, for a droplet of radius 1 μm
and 𝑛𝑟 = 15, average length of a mesh element Δ = 6.67 × 10−8 m.
Figure 3.7 Heat transfer grid independence study (Q vs 𝒏𝒓 where 𝒏𝒓 = # 𝒄𝒆𝒍𝒍𝒔/𝒓𝟏) for 𝒓𝟏 =𝟏 𝛍𝐦 and 𝒓𝟏 = 𝟐𝟓 𝛍𝐦.
To study hydrodynamic grid independence, and also to verify the selected dynamic contact angle
model, an axisymmetric droplet impact simulation is performed for the case of [125], [126]. In the simulation
53
a 2.28 mm diameter spherical water droplet impacts a surface with a velocity of 1 m s−1. The surface properties
are given in Section 3.1.4, and the fluid properties are as follows: dynamic viscosity liquid 𝜇𝑙 =
1 × 10−3 kg m−1 s−1; dynamic viscosity air 𝜇𝑔 = 1.82 × 10−5kg m−1 s−1; 𝜎 = 0.072 N m−1. 𝑛𝑟 for the
axisymmetric study is varied from 19 to 55. Figure 3.8 presents the variation of average diameter D (square
root of the base area) of the impacting droplet as a function of time for different grid resolutions. Experimental
data [125] is also presented. It is observed that the average absolute deviation (AAD) for D from 𝑛𝑟 = 38 to
𝑛𝑟 = 55 is less than 3%. Since, in the parametric study, the droplets merge and oscillate together as a larger
droplet before settling, 𝑛𝑟 = 40 is selected as the minimum resolution for the parametric study (Figure 3.6-a),
which is a stricter requirement than imposed by the heat transfer process alone.
Figure 3.8 Colliding droplet contact patch diameter on surface (D) vs. time (t) for different
grid resolutions and experimental data [125].
54
3.3 Validation of Results
Response time and spatial resolution requirements make it difficult to study the process of droplet
coalescence experimentally. For example, in the present study: 1 μm < 𝑟 < 100 μm, and the times of
coalescence are only 0.5 μs < 4 × |𝜏| < 1.2 ms. During the startup of dropwise condensation on a surface,
droplets could be even smaller. Owing to these challenges, there are no experimental or computational studies
available to validate the time-varying heat-transfer during coalescence. However, the steady state heat transfer
results from this simulation (just before and long after coalescence) can be compared with published steady-
state conduction heat transfer simulation data from Sadhal and Martin [95] to validate the heat transfer
formulation used in the study.
Steady condensation heat transfer rates for water droplets of r = 1 – 100 μm radii (the range of sizes
studied here) from this simulation study are compared with results from [95]. The average absolute deviation
(AD) from that study is less than 3%. The comparison is given in Figure 3.9, where the total heat transferred
through the base of the droplet (Q) is plotted against the Biot number (Bi =ℎ𝑖𝑟
𝑘) of the droplet. Here ℎ𝑖 is the
interfacial heat transfer coefficient, r is the radius, and k is the thermal conductivity of the droplet. Bi
corresponds directly to r in the simulation as hi and k are fixed.
55
Figure 3.9 Comparison of the simulation data (heat transfer rate (Q) vs. Biot number (Bi)) with
that of Sadhal and Martin [95], and conduction-only finite element COMSOL simulations.
A finite element conduction heat transfer analysis is also performed for droplets of r = 1 – 100 μm to
further validate the simulation heat transfer formulation. For the studied range of droplet sizes Marangoni
circulation and other convection effects can be neglected [57]. Therefore, the finite element analysis reduces to
a simple conduction problem in truncated spheres. The droplets are evaluated with 2-D axisymmetric domains
(Figure 3.10). A fixed temperature boundary condition is applied on the base of the axisymmetric droplet
domain (1 K below the saturation temperature), representing the cooled condenser surface temperature. A
convection boundary condition is applied to the liquid-vapor interface to account for interfacial resistance hi.
The conduction studies were performed in COMSOL Multiphysics® (v. 5.2 [98]).
56
Grid sensitivity study was performed for each finite element case where three meshes were utilized to
extrapolate the converged value of total heat transfer Q through the base of any droplet. The method is given
by Celik et al. [100], and is briefly described here.
Figure 3.10 Representative mesh sizes: (a) coarse 𝚫𝟑 = 𝟏. 𝟗𝟕 × 𝟏𝟎
−𝟗𝛍𝐦− 𝟏. 𝟗𝟗 × 𝟏𝟎−𝟓𝛍𝐦; (b)
fine 𝚫𝟐 = 𝟓. 𝟖𝟒 × 𝟏𝟎−𝟏𝟎𝛍𝐦− 𝟓. 𝟗𝟎 × 𝟏𝟎−𝟔𝛍𝐦; (c) finest 𝚫𝟏 = 𝟏. 𝟒𝟖 × 𝟏𝟎
−𝟏𝟎 𝛍𝐦 −𝟏. 𝟓𝟎 × 𝟏𝟎−𝟔𝛍𝐦.
Meshes (Figure 3.10) with average element dimensions: Δ3 = 1.97 × 10−9μm− 1.99 × 10−5μm,
Δ2 = 5.84 × 10−10μm− 5.90 × 10−6μm and Δ1 = 1.48 × 10
−10 μm − 1.50 × 10−6μm are employed.
Subscript 3 refers to the coarsest mesh and subscript 1 refers to the finest mesh. The steps are as follows:
𝜀32 = 𝑄3 − 𝑄2; 𝜀21 = 𝑄2 − 𝑄1; 𝑠 = 𝑠𝑔𝑛 (
𝜀32𝜀21) 3.13
𝑅32 =
Δ3Δ2; 𝑅21 =
Δ2Δ1
3.14
𝑞(𝑝) = ln(𝑅21𝑝− 𝑠
𝑅32𝑝− 𝑠
) ; 𝑝(𝑞) =| ln (|
𝜀32𝜀21|) + 𝑞|
ln (𝑅21) 3.15
𝑄𝑒𝑥𝑡 = (𝑅21
𝑝𝑄1 − 𝑄2)/(𝑅21
𝑝− 1) 3.16
57
𝑒𝑎21 = |
𝑄1 − 𝑄2𝑄1
| ; GCI =1.25𝑒𝑎
21
𝑅21𝑝− 1
3.17
𝑢𝑛𝑐 = Qext × GCI 3.18
The 𝜀 values are the changes in Q between meshes. R values are the mesh refinement ratios. p is the
empirical convergence rate, defined implicitly in terms of parameter q. 𝑄𝑒𝑥𝑡 is the extrapolated value of Q for
Δ → 0 (infinitely fine mesh). 𝑒𝑎21 is the relative error between the two finest meshes. GCI is the grid convergence
index, a relative uncertainty estimate for 𝑄𝑒𝑥𝑡. The absolute uncertainty unc in 𝑄𝑒𝑥𝑡 is determined from Eqn.
3.18.
All studied cases were found to be monotonically converging (𝑠 = 1) and therefore the grid
convergence index (GCI) uncertainty estimate for 𝑄𝑒𝑥𝑡 was reported (<0.2% uncertainty in all cases). Figure
3.9 shows the comparison with simulation results.
The hydrodynamics (Figure 3.8) and steady-state heat transfer rates (Figure 3.7) are both separately
validated. This supports the validity of the simulation approach for predicting transient heat-transfer in
coalescing droplets.
3.3.1 Results and Discussion
Figure 3.1 shows a representative time series evolution of surface temperature profiles as two parent
droplets (r1 = 25 μm; Rt = 1) merge together to form a new child droplet. Heat transfer through the base of the
merging droplets is presented. Initially, the droplets have quasi-steady temperature profiles, which are disturbed
at the start of coalescence. When the two droplets come in contact, a liquid bridge forms (Figure 3.1, 0 μs).
58
The bridge initially has concave curvature, resulting in a local low-pressure zone due to surface tension forces.
Liquid flows from the outer high pressure regions in the parent droplets toward the low pressure region in the
bridge [129] (0 – 20 μs). Due to contact angle hysteresis, the outer portion of the three-phase contact line does
not begin to move until well after the start of coalescence (~25 μs). At this point, the child droplet begins
oscillating on the condensing surface (30 – 200 μs). The oscillations gradually reduce in intensity due to viscous
dissipation within the droplet. The temperature profile within the droplet also decays toward a quasi-steady
profile (Figure 3.1, 500 μs). For these simulations of water condensing at atmospheric pressure, the oscillation
period and the heat transfer decay period are found to be approximately equal.
A decaying exponential function, with time constant 𝜏, can be used to represent the time varying
average heat flux through the child droplet. A parametric study is performed to characterize heat flux trends in
coalescing droplets over a range of sizes (r1) and radius ratios (Rt = r2/r1). Table 3.2 provides the simulation test
matrix and along with the values of 𝜏 obtained from the simulations. The working fluid is saturated water at
atmospheric pressure.
Droplet Radii [𝛍𝐦] Ratio (Rt = r2/r1) Time constant (𝝉) [𝛍𝐬]
r1 r2
1 1 1 0.129
1 2 2 0.357
1 4 4 0.650
6.25 6.25 1 5.80
6.25 12.5 2 16.1
6.25 25 4 28.9
25 25 1 65.9
25 50 2 170
25 100 4 287
Table 3.2 Simulation test matrix for the parametric study of coalescing droplets, with varying smaller
radius (r1) and the ratio between the radii of the merging droplets (Rt). Time constant 𝜏 is obtained
from the simulation results.
59
The following sections characterize the process of coalescence and propose trends for the stabilization
time of coalescing water droplets at atmospheric pressure for the studied range of droplet sizes and surface
parameters. The impact of coalescence on condensation heat transfer is assessed considering direct heat transfer
through a child droplet (direct mechanism), and repeated clearing of the surrounding surface as the droplet
oscillates (indirect mechanism).
3.3.2 Characterizing Coalescence
In dropwise condensation, latent heat from the condensation process transfers through the liquid-vapor
interface (interfacial resistance 𝑅int′′ ) and then conducts through the droplet to the cooled wall (Rcond). Some
studies have argued that the overall single droplet heat transfer rate depends on the three-phase contact line
length [130]–[132], as most of the heat is transferred through the perimeter of the droplet, near the condenser
surface (Figure 3.5). This result applies for droplets with high Bi (Biot number Bi = ℎ𝑖𝑟
𝑘), for which conduction
resistance is much greater than the interfacial resistance. However, for cases where the conduction resistance is
comparable with the interfacial resistance (low Bi), significant conduction occurs through the entire base of the
droplet, and not just the contact-line region. Figure 3.11 shows the dependence of heat transfer on the droplet
radius. The working fluid for the figure is saturated water at atmospheric pressure. The calculations were based
on the heat transfer model of Adhikari et al. [36].
60
Figure 3.11 Total heat transfer rate (Q) vs. droplet radius (r) for water condensation at atmospheric
pressure. The dashed lines indicate the rate of variation of heat transfer with droplet radius for
different size droplets.
The form of the curve in Figure 3.11, for the range r ∈ [3 × 10−10, 3 × 10−4] m (Q ∈
[1 × 10−12, 1] W), is approximately quadratic (i.e., Q ∝ r2). This suggests that for smaller droplets, heat
transfer is proportional to base area.
61
Figure 3.12 compares the time variation of the heat transferred through two equal size coalescing
droplets with 𝑟1 = 𝑟2 = 6.25 μm (Bi ≈ 0.3) with the time variation of the base area of the merging droplets and
the perimeter length of the contact line. It is clear from the comparison that heat transfer rate is proportional to
base area during the coalescence of water droplets of this size range.
62
Figure 3.12 . Comparison of a) condensing droplet heat transfer rate vs. time, b) base area vs.
time, and c) contact line (perimeter) length vs. time for two merging drops of radii 𝟔. 𝟐𝟓 𝛍𝐦
(𝐁𝐢 ≈ 𝟎. 𝟑).
A slight increase in the heat transfer is observed shortly after the beginning of coalescence. This occurs
because the total base area briefly increases while the liquid bridge grows, but the outer contact line sections
63
are still pinned due to hysteresis. Once the hysteresis effect is overcome, the droplet base area and heat transfer
sharply decrease, and then oscillate in-sync.
A Finite Fourier Transform (FFT) can be applied to transient heat transfer traces to extract the principal
frequencies in the oscillating coalescence process. As an example, the principal frequency of the time varying
heat transfer through droplets (𝑅1 = 𝑅2 = 1 μm) is 1 × 105 Hz. The average heat transfer trend can be
approximated with a decaying exponential function (Figure 3.13), neglecting this oscillatory component.
Figure 3.13 Decaying exponential function superimposed over Q vs Time plot of two equal
merging drops of radii 𝟔. 𝟐𝟓 𝛍𝐦 each.
Figure 3.13 presents the best-fit decaying exponential function and the oscillatory time-varying heat
transfer through the base of two merging droplets (𝑟1 = 𝑟2 = 6.25 μm; time constant τ = 5.8 × 10−6 𝑠). This
64
heat transfer decay rate could be incorporated into Lagrangian-type models of the overall dropwise
condensation process [13] to more accurately account for transient heat transfer after coalescence events, rather
than assuming a step change from the quasi-steady heat transfer rate of the two parent droplets to that of the
single child droplet.
3.3.3 Effect of Coalescence on Heat Transfer
If a dropwise condensation model assumes instantaneous merging and recovery of quasi-steady heat
transfer for a pair of droplets, it will under-predict the total heat transfer. The resulting heat transfer deficit is
due to a step-change assumption is:
Def = (
∫ 𝑎𝑒−𝑡𝜏
𝑡coalesce0
𝑑𝑡
∫ 𝑎𝑒−𝑡𝜏
𝑡coalesce0
𝑑𝑡+𝑐𝑡coalesce
) × 100 %. 3.19
Here, a is the difference in the steady state heat transfer between that of the two parent droplets and the
child droplet. c is the steady state heat transfer for the child droplet. The decaying heat transfer is integrated
from time t = 0 to t = tcoalesce – the time for the child droplet to settle to a steady heat transfer rate.
This heat transfer deficit effect is illustrated for two representative cases: (1) 𝑟1 = 1 μm, Rt = 1 amd
(2) 𝑟1 = 25 μm,Rt = 4. For cases 1 and 2, Def1 ~ 8% and Def2 ~ 5%. This deficit could become significant if
many droplets are coalescing on a surface.
Assuming instantaneous droplet coalescence thus causes a modest under-prediction of heat transfer by
neglecting the thermal stabilization time of the child droplet. This could be termed as the direct mechanism of
65
heat transfer enhancement due to coalescence. However, merging droplet hydrodynamics can cause a second
indirect heat transfer enhancement mechanism. As merging droplets oscillate before stabilizing, they repeatedly
clear small droplets from surrounding surface area, and initiate periodic renucleation in this swept area. Local
condensation heat flux can be very high right after droplets are cleared from a region. Therefore, this indirect
heat transfer enhance mechanism may be even more significant than the direct mechanism.
Glicksman and Hunt [28] performed a numerical study of the dropwise condensation process from
initialization on a dry surface, yielding transient heat transfer trends. Data from their study is adopted here in
this paper to estimate the indirect heat transfer enhancement mechanism (curve fit to their data in Eqn. 3.20).
Their data are for saturated water at atmospheric pressure (Tsat = 100°C) with a condensing surface 0.28°C
below the Tsat. Nucleation site density was assumed to be 109 cm−2.
𝑞"𝑆𝐺𝑆(𝑡) = (8.379 × 105 W m−2) (
𝑡
1 s)−0.2356
3.20
Here, t represents the time since the condensing surface was initially dry. The transient heat flux trend
from [28] (𝑞"𝑆𝐺𝑆(𝑡)) is applied as a subgrid scale model on portions of the wall not covered by grid scale
droplets in the VOF simulation. This heat flux represents the heat transfer from condensing small droplets
around the studied grid-scale droplets. Here, 𝑞"𝑆𝐺𝑆(𝑡) is evaluated for every cell face on the wall boundary. A
local time (t) is evaluated for each face as the time since it was last covered by a grid-scale droplet. 𝑞"𝑆𝐺𝑆 is set
to zero when a wall face is covered by a grid-scale droplet. Thus, this model captures the effect of merging
droplets periodically sweeping the surface around them and re-initializing nucleation in the swept area.
66
This subgrid scale model is evaluated for two cases with r1 = 25 μm;Rt = 1. In the first case, two
droplets coalesce and oscillate, as described in Section 3.3.1. The subgrid scale heat flux value on the wall faces
is initialized to 0. A second case represents the results of assuming instantaneous coalescence. Here, a single
stationary droplet (same total liquid volume as in case 1) is placed in the center of the domain with a quasi-
steady temperature distribution. The subgrid scale heat flux value on the wall is initialized to 0, except for the
area under the two parent droplets, which are set to t = 0 s. The heat flux in these cells evolves over time
according to Eqn. 3.20. The difference in heat transfer between the two cases (oscillating interface tracked
droplet coalescence case and the static case) serves as an estimate of the heat transfer underprediction by
assuming instantaneous coalescence. This resulting indirect heat transfer deficit is:
Defindirect = (
∫ 𝑞"𝑆𝐺𝑆_𝐷𝑦𝑛𝑎𝑚𝑖𝑐𝑡𝑐𝑜𝑎𝑙𝑒𝑠𝑐𝑒0
𝑑𝑡 − ∫ 𝑞"𝑆𝐺𝑆_𝑆𝑡𝑎𝑡𝑖𝑐𝑡𝑐𝑜𝑎𝑙𝑒𝑠𝑐𝑒0
𝑑𝑡
∫ 𝑞"𝑆𝐺𝑆_𝑆𝑡𝑎𝑡𝑖𝑐𝑡𝑐𝑜𝑎𝑙𝑒𝑠𝑐𝑒0
𝑑𝑡) × 100% 3.21
Figure 3.14 shows the different trends of subgrid scale heat transfer between the dynamic case and the
static case. Figure 3.14-a shows the dynamic case in which the oscillating child droplet repeatedly clears
sections of the wall, reinitializing nucleation and high heat fluxes. For the static case (Figure 3.14-b), however,
𝑞"𝑆𝐺𝑆(𝑡) is only non-zero on that exposed area that was previously under the parent droplets.
67
Figure 3.14 Differences between subgrid scale heat transfer due to microscale condensing
droplets around two coalescing r = 25 μm droplets: a) the dynamic coalescence case; and b) the
static coalescence case. Top-down view snapshots of the droplet shapes, represented in black,
are presented. The colored area underneath the droplets shows the evolution of subgrid scale
heat flux due to small condensing droplets around the primary coalescing droplets
(𝐪"𝐒𝐆𝐒∗ (𝐭) =
𝐪"𝐒𝐆𝐒−𝐦𝐢𝐧(𝐪"𝐒𝐆𝐒(𝐭))
𝐦𝐚𝐱(𝐪"𝐒𝐆𝐒(𝐭))−𝐦𝐢𝐧(𝐪"𝐒𝐆𝐒(𝐭))).
68
The heat transfer trends for the two cases are presented in Figure 3.15. After 350 μs, when the
oscillations in the droplet have almost died out, the heat transfer deficit approaches Defindirect ~ 11%, which
is greater than the direct deficit. Unlike the direct deficit, the magnitude of this indirect deficit is expected to
increase for larger oscillating droplets.
Figure 3.15 Cumulative heat transfer for an interface tracked dynamic coalescence case and a
static case. Initially, for both cases, both droplets have radius 𝒓 = 𝟐𝟓 𝛍𝐦.
69
3.3.4 Simulation Data and Proposed Correlation for Droplet Coalescence Time Constant
Simulation data for τ(r1, Rt) from the parametric study cases listed in Table 3.2 were used to form a
correlation (Eqn. 3.22) for τ for coalescence of condensing water droplets at atmospheric pressure on the
studied hydrophobic silane surface. τ and r1 were non-dimensionalized (τ∗ = 𝜏𝜌𝜎2/𝜇3; r1∗ = 𝜎𝜌𝑟1/𝜇2) using
the simulation parameters 𝜌, 𝜎 and 𝜇 (for water at 1 atm), and a correlation was formed for τ∗(r1∗, Rt). Table
3.3 lists the constants given in Eqn. 3.22. Correlation τ∗ vs. r1∗ curves for varying Rt are compared with
simulation data (markers) in Figure 3.16.
τ∗ = a × r1∗b × Rtc 3.22
Constant Value
a 0.063
b 1.927
c 1.144
Table 3.3 Constants for the droplet coalescence time constant correlation (Eqn. 3.22)
The absolute average deviation (AAD) between the correlation and the data is 14%, and the maximum
error is 37%. This represents reasonable agreement considering the three order of magnitude range of τ* in the
study space.
70
Figure 3.16 Dimensionless time constant (𝝉∗) vs. dimensionless smaller droplet radius (𝒓𝟏
∗ ) and
droplet radius ratio Rt. The curves are generated using the correlation given in Eqn. 3.22 and the
simulation data is represented by the markers.
Two additional test cases: (1) 𝑟1 = 3 μm, Rt = 1, and (2) 𝑟1 = 12.5 μm,Rt = 2, were simulated to
assess the correlation given in Eqn. 3.22. The comparison is provided in Table 3.4. The relative deviation for
both cases is less than 30%.
𝒓𝟏 [𝛍𝐦] Rt 𝛕 𝐒𝐢𝐦𝐮𝐥𝐚𝐭𝐢𝐨𝐧 [𝛍𝐬] 𝛕 𝐜𝐨𝐫𝐫𝐞𝐥𝐚𝐭𝐢𝐨𝐧 [𝛍𝐬] AD (%)
3 1 1.36 1.11 18
12.5 2 53.61 38.34 28
Table 3.4 𝝉 for representative test cases obtained from best-fit exponential curves to the experimental
data and the proposed correlation (Eqn. 3.22).
71
3.4 Conclusions
In this study, a VOF formulation was proposed and validated for interface dynamics predictions
(droplet impact) and steady state droplet condensation heat transfer. This method was then applied to simulate
the process of water droplet coalescence during dropwise condensation for 9 cases over a range of radii (1 μm ≤
r ≤ 100 μm) and radius ratios (1 ≤ Rt ≤ 4). Results were used to determine the time constant for droplet
stabilization and transient heat transfer trends after dropwise condensation. The working fluid in this study was
saturated water at atmospheric pressure. The surface properties (hydrophobic silane) were adopted from the
study of Yokoi et al.[125]
Two mechanisms (Section 3.3.3) were identified for heat transfer enhancement due to droplet
coalescence. The direct mechanism is due to the gradual decay of heat transfer from the quasi-steady rate for
the two parent droplets to that of the larger child droplet. Instantaneous droplet coalescence models neglect this
thermal stabilization time for the child droplet, and therefore under-predict droplet heat transfer by 5-10%
during the period of coalescence. The indirect mechanism is due to the oscillating child droplet repeatedly
clearing small droplets from the surrounding wall area, reinitializing nucleation in those areas. For two
coalescing droplets of equal radii (r1 = 25 μm), heat transfer enhancement due to the indirect mechanism is
~11%. This indirect effect may be even greater in larger droplets that oscillate over larger areas and have more
momentum (longer oscillation periods).
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A correlation for τ(r1, Rt) was presented for the studied range of droplet sizes, fluid properties, and
surface conditions. Two test cases (r1 = 3 μm,Rt = 1 and r1 = 12.5 μm,Rt = 2) were simulated to test the
accuracy of the correlation, and satisfactory agreement was found.
Prior theoretical and numerical studies of dropwise condensation have generally assumed droplet
coalescence to be instantaneous, and that child droplets immediately recover quasi-steady internal temperature
profiles (no history effect) [28], [34]. The present study used a direct interface tracking simulation technique to
characterize the process of droplet coalescence and post-coalescence heat transfer trends. Findings suggest that
the assumption of instantaneous coalescence underestimates the total heat transfer (direct underprediction +
indirect underprediction due to the oscillating child droplet sweeping the area underneath) in the time shortly
after droplet coalescence by 15-20%. Results from this study provide an improved closure model for droplet
coalescence that can be incorporated into Lagrangian dropwise condensation models to improve overall heat
transfer predictions. Such improved dropwise condensation models may provide insights into the effects of
coalescence and oscillating droplet dynamics on overall surface heat transfer.
Future investigations are warranted to expand results for coalescing droplet dynamics and heat transfer
to a broader range of fluids and surfaces. In order to achieve that, a better surface description is needed. For
example, Biot number of the substrate, Bisub can be introduced to account for heat transfer on a rough surface.
Investigation of more complex coalescence dynamics is also needed, such as a cascade of coalescence events
initiated by merging of a single droplet pair. If these complex droplet coalescence processes are better
understood, it may be possible to enhance dropwise condensation heat transfer through surface engineering.
For example, if contact angle hysteresis can be reduced, the oscillation time and spatial span of coalescing
droplets may increase, amplifying the two heat transfer enhancement mechanisms identified in this study.
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Chapter 4
Heat Transfer Measurements of Fast
Transient Startup of Dropwise Condensation
74
Dropwise condensation (DWC) was first studied rigorously over 80 years ago by Schmidt et al.[42],
and has received sustained research interest because it can yield heat transfer coefficients (HTC) an order of
magnitude higher than in filmwise condensation [34], [59], [103], [133]. Industrial and commercial applications
have been explored, including thermal management [8], [134], power generation [135], desalination [7], and
fogging on transparent surfaces [136]. Many experimental [14], [49], [103], [137], [138] and numerical [28],
[36], [60], [74], [86], [90] studies have been performed to characterize fundamental aspects of DWC, including
initial droplet formation [92], [93], droplet size distribution and density statistics [104], [138]–[140], and critical
sizes for droplet growth and departure [28], [141]. Numerical studies by Glicksman and Hunt [28], and Burnside
and Hadi [1] predict that instantaneous HTCs during the startup period of dropwise condensation could be much
greater than steady state values. Steady state description is not adequate for some applications, such as vapor
chambers for thermal management of devices with pulsed heat fluxes [142], or refrigeration cycles [143] that
operate for short periods of time to hold some temperature. Therefore, it is imperative to characterize transients
during DWC.
Both numerical and experimental studies face different sets of problems dealing with transient DWC.
Numerical studies in the past have primarily focused on droplet tracking [28], [144] and drop-size distribution
on a surface as fractals [33], [145]. These studies have treated droplets as rigid spheres with quasi-steady
temperature profiles. Coalescence events have generally been treated as being instantaneous. As such, both the
droplet temperature history and the complex hydrodynamics during coalescence have been neglected, which
may have significant effects on the heat transfer response of the condensing surface [146]. It is unclear whether
such modeling approaches are suitable for the startup process.
Most of the experiments in literature have employed temperature sensors in the substrate, under the
condensing surface, at varying depths[14], [26], [38]–[40]. Surface temperature can be extrapolated from such
75
measurements using Fourier’s law. However, this approach may be difficult to adapt to transient heat transfer
measurements of DWC startup. During startup, the temperature profile through the substrate will be non-linear,
making instantaneous heat flux estimation difficult. Further, conventional DWC heat transfer test sections have
relatively thick substrates, which may incur conduction timescales comparable to that of the overall DWC
startup process. For example, in the seminal experimental study by Hannemann and Mikic [40], flux is
measured using thermocouples (spaced along th = 31.75 mm) in a copper rod, which is pressed against a
stainless-steel condenser plate. The conduction time scale τcond can be estimated as th2
αcopper, where αcopper is
the thermal diffusivity of copper. Here, 𝜏𝑐𝑜𝑛𝑑 ~ 𝑂(10) s, which makes transient flux measurements for
significantly shorter periods infeasible.
Owing to the above-mentioned difficulties, almost all of the experimental and the numerical studies
focus on the steady state description of DWC. Limited data are available in literature for transient processes,
which are often encountered in practice. For example, some researchers [90], [146] are working on interface
resolving numerical techniques, like Volume of Fluid (VOF), to overcome the drawbacks associated with
previously mentioned techniques. Since, the drop size distribution spans 6 orders of magnitude, the latter
approach cannot resolve all the droplets. Therefore, transient heat transfer data from experimental studies is
needed to supplement these numerical studies.
Parin et al.[41] performed a quasi-transient experimental study of DWC on nano-structured aluminum
surfaces. They utilized the inlet-outlet temperature difference of the coolant to calculate the heat flux. Surface
temperatures were calculated using Fourier’s law. They report HTCs as a function of time, but their temporal
resolution was only one reading per ~80 s. Numerical studies[1], [28] indicate that such coarse resolution might
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be inadequate to capture the variation of transients during DWC at industrially relevant heat fluxes. Results
from the present study, which uses high-intensity spray cooling, indicate start-up times of only a few seconds.
The objective of this investigation is to provide the first experimental measurements of transient heat
transfer during the startup of DWC at relatively high heat fluxes (250 – 350 kWm−2). A uniquely instrumented
test section and facility are designed to collect these transient measurements (c.f. Section 4.1.2). Time varying
surface temperature measurements are then used as boundary conditions for an inverse heat transfer code to
determine instantaneous heat fluxes and the HTCs. Steady-state portions heat data are compared with available
models from the literature and reasonable agreement is obtained, indicating validity of the experimental facility.
Transient data are then interrogated and reveal three distinct phases of the startup period, which emphasizes the
importance of this characterization for devices operating in transient mode.
The above mentioned three distinct phases characterize the startup in dropwise condensation and the
minimum HTC period in phase II emphasizes the importance of this characterization for devices operating in
transient mode.
4.1 Experimental Design and Setup
An experimental facility was developed to record transient space-averaged heat fluxes and high-speed
video of the DWC startup process. The overall facility is designed to closely reproduce the conditions assumed
in prior numerical models of the DWC startup process [28], [147]. Typically, these studies assume that an
initially dry surface is exposed to a step change in cooling, causing instantaneous startup of DWC on a face
exposed to a constant pressure pure steam atmosphere. In this experiment, condensation occurs on a thin
composite test plate (~1.5 mm aluminum core), which yields a short characteristic conduction time scale
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(τ ~ δ2/α = 58 ms). One side of the plate is exposed to pure condensing steam supplied from a boiler with
automated temperature control. The plate is resistively heated and dried between runs, and a solenoid valve can
be opened to rapidly deliver high intensity spray cooling on the back face to initiate DWC. Thin resistive
temperature devices (RTDs) printed on both faces provide high-frequency temperature measurements, which
are supplied to an inverse 1-D conduction model to determine transient heat fluxes and HTCs.
Conceptually, the test plate (c.f. Section 4.1.3) must be thin enough to limit transient conduction effects
and delay, but thick enough to yield an appreciable temperature difference between faces for accurate heat flux
and HTC measurements. Non-condensable gases can concentrate near condensing surfaces and significantly
lower the heat fluxes [55], [62], [148]. Therefore, the facility must also incorporate provisions to remove these
gases during operation (c.f. Section 4.1.1).
4.1.1 Apparatus
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Figure 4.1 Experimental setup to capture transient heat transfer data during DWC. a) Schematic
of the setup, b) Photograph of the full test stand, and c) photographs of major components: (1)
Steam generator; (2) Test section; (3) Test plate with resistance temperature detectors (RTDs)
printed on both sides and heater for drying between tests; (4) Full cone spray nozzle; (5) Primary
pressurized cooling water tank; (6) Auxiliary water collection tank; (7) Venting and vacuum
arrangement for steam; (8) Peripheral heater for the test plate; (9) constant current source for
RTDs and signal amplifier; (10) Data acquisition system (DAQ); (11) View-port for camera.
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The experimental facility is shown schematically in Figure 4.1-a, photograph of the full test setup is
shown in Figure 4.1-b, and photographs of key components are shown in Fig. 4.1-c. Provisions for high speed
photography of the condensation process are detailed in Section 4.1.2. The test plate (3, c.f. Section 4.1.3) acts
as a pressure barrier that separates the test section (2, c.f. Section 4.1.2) into a sealed steam side and an open
spray-cooling side. Distilled water is boiled in the steam generator (1), which has an electronic process
controller to maintain delivery pressure. Riser and downcomer tubes passively supply steam and return
condensate to the test section. Non-condensable gases are purged from the steam loop via the valve at point (7).
Between tests, the steam generator is heated while a dry-scroll vacuum pump extracts vapor and gas through
this valve. A degassing cycle is repeated several times until recorded steam temperatures and pressures in the
loop correspond to saturated states (i.e., for the pressure transducer and temperature sensor in the test chamber,
Pmeasured = Psat(Tmeasured). The steam side also has a vent valve, which is open to the atmosphere and continuously
exhausts during experimental runs to minimize build-up of noncondensables. Resistive heaters (8) are used to
dry the test plate between runs.
Filtered tap water coolant is stored in a primary tank (5) pressurized with compressed air. A solenoid
valve can be actuated to rapidly initiate intense spray cooling via a full cone spray nozzle directly behind the
test plate (4). The spray cooling HTC during the experimental run is 10 − 20 kW m−2 K−1 (c.f. Section 4.2.4).
The spent coolant is collected in an auxiliary tank (6) and pumped back to the primary tank (5) between tests.
Temperature measurements from the two sides of the test plate are acquired using a purpose built 4-
wire RTD amplifier board (9). These readings and signals from other instruments are digitized with a high-
speed data acquisition system (10, LabJack® T7), which can sample up to 100,000 signals per second.
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4.1.2 Test Section
Figure 4.2 Test Section: 3 – Test plate; 4 – Full cone spray nozzle; 2.1 – Coolant supply line;
2.2 – Coolant discharge port to the auxiliary tank; 2.3 – Condensate return port; 2.4 – Steam
inlet; 2.5 – Sight glass assembly; 2.6 – Port for venting, vacuum, and pressure transducer; 2.7 –
Temperature sensor port. (Dimensions in mm)
Figure 4.2 presents a detailed cross section view of the test section. (2.1) is the coolant supply line for
spray cooling, (2.2) is the discharge port to the auxiliary water tank (Figure 4.1 (6)), (2.3) is the return line for
steam, (2.4) is the steam inlet, (2.5) is the sight glass assembly for a view port, (2.6) is the port used in venting,
measuring pressure and pulling vacuum in the chamber, and (2.7) is the port for installing the calibrated
thermocouple. The main body of this chamber is constructed from stainless steel pipe and flanges. The test plate
(3) is sandwiched between two pipe stubs, and is sealed on its faces with custom annular cup-type silicone
gaskets. The right side of the test section is a pure steam and water environment (steam side). The left side of
the assembly is open to the atmosphere and cooling system (coolant side). The entire test section, steam inlet
and condensate outlet are insulated to minimize heat loss on the steam side. Additionally, a resistive heater (not
shown) dries the test plate between runs. The sight glass is also heated to remove condensation. Calibrated
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pressure measurements (± 1 kPa) on the steam chamber are used to determine instantaneous saturation
temperatures during DWC. A calibrated thermocouple (± 0.2 K) is also installed in the chamber for redundant
measurements of the vapor state and to detect the presence of non-condensable gases in the steam environment.
Figure 4.3 a) Evolution of steam saturation temperature 𝐓𝐄 with time; b) Evolution of steam
pressure 𝐏𝐄 with time.
Figure 4.3 shows both saturation temperature and pressure, as functions of time, during the experiment.
The saturation temperature (Figure 4.3-a) is derived directly from the pressure (Figure 4.3-b). The pressure
data is collected at a frequency (f) of 4000 Hz, and later down-sampled using a moving average filter (period
𝑛 = 100) to smooth out the data. The vent valve (Figure 4.2, (8)), as mentioned earlier, is kept open during the
entire run to limit buildup of non-condensable gases in the steam chamber. The pressure gauge (Figure 4.2
(2.6)) is calibrated with a thermocouple (Figure 4.2 (2.7)), which itself is calibrated (±0.2 K) with a high
accuracy RTD thermometer. The calibration process is as follows. The vent can be partially closed to different
degrees to maintain different steady state pressures in the test section as the steam is vented. Temperature
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readings from the calibrated thermocouple are recorded at different pressure points, which are later used to
calibrate the pressure gauge. Since this calibration process does not involve significant condensation, the effect
of non-condensable gases is negligible. Readings from the pressure gauge are used to calculate the steam
saturation temperature during condensation experiments as it responds much faster than the thermocouple.
Pressure and thermocouple readings from condensation tests are also compared with the saturation curve for
steam and the mean deviation is less than 0.3 K. Therefore, it is safe to conclude that the steam side is a nearly
pure steam environment.
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Figure 4.4 a) High speed camera arrangement; b) Camera view during the experimental run.
Figure 4.4-a shows the high-speed camera aimed directly at the condensing surface of the test plate,
and Figure 4.4-b shows its view. The camera is rigidly mounted to the experimental facility to minimize jitter.
An LED panel behind the camera provides a diffuse light source. The camera (Phantom® Miro 340,
84
monochrome) is equipped with a macro lens and can capture images at up to 800 frames per second (FPS) at
1600 × 1600 resolution. In this imaging arrangement, pixels in the image correspond to 10 μm squares of the
test plate. Video data are used to qualitatively assess the condensation process and measure departing droplet
sizes (c.f. Section 4.2.3).
4.1.3 Sensor Plate and Amplification Module
Figure 4.5 Test plate: 1 – Gold-plated copper track used as an RTD in a 4-wire configuration; 2
– Gold finger contacts for connection to the RTD processing circuits; 3 – Area visible to the
camera (Figure 4.4-b).
Figure 4.5 presents a rendering of one side of the condenser test plate (Figure 4.1 (3)); the patterning
is identical on the back face. This RTD arrangement effectively provides space-averaged temperatures for 10
85
zones on each side of the test boards. Temperature differences between corresponding sensor regions on both
faces of the boards can be used to determine average heat fluxes and HTCs for each zone.
Figure 4.6 a) Test plate dimensions in 𝐦𝐦. b) Stack-up of layers over the aluminum substrate
on the condensing side of the plate: (1) 100 nm thick conformal coat of parylene-C; (2) 80 nm
thick (nominal) ENIG® finish layer (3) 8 𝛍𝐦 thick copper tracks; (4) 85 𝛍𝐦 thick
COBRITHERM (Benmayor Aismalibar. proprietary) dielectric layer.
86
Figure 4.6-a provides the dimensions of the test plate and Figure 4.6-b shows the stack-up of different
layers on the condensing side of the test plate. The stack-up is identical on the cooling side without the
conformal coat of hydrophobic parylene-C. The test plate is made out of 1.45 mm thick Aluminum 5052, which
is coated with an 85 μm nominal-thickness, high thermal conductivity dielectric on both sides (COBRITHERM
from Benmayor Aismalibar®, (4)). Measurements of the effective dielectric thermal conductivity are discussed
in section 4.1.4. The dielectric also serves as an adhesive that bonds the copper conductor layers to the aluminum
core. The copper tracks (3) that function as surface RTDs are nominally 8 μm thick and 127 μm wide. The net
resistance of each 4-wire RTD trace ranges from 25 Ω to 30 Ω. To prevent corrosion, the tracks are finished
with an Electroless Nickel Immersion Gold (ENIG®, (2)) process. A 100 nm thick layer of hydrophobic (θ =
94.8° [149]) Parylene-C (1) is applied on the condensing side to promote DWC using vapor deposition.
87
Figure 4.7 a) Constant current supply and signal amplification board: 1 – Edge connector, which
mates with the test plate terminals; 2 – Signal amplification block; 3 – Constant current supply
block; 4 – Signal cable connector to the DAQ; b) Schematic of a single RTD, current and
amplifier loop: 1 – Constant current supply; 2 – Voltage signal amplification.
Figure 4.7-a shows a rendering of the constant current supply and signal amplification module, and
Figure 4.7-b shows a schematic of a single RTD, current supply, and voltage amplification loop. The gold
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fingers shown in Figure 4.5 mate directly into the edge connector. Two current and amplification boards are
installed, one on each side of the plate (10 RTD channels each). The current supply provides each RTD with
constant 5 mA current. This current is low enough that RTD self-heating (estimated at 300 W m−2) should be
less than 1 % of the steady state heat flux due to DWC. The amplifier scales the temperature dependent voltage
from each RTD by 40 × for processing. The amplified signal is sampled by the DAQ at 80,000 samples per
second (4,000 samples for each RTD per second). The current supply and amplifier circuits both employ passive
temperature compensation. The test plate, current supply, and DAQ are calibrated as a full integrated
temperature measurement system before tests.
4.1.4 Dielectric Thermal Conductivity Measurement
The test plate has thin layers of thermally conductive polymer dielectric on both faces to electrically
isolate the RTDs from the aluminum core. While the dielectric layers are relatively thin (85 μm, reported
𝑘D ~ 2 W m−1K−1), they still account for the major portion of overall thermal resistance through the test plate.
The manufacturer only reports a typical room temperature thermal conductivity of the dielectric. The high
temperature thermal conductivity of the dielectric and film thickness variation were unknown. To enable
accurate DWC heat fluxes and uncertainties, the following in-situ calibration study is performed to measure
effective thermal conductivity of the dielectric layers.
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Figure 4.8 Conduction resistance network schematic between surface temperature
measurements on the cold and the condensing faces of the test plate.
Figure 4.8 presents the thermal resistance network schematic between the surface temperature
measurements on the two sides of the test plate. Here, TC is the measured surface temperature on the cooled
face, RD,1 is the thermal resistance through the dielectric layer on the cooled side, RAl is the thermal resistance
through the aluminum core, RD,2 is the thermal resistance through the dielectric layer on the condensing side,
and Tsurf is the measured surface temperature on the condensing face.
A second test plate is used to measure the effective value of RD,2 (from the same production batch as
the main test plate). RTD temperature sensors are first calibrated per the procedure in the Appendix (A1). A
thermopile-based heat flux sensor (Fluxteq® PHFS − 01, 25 mm× 30 mm) is bonded to the condensing side
of the test plate with high thermal conductivity silicone adhesive, covering one of the central RTD elements.
This sensor is comprised of thin film metal sandwiched between layers of polyimide film. Equal thickness
polyimide film is applied around the periphery of the sensor to ensure nearly 1D conduction through the sensor.
Dropwise condensation experiments are then performed with this second plate. Local heat flux data from the
sensor are combined with surface RTD measurements to determine the thermal conductivity of the dielectric
layer kD through an inverse numerical method.
90
Figure 4.9 COMSOL® Multiphysics mesh to determine thermal conductivity of the dielectric
layer: 1 – Coarse mesh; 2 – Fine mesh; 3 – Finest mesh.
COMSOL® Multiphysics [98] is used to perform the numerical portion of the calibration procedure. A
three-dimensional disc domain is used to simulate the aluminum substrate and the dielectric layers. The thin
copper traces are determined to introduce minimal thermal resistances, and are neglected. Figure 4.9 shows the
meshed domain, with different degrees of refinement. Steady state temperature readings from all the RTDs are
applied as fixed temperature boundary conditions over corresponding portions of the test plate faces. However,
the rectangular region covered by the heat flux sensor is instead modeled using a constant heat flux boundary
condition equal to the measured value.
The temperature dependent thermal conductivity of the aluminum substrate, kAl is provided as an input
to the numerical simulation. kD is then estimated by solving for the value that minimizes the absolute difference
between the measured RTD temperature under the flux meter and the average simulation value in the
corresponding region. The global optimal value of kD is found to be 0.937 W m−1 K−1, significantly lower than
the value reported by the manufacturer. Each enforced boundary condition is individually perturbed and the
corresponding sensitivities to kD are determined, yielding a propagated uncertainty accounting for RTD and
flux sensor errors of 0.046 W m−1K−1. Grid independence of the results is ascertained using the refinement
91
process illustrated in Figure 4.9. It should be cautioned that this kD represents an effective value based on in-
situ calibration of the test plate at operating conditions and temperatures. The single value represents an average
for the cooled- and condensing-side dielectric layers, which are at different temperatures. This effective
conductivity may also incorporate effects due to variations in the dielectric layer thickness.
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4.2 Results and Discussion
4.2.1 Transient Heat Transfer Measurements
The experimental procedure begins with degassing of the steam loop using a vacuum pump and heating,
as described in section 4.1.1. The steam loop is sealed off to the outside environment after degassing. A port on
the test chamber is opened to the surroundings when the system pressure is greater than the ambient to facilitate
removal of non-condensables during tests. The test plate is electrically heated above the saturation temperature
and dried between tests. The sight glass is also heated to remove condensate for visual access.
Before each run, coolant water is pumped from the auxiliary tank (Figure 4.1 (6)), filtered, and
transferred to the main coolant tank (Figure 4.1 (5)). The main coolant tank is then sealed and pressurized with
compressed air (~ 400 kpa − gauge). Test plate temperatures, coolant temperature, steam generator
temperature and pressure, and steam side test section pressure and temperature are continuously monitored
during pre-experimental run preparations. The test plate is visually checked to confirm dryness. Each
condensation start-up experiment is not started until all the test plate is visibly clear of droplets and the plate
temperature is at least 5°C above Tsat.
Just before starting a condensation test, the test plate heater is turned off and test plate surface
temperature data logging is initiated. The solenoid valve (Figure 4.1) is opened to rapidly initiate spray cooling.
The high-speed camera (Figure 4.4) is triggered simultaneously with the solenoid valve to capture the DWC
startup.
93
Figure 4.10 DWC on the vertical test plate at different times after startup (pressure = 1 atm)
94
Figure 4.10 presents frames from high speed video of developing DWC, with cooling beginning at 𝑡 =
0 s. Images are of the portion of the test plate indicated in Figure 4.5. The condensing surface is vertical and
the system pressure is ~1 atm (103 kPa). At t = 0.50 s, the average wall temperature is Tsurf ~ 99.1 °C with
condensing-side heat flux q"surf ~ 250 kWm−2. The typical visible droplet radius is r ~ 150 μm. At t =
1.10 s, Tsurf ~ 97.2 °C, q"surf ~ 290 kWm−2, and r ~ 0.7 mm. Once droplets reach 𝑟max ~ 1.4 mm−
1.6 mm, they begin to slide down and depart the surface. In the t = 1.50 s and t = 1.90 s frames, it is apparent
how large sliding droplets “sweep” the surface of small droplets, allowing renucleation of small droplets in their
wake. These small droplets yield much higher heat fluxes per area than large droplets.
4.2.2 Heat Flux and Heat Transfer Coefficient Calculation Methodology
Figure 4.11 Schematic representing 1 dimensional discretization of the test plate, which is
divided into separate material zones with different material properties and discretization lengths:
1 – Dielectric zone (discretization length 𝚫𝐱𝐝); 2 – Aluminum zone (discretization length 𝚫𝐱𝐀𝐥); 3 – Dielectric zone (discretization length 𝚫𝐱𝐝); 4 – Boundary; 5 – Aluminum-dielectric interface;
6 – Aluminum-dielectric Interface; 7 – Boundary.
95
An inverse approach is used to infer instantaneous surface heat flux and HTCs from the high frequency
surface temperature measurements. A finite volume solver is implemented to solve one-dimensional unsteady
heat transfer (Eqn. 4.1) for the test plate, comprising two dielectric layers sandwiching one aluminum layer
(Figure 4.11).
∂T
∂t=𝜕
𝜕𝑥(α∂T
∂x) 4.1
Here T is temperature, t is time, α is the temperature-dependent thermal diffusivity and 𝑥 is the spatial
coordinate through the plate thickness. The discretization is 2nd order accurate in space and 4th order accurate
in time (using the ODE45 integrator in MATLAB® [150]). The thermal conductivity k is harmonically
weighted at the interface using the discretization lengths on both sides of the interface (Eqn. 4.2).
kinterface =
Δxleft + Δxright
kleftΔxleft
+krightΔxright
4.2
Here left and right are with respect to the interface between the material zones in Figure 4.11.
Temperature readings are collected at 80,000 Hz for the RTDs (4,000 Hz for each RTD). The heat
transfer process is modeled as 1-D; therefore, measurements from two corresponding RTDs (one on the cooled
face and the other on the DWC face, covering the same areas) are used as transient temperature boundary
conditions for the finite volume solver. A steady state temperature profile based on the boundary temperature
96
measurements from the RTDs is chosen as the initial condition. Temperature data are smoothed to remove noise
using a moving average filter (100-point window).
Heat flux, anywhere in the domain (Figure 4.11) can be calculated with
q" = k
Tn − Tn−1xn − xn−1
4.3
Here, Tn represents the temperature in a particular cell and xn represents its position.
HTCDWC is calculated using the following expression
HTCDWC = [TE − TSurfq"Surf
−thCkC]−1
4.4
It is important to note that the HTCDWCextracted from the experimental measurements includes the
effect of two additional physical resistances, namely: the promoter layer resistance and the constriction
resistance described in section 4.2.3.
Here q"surf is calculated using Eqn. 5.2 at the surface, TE (steam temperature) is derived from the
pressure readings described in section 4.2.1, Tsurf comes from direct measurements on the test plate surface,
thC is the parylene-C promoter layer thickness and kC is its thermal conductivity.
A grid independence study is performed to ensure sufficient resolution (Table 4.1).
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𝐍𝐃 𝐍𝐀𝐥 𝐀𝐛𝐬𝐨𝐥𝐮𝐭𝐞 𝐀𝐯𝐞𝐫𝐚𝐠𝐞 𝐃𝐞𝐯𝐢𝐚𝐭𝐢𝐨𝐧 % (𝐇𝐓𝐂𝐃𝐖𝐂) 5 15 -
10 30 0.5
20 60 0.1
Table 4.1 Grid independence calculation summary for the 1D, unsteady conduction equation
solver.
Here ND, in the first column, is the number of cells in the dielectric domain (Figure 4.11 (1,3)) and
NAl, in the second column, is the number of cells in the Aluminum substrate domain (Figure 4.11 (2)). Relative
deviation of HTCDWC, with respect to the previous grid resolution, is calculated at each time step. The Absolute
Average Deviation is then calculated using that data. Based on the summary given in Table 4.1, ND = 10 and
NAl = 30 are selected for the actual numerical calculations.
Figure 4.12 Temperature profile through the plate for time t = 0.3 s.
98
Figure 4.12 presents the predicted temperature profile through the test plate at t = 0.3 s. The temperature
profile is non-symmetric, and therefore still evolving in time. This illustrates the need to use this inverse method
to extract instantaneous heat flux and HTC values instead of simpler steady state resistance formulae.
The test plate RTDs and the test chamber thermocouple (Figure 4.2 (9), which is used to calibrate the
pressure sensor – Figure 4.2 (8)) were calibrated using a high accuracy reference sensor (Accumac® AM8040,
±0.01 K). A conservative uncertainty is assumed for these instruments of ±0.2 K. A Monte Carlo method was
used to propagate these uncertainties along with the ~5% uncertainty in the dielectric thermal conductivity (kD,
c.f. Section 4.1.4) and ~5% uncertainty in the promoter layer thickness (thC) into the numerical model. 3 sets
of simulations (100, 200 and 400) with a random distribution of uncertainties (95% confidence) applied to the
temperature measurements, the dielectric thermal conductivity and the promoter layer thickness are run.
HTCDWC is calculated for each time step in each simulation. The bounds of 95% confidence are recorded (for
each time step) from the distribution of HTCs across the simulations. The uncertainty bounds are checked for
convergence. They converge sufficiently from 200 to 400 simulations. Uncertainty in HTCDWC, from the
condensation initiation (0.25 s) to 0.5 s, ranges from 20% to 8% and ranges from 8% to 5% after.
4.2.3 Validation Using Constriction Resistance and Theoretical Drop-Size Distribution Model
To validate the experimental system and approach, the steady state heat flux (q"surf) and the heat
transfer coefficient (HTC DWC) are compared with a theoretical drop-size distribution model, which accounts for
the constriction resistance due to the finite thermal conductivity of the condensation substrate and the thermal
resistance due to the presence of a promoter layer, which induces dropwise condensation on the surface.
99
Constriction resistance in DWC was first discussed in detail by Mikic [151]. This theory argues that
the surface area under microscopic droplets experiences high heat transfer while that under larger droplets has
low flux except near the contact-line [36]. This uneven distribution of heat flux results in 3D conduction effects
in the substrate and constriction of the heat flow lines near areas populated with small droplets. The effective
conduction cross-section area is therefore smaller than the overall surface, analogous to contact resistance in
solids. An additional constriction resistance term can be defined in series with other transport stages to account
for this effect. Constriction resistance in DWC depends primarily on the thermal conductivity of the substrate
[63], [65], [152].
Figure 4.13 Schematic showing the constriction resistance 𝐑𝐬 and the promoter layer resistance
𝐑𝐂 between the surface temperature measured by the RTDs on the condensation side (𝐓𝐬𝐮𝐫𝐟) and
the Steam temperature (𝐓𝐄).
Figure 4.13 shows the schematic representation of thermal resistances (Constriction resistance RS,
Promoter layer resistance RC) between the temperature measured by the RTDs Tsurf on the test plate surface
(condensation side) and the steam temperature TE (TE = Tsat(PE)). These resistances have a considerable effect
on the overall heat transfer performance of a surface during DWC and the effect is discussed in this section.
These resistances are evaluated based on the theoretical models available in literature, which also serves to
demonstrate agreement between findings from this experiment and relevant prior work.
100
The condensing substrate comprises of, from exposed face to bottom: a 100 nm thick conformal coat
of Parylene-C (DWC promoter), 100 nm thick ENIG® surface finish on the copper traces, 8 μm thick copper
traces, 85 μm thick dielectric layer, and 1.45 mm thick Aluminum 5052 substrate. Thermal resistance due to
the copper traces and their surface finish can be neglected as it is 2 orders of magnitude smaller than the smallest
other thermal resistance due to Parylene-C. To compare with the available data in literature, apparent thermal
conductivity of the condensing surface is calculated using the following expression
kA = (thAlkAl
+thdkd
+thCkC)−1
× (thAl + thD + thC) 4.5
Here thAl is the thickness of the aluminum substrate, thD is the thickness of dielectric layer on the
condensing side, and thC is the promoter layer thickness. kAl, kd, and kC are the thermal conductivities of
aluminum, dielectric and Parylene-C, respectively. The apparent thermal conductivity kA ~ 15 Wm−1K−1, is
comparable to that of stainless steels. Hannemann and Mikic [40] have reported HTCDWC ss = 62 kWm−2K−2.
Using HTCDWC Cu = 250 kWm−2K−2 [153] the value of additional constriction resistance for stainless steel
Rsss ~ 1.2 × 105 m2KW−1, which is used for calculations in the present study.
Beside constriction resistance, thermal resistance due to the promoter layer Rc, which induces DWC
on the surface, also has a significant impact on the heat transfer coefficient. To study the impact of Rc, we need
to look into heat transfer through a single droplet [36] in conjunction with the population balance theory [17],
[103], [133], which implies that at steady-state the number of droplets entering a size range must be equal to
the number of droplets leaving the size range.
101
The steady-state heat flux through the condenser surface q"surftheory
can be calculated using the following
expression:
q"surftheory
= ∫ Qdrop(r)n(r)drre
rmin
+∫ Qdrop(r)N(r)drrmax
re
4.6
Here Qdrop(r) [W] is the heat transfer rate through a droplet of radius r, n(r) is the number of droplets
of radius r per unit area, per unit r, which grow directly due to condensation, N(r) is the number of droplets of
radius r per unit area, per unit r, which grow as a result of direct condensation as well as coalescence, and the
effective radius re is the transition radius between the two regimes of droplet growth. Heat transfer through a
single drop can be expressed as [154]:
Qdrop =
ΔTπr2(1 − rmin/r )
(1
2hi(1 − cosθ)+ F
θrk ∙ sinθ
+thC
kc sin2 θ)
4.7
Here ΔT = TE − Tsurf is the degree of surface sub-cooling, rmin is the minimum thermodynamically
viable radius for a droplet to grow, hi is the interfacial heat transfer coefficient, hi ~ 15.7 MWm−2K−1 at 1 atm
assumed here[24]. F(θ, r)[36] is a contact angle (θ) and droplet radius dependent function, which accounts for
the conduction heat transfer through a droplet. k is the thermal conductivity of the condensate, thC is the
thickness of the promoter layer, and kc is the thermal conductivity of the promoter layer.
The minimum drop radius, which depends on the degree of subcooling is given by [133], [153]
102
rmin =
2σTsatρHlvΔT
4.8
Here σ is the surface tension of the condensate, Tsat is the saturated steam temperature, ρ is the density
of the condensate and Hlv is the liquid-vapor phase change enthalpy.
The effective radius re is defined as half the mean spacing between the active nucleation sites on the
condensing surface, assuming the nucleation sites form a square array. It is given by
re = (4Ns)
−1/2 4.9
where Ns is the number of active nucleation sites. Nucleation site density has been reported to be in
the range of 109 − 1015 and depends on the surface subcooling[133]. Rose[3] has proposed the following
expression for the site density
Ns =
0.037
rmin2 4.10
n(r) is the size distribution of small droplets growing as a result of direct condensation, and is calculated
using the population balance theory (Appendix, A2). N(r) for larger droplets was given by Le Fevre and Rose
[14]:
N(r) =
1
3πr2rmax(r
rmax)−2/3
4.11
103
Here rmax is the experimentally determined departing droplet radius. Images from the high-speed
photography indicate rmax ~ 1.4 – 1.6 mm for the studied substrate and conditions.
Heat transfer coefficient HTCDWCtheory
can then be calculated as
HTCDWCtheory
=q"surftheory
ΔT 4.12
Here the minimum droplet radius rmin is given by Eqn. 4.8, which depends on the degree of
subcooling ΔT. For the same nucleation site density Ns ~ 2 × 1015 (Eqn. 4.10), δ = 100 nm, kc =
0.084 Wm−1K−1 (thermal conductivity of Parylene-C [149]), and adding an additional constriction resistance
Rsss, q"surf
theory and HTCDWC
theoryare calculated. Table 4.2 provides a comparison between the theoretically
calculated quantities and the mean steady state quantities from the experiment.
𝐪"𝐬𝐮𝐫𝐟𝐭𝐡𝐞𝐨𝐫𝐲
282 kWm−2
𝐪" 𝐬𝐮𝐫𝐟 268 ±10 kWm−2
𝐇𝐓𝐂𝐃𝐖𝐂𝐭𝐡𝐞𝐨𝐫𝐲
48 kWm−2K−1
𝐇𝐓𝐂 𝐃𝐖𝐂 46 ± 2 kWm−2K−1
Table 4.2 Comparison between theoretically calculated heat flux and
heat transfer coefficient with mean steady state quantities from the
experiment.
This provides a degree of validation for the experimental and post-processing approach developed in
this investigation.
104
4.2.4 Discussion on Transient Heat Fluxes and Heat Transfer Coefficients
Figure 4.14 presents temperature readings from two corresponding RTDs on the two faces of the test
plate. The steam temperature TE is obtained from the pressure transducer in the test chamber (TE = Tsat(PE)).
At the high heat flux levels found in this representative test, quasi-steady temperature and heat flux are observed
after ~10 s (Figure 4.15-a). Quasi-steady readings after this startup period were used for comparison with
theoretical models in Section 4.2.3.
Transient data from this study has revealed three distinct phases of the startup period. In phase I, when
DWC initiates, the surface experiences high heat transfer due to the presence of small microscopic droplets.
The instantaneous HTC (Figure 4.16) reaches as high as 400 ± 50 kWm−2K−1. The HTC sharply declines
during this period as the mean droplet size grows.
In phase II, the droplets continue to grow, and the HTC declines, but at a more gradual rate than in
phase I. This transition might be attributed to a shift in droplet growth mechanism, i.e., from growth primarily
as a result of direct condensation to growth primarily due to coalescence. During this phase, droplets are still
small enough to be pinned by contact-line forces, and do not slide or depart. As the surface becomes covered
by relatively large droplets, the HTC falls to a minimum value of 27 ± 2 kWm−2K−1, which is even lower than
the mean steady state value of 46 ± 2 kWm−2K−1.
Phase III begins as the first droplets reach the critical departure size (1.4 – 1.6 mm) and begin sliding
downward. These droplets cause an avalanche effect, in which they collect other small to medium droplets in
their path. This clear the surface and initiate fresh nucleation of droplets on the surface. The first droplet
departures are somewhat synchronized, resulting in a spike in HTC up 90 ± 4 kWm−2K−1.
105
Figure 4.14 a) Surface temperature on the Condensing side 𝐓𝐬𝐮𝐫𝐟, surface temperature on the
cooling side 𝐓𝐂 and the steam temperature 𝐓𝐄 during the experiment: 1 – First inflection point;
2 – Second inflection point; b) Magnified presentation of the chart area indicated above it.
106
Figure 4.14-b presents a more detailed view of the surface temperature evolution during the early
stages of this DWC process (𝑡 = 0 − 3.5 s). This startup time is hypothesized to be a function of cooling
intensity. The present case corresponds to relatively high heat flux water spray cooling. An interesting question
to study in future would be the correlation between total cooling per unit area Qstartup, which is approximately
the amount of cooling (heat transfer [J m−2]) needed to condense the amount of liquid on the wall during steady
state, and the startup time. Integrating the heat flux on the cooling side q"C over the startup period for the present
case results in Qstartup ~ 1.1 MJ m−2. This is equivalent to 0.051 mL cm-2 of condensed liquid water (i.e., the
volume of water in a uniform film 510 μm thick).
The instantaneous heat flux (Figure 4.15), and the heat transfer coefficients (Figure 4.16), are inferred
using the computed unsteady temperature profiles through the substrate (Figure 4.12) at each time step. It is
important to note that the heat fluxes did not vary by more than 10% in different sections of the plate at any
time step, which confirms that the heat transfer is nearly 1 dimensional.
Figure 4.15 Solved heat fluxes on the cooled and condensing faces of the test plate.
107
Figure 4.16 Evolution of heat transfer coefficients on the condensing face (a, 𝐇𝐓𝐂𝐃𝐖𝐂) and
cooled face (b, 𝐇𝐓𝐂𝐂). The insets show the uncertainty bounds for the HTCs.
Initially, the surface temperature is greater than TE due to the heating applied to dry the plate between
tests. Spray cooling is initiated on the back side of the plate at t = 0. The thermal conduction front reaches the
steam side of the plate after about 50 ms, which is comparable to the conduction time scale of the plate
(thAl2 αAl⁄ ) + 2(thD
2 αD⁄ ) = 58 ms. During the first ~ 0.25 s, the steam-side of the test plate is above the steam
saturation temperature, and no condensation is visible. The heat transfer coefficient on the cooling side varies
during this period as the spray flow stabilizes. The average HTCC = 1.5 ± .05 kW m−2 K−1 until t ~ 0.15 s.
HTC climbs to 27 ± 1 kW m−2 K−1 at t ~ 0.175 − 0.2 s, and begins to reduce toward the steady-state value
of 12 ± 0.5 kW m−2 K−1.
108
At t ~ 0.25 s (Figure 4.14-b, point 1) Tsurf drops below Tsat and condensation begins. This can be
inferred from the decrease in the rate of surface temperature reduction. Three distinct phases are observed during
the startup. Based on the form of the surface temperature evolution (Figure 4.14-b), there is a period (0.25 –
1.15 s, points 1 → 2) of high condensation heat flux (phase I).In this phase HTCDWC goes from 360 ±
60 kWm−2K−1 to 60 ± 3 kWm−2K−1. It is followed by a period (1.2 – 2.5 s) of lower heat flux (phase II) in
which the HTC drops to 27 ± 2 kWm−2K−1. The HTC recovers (phase III) and reaches another local
maximum of 90 ± 4 kWm−2K−1 at ~ 3.2 s.
We hypothesize that the initially high heat flux is due to the surface being covered by microscopic
droplets, which yield higher heat transfer coefficients. The average droplet size increases, lowering heat transfer
from 1.2 – 2.5 s. Once some droplets become sufficiently large, their weight overcomes contact-line pinning
forces. Those droplets then slide down the surface, clearing small-to-medium size droplets and restarting
nucleation of microscopic droplets. The delay before droplets begin to slide, during which time the surface has
few droplets of the smallest size range, may explain the depression in heat flux between 1.2 – 2.5 s. This
inflection period highlights the need to characterize dropwise condensation start-up for thermal systems
operating in transient mode. Counter-intuitively, DWC heat transfer can be poor during part of the start-up
phase, because even though maximum droplets sizes are small, the departure and sweeping process has not yet
initiated.
An extension of our hypothesis implies that the instantaneous HTC could be inversely proportional to
the mean droplet radius (MDR) on the surface. Figure 4.17 shows MDR−1 plotted against the simulation time.
The data is from a numerical study analyzing growth of breath figures by Meakin [144]. The study uses Monte-
Carlo simulation technique to deposit fixed size droplets on a surface, at a random location one by one. If the
droplets touch each other, they merge. Droplets are merged till the surface has a droplet big enough to depart.
109
The departing droplet slides down and captures any droplet in its path. No new droplets are added till the
departing droplet clears out of the surface. droplet deposition continues afterward, as described earlier. It is
important to note that the physics involving heat transfer and hydrodynamics of the droplets is not included in
the study. Nonetheless, it points at the three distinct phases of the startup process. The present work has captured
the first experimental verification of startup in DWC.
Figure 4.17 Inverse of the mean droplet radius S plotted against simulation time 𝐭∗. Adapted
from the numerical study by Meakin [144].
At the quasi-steady condition, the condensing side of the test plate averages about 6°C below TE
(average Tsurf ~ 94.4 °C and TE ~ 100.5 °C).
110
4.3 Conclusions
Transient heat transfer process during dropwise condensation is studied experimentally. High temporal
resolution surface temperature and steam temperature measurements are carefully recorded. Unsteady heat flux
is calculated using a 1-D numerical inverse approach. Heat transfer coefficients are then determined, using
condensing side surface temperature and steam temperature. The main results from this study are as follows
• Transient heat transfer during dropwise condensation startup process, with high temporal resolution, is
studied for the first time. A uniquely instrumented test section and facility are designed to collect these
transient measurements.
• A high heat transfer spray cooling process is utilized in the present study and as a result, the startup period
lasts for only about 5 seconds. In contrast, another study by Parin et al.[41], which also reports HTC as a
function of time and utilizes a less effective cooling mechanism, doesn’t achieve steady state even after 700
s. Several studies [13], [24] in the past have reported the time to achieve steady state DWC on the order of
minutes.
• Three distinct phases of DWC startup process are identified and compared with the numerical data from
Meakin [144]. It is observed that the heat transfer can be poor during part of the startup process. Use of
DWC models for steady state operation would overestimate heat transfer during this period, leading to
undersized hardware designs.
111
Chapter 5
Preliminary Study on Multi-Scale Simulation
of Dropwise Condensation
112
One of the primary research challenges in studying droplet conduction is numerically predicting the
overall heat flux through the condensing surface. The active droplet size spectrum is wide, making direct
interface resolving simulations computationally unfeasible. For example, for water vapor condensing at Patm on
a surface cooled 5 K (Δ𝑇) below the saturation temperature, the smallest droplet radius 𝑟𝑚𝑖𝑛 =2𝜎𝑇𝑠𝑎𝑡
𝜌ℎ𝑙𝑣Δ𝑇=
2.59 × 10−9 m, and the largest droplet radius is ~O(10−3 m). There is a difference of 6 orders of magnitude
between the smallest and the largest drop, making direct simulation prohibitively expensive. Some investigators
have therefore developed fractal approaches to model the droplet distribution [112], [155], which allow
simulation of a smaller range of scales to predict the overall heat transfer in the larger domain. Others have used
a Lagrangian based droplet tracking framework [28] in which each droplet is treated like a rigid hemisphere
and individually tracked until it touches other droplets. Once the droplets come in contact they are
instantaneously replaced by a larger droplet (conserving the volume) in the next time step. This approach
requires tracking of only a few variables per droplet (e.g., position and radius). Lagrangian computing costs
may still be extreme for a practical-scale simulation as there are typically 107 – 109 active nucleation sites per
mm2 of the condensing surface [13]. One approach in Lagrangian studies has been to successively expand the
simulation area while reducing resolution in stages, as dropwise condensation evolves [28]. Heat transfer results
from finer stages can be applied to predict average heat transfer in the wall spaces between droplets in coarser
simulations. There can be multiple steps in between tracking the droplets on the smallest portion of the domain
to the entire domain. To date, none of these numerical studies of dropwise condensation have considered droplet
hydrodynamics during condensation. Such level of detail would require solution of fluid flow and heat transfer
equations with many individual degrees of freedom per resolved droplet. Interface tracking techniques like VOF
[121] or level set [156] can theoretically provide this level of detail, but due to computational cost, have only
been applied to study dynamics of few droplets [88], [90].
113
The objective of the present study is to develop a modeling approach that captures the multi-scale
process of dropwise condensation to predict the overall transient heat transfer rate with acceptable
computational costs. In this approach, droplet dynamics for the larger (grid scale) droplets will be resolved with
an interface capturing approach (VOF). Heat transfer for the smaller (subgrid scale, SGS) droplets will be
predicted with an Eulerian averaged or Lagrangian SGS heat flux model (Figure 5.1). Smaller droplets are not
very mobile and do not directly affect droplets nearby on a large scale. Therefore, a relatively simple modeling
approach that captures SGS heat transfer, but neglects SGS hydrodynamics may be successful. Larger droplets
only directly contribute a small portion of the overall heat transfer (less than 10% of the total), but sweep clear
surface area during coalescence and sliding, effectively resetting the surface for fresh nucleation. Therefore, it
is important to capture their hydrodynamics.
Figure 5.1 Illustrative distribution of grid scale and subgrid scale drop sizes. Grid scale droplets
are resolved using VOF, and an average heat flux model is applied instead of resolving subgrid
scale droplets.
114
Additional numerical and experimental data needs to be generated to develop and validate a robust SGS
average heat flux model. Preliminary simulation results for the approach are presented here. Challenges ahead
and additional work for the future are also discussed.
As described in Section 1.2 the current numerical approaches do not account for droplet history, and
the hydrodynamics involved in dropwise condensation. Techniques based on fractals are sensitive to specified
�� values (maximum droplet radius on the surface). They are quasi-steady in nature and therefore, not suitable
for predicting transient heat transfer, such as that during the DWC startup process. Techniques which are based
on droplet tracking have been shown to be insensitive to the initial droplet size distribution; Wu et al. [34] have
shown that their steady state drop size distribution and average heat flux predictions were the same whether the
initial droplets were of the same size or different. Predictions made by such models about the steady state drop
distribution may only be an emergent phenomenon, and they may not reliably predict transient behavior. The
proposed approach aims to combine empirical results from the experiments with direct simulation interface
tracking techniques to model dropwise condensation inexpensively, and accurately.
5.1 Modeling Approach
5.1.1 Grid-Scale Modeling
Grid scale droplets are resolved using VOF which is described in Section 3.2.1. However, in the
present study, the fluid generation term 𝜙 𝑝𝑐 which is given in Eqn. 3.1 is non-zero, as droplets are allowed to
115
grow due to condensation. Eqn. 5.1 describes the dependence of the fluid generation term on the total heat
transfer through the surface.
��𝑝𝑐 =
(��𝑔𝑠 + ��𝑠𝑔𝑠)
𝜌ℎ𝑓𝑔
��𝑔𝑠 = ��𝑝𝑐
5.1
Here ��𝑔𝑠 is the grid scale volumetric heat transfer and ��𝑠𝑔𝑠 is the subgrid scale (SGS) volumetric heat
transfer (only applied in cells adjacent to the wall). The surface tension model, phase change model, and
dynamic contact angle model are the same as described in Section 3.2.
5.1.2 Subgrid Scale Modeling
Limited experimental transient heat flux data are available in dropwise condensation literature to guide
the modeling of subgrid scale average heat flux correctly. Experimental work in this dissertation will help
address this need in the future Chapter 4. In this preliminary proof of concept study, a representative SGS
model, adapted from the numerical study by Glicksman and Hunt [28], is used for the purpose of illustrating
the multi-scale simulation approach. Figure 5.2 shows the time dependent average heat flux data (𝑞"𝑠𝑔𝑠: curve
with the initial site density 108 cm−2) from [28] used to inform this SGS model.
116
Figure 5.2 Average heat transfer coefficient vs time of a surface (given by Glicksman and Hunt
[28]).
The time dependent SGS model is applied to each cell neighboring the condensing surface, individually.
Each cell has a local clock for tracking of the evolving heat flux given by the model. The clock for each cell
which is swept over by a sliding grid scale droplet is reset to time = 0. ��𝑠𝑔𝑠 is also weighted by the phase fraction
term in each cell (Eqn. 5.2) so that it is only applied in the portion of each wall cell not covered by a grid-scale
droplet.
��𝑠𝑔𝑠 = ��𝑎𝑣𝑔 × 𝜙
��𝑎𝑣𝑔 = 𝑞"𝑠𝑔𝑠 × 𝐴𝑤𝑎𝑙𝑙 5.2
117
Here 𝐴𝑤𝑎𝑙𝑙 is the area of the cell face neighboring the condensing surface boundary. The concept behind this
method is that, as the mesh is refined, the SGS spaces would be uncovered by grid-scale droplets for shorter
intervals of time. The SGS model would therefore account for a smaller range of the droplet size spectrum as
the mesh resolution is refined. If implemented consistently, this approach could yield mesh convergent
predictions for overall DWC heat transfer.
5.2 Illustrative Domain and Case Setup
A 10 cm × 10 cm× 1 cm domain is shown here as the test case (Figure 5.3). The test simulates a
condensing surface cooled 2 K below 𝑇𝑠𝑎𝑡 (373.15 K) at 𝑃𝑎𝑡𝑚 exposed to saturated steam. Droplet departure is
not considered. All boundaries are labelled in the figure, and the boundary conditions are given in Table 5.1.
Figure 5.3 Representative VOF domain for multi-scale simulation test case.
118
Boundary Boundary condition
Symmetry
∇𝑇 ∙ �� = 0
∇𝑃 ∙ �� = 0
𝑈 ∙ �� = 0; ∇𝑈 ∙ �� = 0
∇𝜙 ∙ �� = 0
Atmosphere
∇𝑇 ∙ �� = 0
𝑃 = 𝑃𝑎𝑡𝑚
∇𝑈 ∙ �� = 0
∇𝜙 ∙ �� = 0
Wall (condensing surface)
𝑇 = (𝑇𝑠𝑎𝑡 − 1) K
∇𝑃 ∙ �� = 0
𝑈 = 0
�� ∙ ��𝑤 = cos(𝜃𝐷)
Table 5.1 Boundary conditions for the VOF domain.
Here �� is the normal vector (pointing in the liquid phase) to the interface, ��𝑤 is the normal vector
(pointing in the solid boundary) to the wall and 𝜃𝐷 is the dynamic contact angle, which is dependent on the 3-
phase contact line velocity.
A uniform initial temperature (𝑇𝑠𝑎𝑡) is applied to the entire domain. Here x and y are the directions
along the wall, and z is normal to the wall. Uniform grading 𝑔𝑟 =𝐿𝑛
𝐿1 is applied in the z direction. Here 𝐿1 is the
height of the layer of cells neighboring the condensing surface and 𝐿𝑛 is the height of the layer at the
“Atmosphere” boundary.
119
5.3 Preliminary Results and Discussion
The simulation starts with a bare patch of wall. Liquid, in terms of phase fraction 𝜙, is initially
generated in each cell due to ��𝑠𝑔𝑠. As the simulation progresses the liquid accumulated in the cells is pulled
together to form tiny droplets due to the volumetric surface tension force implementation and the numerical
interface compression scheme[157]. These droplets grow due to condensation (Eqn. 5.1) and coalescence. Grid
scale droplets collect the liquid generated due to ��𝑠𝑔𝑠 on the bare surface between them. This is analogous to
tiny subgrid scale droplets coalescing with them. Figure 5.4-a shows the liquid coverage on the condensing
wall after about a minute of real time in the simulation. Several of the droplets in Figure 5.4-a are captured
during sliding and merging processes. Figure 5.4-b shows a 3D rendering of dropwise condensation on a
hydrophobic surface.
Figure 5.4 a) Liquid coverage on the condensing surface after the elapse of 1 minute in the
multi-scale simulation b) 3D rendering of another test simulation with contact angle set to
𝟏𝟎𝟎°.
120
Figure 5.5 shows the cumulative surface coverage as percentage of the total condensing surface area
as a function of the droplet radius. The domain in this simulation is from a separate study and measures 0.5 cm
× 0.5 cm × 0.1 cm and the grid resolution is 500 × 500 in the XY plane. The drop distribution in the figure
evolves and approaches the distribution found in literature [14], [17].
Figure 5.5 Cumulative surface coverage by the grid scale droplets, of the condensing surface
vs droplet radii for different times.
Figure 5.6 shows the heat transfer maps from one of the many trial runs for the grid scale and the
subgrid scale separately. The grid scale heat transfer is mostly concentrated around the three-phase contact line
121
of the droplets and therefore, the grid scale heat transfer map (Figure 5.6-a) traces the droplet outlines, as
expected. Since, the subgrid average heat flux model is time dependent, the subgrid scale heat transfer map
(Figure 5.6-b) traces droplet movement on the surface.
Figure 5.6 a) Grid scale heat transfer map; b) Subgrid scale heat transfer map.
This approach has to be refined and more rigorously assessed in conjunction with the experimental
results in Chapter 4. Robust, widely applicable VOF simulations for dropwise condensation can be set up with
the following steps:
• Apply the subcooling on the condensing surface.
• Apply the correct dynamic contact angle formulation, which depends on the surface-liquid pair, using
experimental data (c.f. section 3.1.4).
• Apply the transient heat flux from the experimental data (c.f. section 4.2.3) as an SGS model (section 5.1.2)
or use this to inform the selection of an SGS model. As the proposed SGS model form captures the evolution
122
of small droplets on a portion of the surface after it is dried or cleared, transient startup heat flux data are
needed as a basis for this approach.
5.4 Conclusions
Earlier models for predicting the overall heat transfer through a surface treated the droplets as rigid
hemispheres or fractals, and did not resolve the fluid dynamics of the droplets. Due to the complex
hydrodynamics involved in dropwise condensation an interface capturing technique is needed to resolve the
droplets. Owing to the large spectrum of drop sizes, such a technique becomes quite expensive for resolving all
the scales.
A multi-scale approach is proposed here in which larger grid-scale droplets are resolved with the VOF
interface capturing technique and a subgrid scale average heat flux model is applied alongside to capture the
effect of smaller droplets. This approach needs a robust, widely applicable, and thoroughly validated subgrid
scale model. Experimental results described in this dissertation can serve as the primary subgrid scale heat
transfer model for such simulations. But some more work needs to be done in order to make these experimental
results generally applicable. Parametric studies, which characterize the effects of bulk substrate properties,
working fluid, surface properties and operating conditions need to be performed within the experimental
paradigm defined in chapter 4. For example:
• A parametric study to characterize the effect of cooling rate on DWC startup.
• A parametric study to catalogue dynamic contact angles for various solid-liquid pairs.
• A parametric study of startup at different pressure conditions etc.
123
Chapter 6
Conclusions and Recommendations for
Future Research
124
In this chapter major findings of this PhD dissertation on heat transfer in dropwise condensation are
summarized. Recommendations for future extension of this research are also suggested.
6.1 Numerical Simulations of dropwise condensation
Numerical techniques for predicting dropwise condensation heat transfer have long been sought.
Simulating the entire process using a fixed Eulerian grid is challenging as the drop size distribution spans 6
orders of magnitude and resolving all the length scales can get computationally expensive. Owing to these
challenges most of the studies in the past have either employed droplet tracking Lagrangian schemes simulating
a small (O(1 mm)) physical condensing surface area, or they have used fractal techniques, which assume the
drop size distribution to be similar at every length scale. Nearly all of these studies have assumed hemispherical
droplets with quasi-steady temperature profiles and have neglected the history of the droplets as they grow and
merge. Such approaches have generally used simplistic formulations for heat transfer through individual drops
and have treated the coalescence of drops as instantaneous. Chapter 2 of this dissertation presents the
development of an improved analytic model for predicting the quasi-steady heat transfer through individual
droplets valid for wide ranges of contact angles and relative interfacial and conduction resistances. Chapter 3
characterized the effect of droplet hydrodynamics during coalescence on condensation heat transfer. Specific
contributions and findings include:
• With the correct formulation of heat transfer through a drop, which takes into account the droplet size (Biot
number, Bi) and its contact angle (𝜃), steady state heat transfer predictions become more accurate. This
formulation can be used with a drop size distribution model to improve overall heat transfer predictions for
a condensing surface.
125
• This formulation for individual droplet heat transfer can also be used to predict correct growth rates for
condensing droplets. It is shown in Chapter 2 that a prior widely used model for droplet heat flux resulted
in considerable error for droplet growth.
• Chapter 3 employed direct interface tracking simulations to explore the coalescence events during dropwise
condensation. This study illustrates how the complex hydrodynamics during coalescence of just 2 droplets
can have a significant effect on the overall heat transfer during dropwise condensation. Merging droplets
go through oscillations before they finally settle as 1 drop. During this period, they constantly sweep their
surrounding area and initiate fresh nucleation. Previous studies for modeling DWC have generally treated
coalescence as an instantaneous event without consideration oscillations or hydrodynamics. Findings here
suggest that such approaches significantly underpredict heat transfer during coalescence.
• Chapter 5 presents a multi-scale numerical framework for simulating dropwise condensation using a direct
interface tracking technique. Qualitative results are also presented to illustrate the grid-scale and the subgrid
scale heat transfer contributions during simulations. A path to reaching a complete framework is outlined.
6.2 Experimental Measurement of Heat Transfer During Dropwise Condensation
Startup
Prior to this investigation, there have been no reported experimental data for the transient process of
dropwise condensation startup at high heat fluxes relevant to industrial applications. Steady state descriptions
of DWC may be poorly suited for design of systems with cyclic or intermittent DWC processes, such as vapor
chambers for thermal devices with pulsed heat fluxes or refrigeration cycles operating for short periods of time.
Chapter 4 provides first experimental measurements during dropwise condensation startup.
126
• Direct surface temperature measurements in conjunction with a finite volume discretization based, unsteady
conduction equation solver is used in this study to extract the transient heat transfer data (heat flux and
HTC) during DWC.
• Different aspects of the DWC startup process, like DWC initiation, droplet sliding initiation, and the quasi
steady state are observed and correlated with measured and calculated quantities, like surface temperature,
heat flux and HTC. Three distinct phases of the startup period are identified. In phase I, when DWC
initiates, the surface experiences high heat transfer due to the presence of small microscopic droplets ~O
(10−1) 𝜇𝑚 covering it. The drop in HTC is fast during this phase as the droplets grow rapidly. As the
droplets start to grow in phase II, the HTC goes down further. This drop in HTC is slower than phase I.
Sliding doesn’t begin in this phase and the droplets continue to grow, pinned to the surface, before reaching
their critical departure size. Phase III begins with the first sliding event quickly followed by the avalanche
of large droplets, which slide down colliding with other small to medium droplets in their path. They clear
the surface and initiate fresh nucleation of droplets on the surface. The above mentioned three distinct
phases characterize the startup in dropwise condensation and the inflection period in phase II emphasizes
the importance of this characterization for devices operating in transient mode.
• The results from the study can be used in guiding design decisions involving DWC applications. They can
also be utilized in formulating closure models for multi-scale VOF simulations.
6.3 Future Research Work
This dissertation provides important submodels, experimental data and methods, and a basic framework
to develop multi-scale simulation methods for dropwise condensation. Such simulations would directly resolve
127
larger droplets using an interface tracking method and capture the effect of smaller droplets is captured using a
subgrid scale model. This approach should be advanced in the following ways:
• A robust, widely applicable, and thoroughly validated subgrid scale model is needed for DWC. Transient
experimental results described in this dissertation can serve to inform subgrid scale heat transfer models for
such simulations. More work is needed to expand the range of data on transient DWC startup. Parametric
studies, which characterize the effects of bulk substrate properties, working fluid, surface properties and
operating conditions need to be performed within the experimental paradigm defined in chapter 4.
• State of the art contact hysteresis models and surface tension force models can be integrated in this
framework to improve the fidelity of simulations. Implementation of these models is described in chapter
3. For example, the implementation of continuum surface tension force model developed by Brackbill et
al.[124] and contact angle hysteresis model described by Yokoi et al.[125].
• Chapter 4 hypothesizes the direct relation between the rate of cooling and the dropwise condensation startup
period. This prediction is based on observation. A parametric study to characterize the effect of cooling rate
will be a very useful design guideline for dropwise condensation applications.
128
Appendix
A 1 Calibrating the Test Plate RTDs
Figure A.1 presents a schematic of the setup used to calibrate the RTDs on the test plate. The RTDs
are spread over a 75 mm diameter circular area, and a careful approach is needed to achieve nearly uniform
temperature over the whole plate during calibration to achieve the target ± 0.2 K uncertainty.
Figure A.1 Schematic cross-section representing the test plate RTD calibration setup: (1)
Reference temperature probe; (2) High thermal conductivity aluminum block; (3) intermediate
thermal conductivity stone block; (4) Temperature controlled heated plate; (5) – Thermal
insulation; (6) Signals to data acquisition system; (7) Test plate.
If the test plate were mounted directly on a temperature controlled hot plate, variations in heated plate
temperature (e.g., due to locations of individual heater elements) would be imposed on the test plate. To
minimize temperature variations in the test plate, a stack-up is made of a temperature controlled heated plate
(4), intermediate thermal conductivity stone (marble) plate (3), high conductivity thick aluminum plate (2),
129
and the test plate (4).There’s thermal interface material (not shown in the figure) between each layer. The
whole system is insulated with thick mineral wool (5). In this arrangement, the stone plate acts a thermal
diffuser to spread out hot spots from the heated plate and the aluminum plate acts as a “ground plane” to
distribute heat over the stone plate, resulting in a nearly uniform temperature near the RTD and test plate. A
2D conduction finite element study was performed for this arrangement. The worst-case scenario of a linearly
increasing temperature profile (2 K range) is applied at the heated plate – stone plate boundary. The
temperature variation at the Aluminum block – test plate interface did not exceed 0.1 K.
This system was used to form piecewise linear calibration curves for the test plate RTDs with the
amplifier cards and data acquisition system at 5 temperatures from 30°C − 120°C, ensuring sufficient settling
time after each temperature adjustment.
130
A 2 Population Balance Theory
Population balance implies that at steady state, the number of droplets entering a size range must be
equal to the number of droplets leaving the size range. For an arbitrary range of droplet radii, let r1 be the lower
bound and 𝑟2 be the upper bound (Δr = r2 − r1). Individual droplets grow at rate G = dr/dt. Population density
n(r) is defined as the number of droplets of base radius r per unit area per unit drop radius. The number of
droplets entering an arbitrary size range is A′n1G1dt and the number of droplets leaving it is A′n2G2dt. Here A′
is an arbitrary area on the condenser surface. Drops are also removed by the action of larger droplets sweeping
the area clean, and restarting the nucleation-growth process. The number of droplets swept off in a size range
is Sn12Δrdt, where S is the sweeping rate at which the condensing surface is renewed and n12 is the average
population density. To conserve the number of droplets in the size range
A′n1G1dt = A′n2G2dt + Sn12Δrdt 6.1
Eqn. 6.1 can be simplified as
A′(G2n2 − G1n1) = −Sn12Δr 6.2
As 𝛥𝑟 → 0, 𝑛12 → 𝑛. Therefore Eqn. 6.2 becomes
d(Gn)
dr+n
τ= 0 6.3
where sweeping period τ = A′/S .
131
The heat transfer rate through a droplet (Qdrop) depends on interfacial resistance [122], [158],
conduction resistance [13], [28] through the drop, and the resistance of the promoter layer [60] under the droplet.
The combined effect of all the resistances on the heat transfer rate is given in Eqn. 4.7.
Heat transfer rate through a drop is also equal to the phase change enthalpy rate
Qdrop = ρHlv2πr
2(1 − cosθ)G 6.4
Using Eqns. 4.7 and 6.4 the droplet growth rate G can be written as a function of the droplet radius r
G = B1(1 − rmin/r)
B2r + B3
B1 =ΔT
2ρHlv
B2 =θ(1 − cosθ)
4k ∙ sinθ
B3 =1
2hi+δ(1 − cosθ)
kc sin2 θ
6.5
As G is a function of r, Eqn. 6.3 can be integrated with respect to r
′ ∫
d(Gn)
Gn= − ∫
dr
Gτ
r
rmin
Gn
Gnmin
6.6
The minimum droplet radius rmin is given in Eqn. 4.8. Solving Eqn. 6.6 gives
132
n(r) =(Gn)min
Gexp [−(
B2τB1
{(r2 − rmin
2 )
2+ rmin(r − rmin)
+ rmin2 ln(r − rmin)}
+B3τB1
{r − rmin + rmin ln(r − rmin)})]
6.7
The preceding expression of population density n(r) applies for small droplets that grow as a result of
direct condensation. For larger droplets, growing mainly due to coalescence, Le Fevre and Rose[14] gave an
expression for N(r) given in Eqn. 4.11. rmax is the radius of the largest droplet before it is removed by the body
forces. The latter is given by [2], [18]
rmax = K(σ
ρg)1/2
6.8
K is a parameter accounting for contact-angle hysteresis, which can be fit to experimental data (high-
speed photography data from the experimental run is used in its determination, K ~ 0.5 – 0.6), and g [ms−2] is
the gravitational acceleration.
The effective radius re is given by Eqn. 4.9.
In order to maintain continuity n(re) = N(re). Using this equality with Eqn. 6.7 and 4.9 gives (Gn)min.
Substituting (Gn)min into Eqn. 6.7 gives
n(r) =1
3πre3rmax
(rermax
)−23∙r(re − rmin)
(r − rmin)
(B2r + B3)
(B2re + B3)∙ exp[C1 + C2]
6.9
133
C1 =B2τB1
{(re2 − r2)
2+ rmin(re − r) + rmin
2 log (re − rminr − rmin
)}
C2 =B3τB1
{re − r + rmin log (re − rminr − rmin
)}
At re d(log(n(r)))
d(log(r))=
d(log(N(r)))
d(log(r))= −
8
3 [60]. Using this expression, the sweeping period can now be
expressed as
τ =
3re2(B2re + B3)
2
B1(11B2re2 − 14B2rermin + 8B3re − 11B3rmin)
6.10
The steady state heat flux q"surftheory
, and the heat transfer coefficient HTCDWCtheory
can now be calculated
using Eqn. 4.6 and 4.12, respectively.
134
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Vita
Sanjay Adhikari
Education
Ph.D., Mechanical Engineering, The Pennsylvania State University August 2020
M.S., Carnegie Mellon University December 2015
B.Tech., National Institute of Technology - Kurukshetra July 2011
Work Experience
Research Assistant, The Pennsylvania State University, University Park, PA Aug. 2015 – Aug. 2015
Graduate Intern, MAHLE Engine Components USA, Inc., Farmington Hills, MI Jan. 2015 – Jul. 2015
Systems Engineer, BrahMos Aerospace Pvt. Ltd., Hyderabad, India Jul. 2011 – Jul. 2013
Selected Publications
1. Adhikari, S, Nabil, M, Rattner, A S, “Condensation Heat Transfer in a Sessile Droplet at Varying Biot
Number and Contact Angle,” International Journal of Heat and Mass Transfer, 2017, Vol. 115, pp. 926-
931, DOI: 10.1016/j.ijheatmasstransfer.2017.07.077
2. Adhikari, S, Rattner, A S, “Heat Transfer During Condensing Droplet Coalescence,” International
journal of Heat and Mass Transfer, 2018, Vol. 127, pp. 1159-1169, DOI:
10.1016/j.ijheatmasstransfer.2018.07.005
