Development and assessment of a physics-based model for subcooled flow boiling with application to CFD

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Development and assessment of a physics-based model for

subcooled flow boiling with application to CFD

by

Ravikishore Kommajosyula

B.E.(Hons.), Birla Institute of Technology and Science, Pilani, 2010 M.Sc.(Hons.), Technische Universität München, 2015

Submitted to the Department of Mechanical Engineering in partial fulfillment of the requirements for the degree of

Doctor of Philosophy in Mechanical Engineering and Computation

at the

MASSACHUSETTS INSTITUTE OF TECHNOLOGY

September 2020

Author . . . . Ravikishore Kommajosyula Department of Mechanical Engineering July 21, 2020

Certified by . . . . Emilio Baglietto, Ph.D. Associate Professor of Nuclear Science and Engineering Thesis Supervisor

Accepted by . . . . Nicolas Hadjiconstantinou, Ph.D. Professor of Mechanical Engineering Thesis Committee Chair Chairman, Department Committee on Graduate Theses

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Development and assessment of a physics-based model for subcooled

flow boiling with application to CFD

by

Ravikishore Kommajosyula

Submitted to the Department of Mechanical Engineering on August 24, 2020, in partial fulfillment of the

requirements for the degree of

Doctor of Philosophy in Mechanical Engineering and Computation

Abstract

Boiling is an extremely efficient mode of heat transfer and is the preferred heat removal mecha-nism in power systems in general and, more recently, in electronics cooling. Physics-based models that describe boiling heat transfer, when coupled with Computational Fluid Dynamics (CFD), can be an invaluable tool to increase the performance of such systems. Existing modeling approaches do not incorporate all relevant heat transfer mechanisms at the wall, limiting their predictive capability and general applicability. These shortcomings restrict the application of CFD in the design process. For the nuclear industry, this means having to rely on expensive experimental campaigns to develop and license new reactor designs.

A second-generation mechanistic heat flux partitioning framework developed in our group provides an enhanced physical description of flow boiling. It introduces several mechanisms not accounted for in previous formulations, such as 1) bubbles sliding on the heater surface, 2) interaction of nucleation sites and 3) microlayer evaporation. The framework requires describ-ing the complete bubble ebullition cycle, includdescrib-ing bubble nucleation, growth, and departure through closure models, which are currently lacking. This thesis extends the framework into a closed-formulation by developing closure models that adequately represent the underlying physics. New models for predicting the bubble departure diameter and frequency are developed based on insights gathered from experiments and direct numerical simulations.

An assessment against existing approaches to model boiling heat transfer demonstrates the model’s ability to predict over 80% of the boiling curves within a 20% error, while also capturing the correct trends with flow conditions. The model implementation in a commercial CFD software is demonstrated using data from the Bartolomei experiment. The extendability of the model to novel heater surfaces is further demonstrated for a sapphire heater substrate, where fewer cavities for nucleation shift the boiling curves to considerably higher wall superheats. This mechanistic representation of boiling heat transfer has the potential to support predictive design with optimal boiling heat transfer for improved system efficiency, with the specific objective to accelerate the development of novel nuclear fuel concepts.

Thesis Supervisor: Emilio Baglietto, Ph.D.

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Acknowledgements

This thesis is a result of five years of intellectually challenging and rewarding research and was made possible through the support and assistance at various levels by a large number of people, whom I would wish to thank. I am extremely grateful to my advisor, Professor Emilio Baglietto, who has been steadfast in his support. This work in its current form would not have been possible without all the guidance, technical conversations, and the freedom to explore my own research ideas. In addition to providing an excellent work environment and access to resources, he was always available as a mentor to provide invaluable career and life advice.

I would like to thank my Ph.D. committee members, Professors Evelyn Wang, John H. Lien-hard V, and Nicolas Hadjiconstantinou for providing unique perspectives, feedback, and insights towards my research progress throughout my Ph.D. I would also like to acknowledge the financial assistance received from the Consortium for Advanced Simulation of Light Water Reactors (CASL).

I am truly lucky to have great mentors over the years, who were always ready to share their wisdom and experiences to advise me on a variety of situations. I thank Professors Michael Bader and Hans-Joachim Bungartz from the Technische Universität München, and Professors Matteo Bucci, Koroush Shirvan, Youssef Marzouk, and Neil Todreas from MIT for mentoring me over the years.

The modeling efforts were greatly enriched by the access to high-quality experimental data from the group of Professor Matteo Bucci. I would like to thank my experimental collaborators Dr. Andrew Richenderfer, Jee Hyun Seong, Dr. Guanyu Su, Dr. Bren Phillips, Artyom Kossoloapov, and Chi Wang for providing these experimental insights.

The most productive collaboration of my Ph.D. came in the form of my labmate, Dr. Etienne Demarly. He was incredibly helpful to get me up to speed at the start of my Ph.D., and we went on to spend countless hours brainstorming and bouncing ideas off each other. Over the past five years, Etienne became one of my best friends outside of work. Thank you, Etienne, for all those discussions and good memories.

I am quite fortunate to have extremely supportive labmates who formed the CFD team, referred to fondly as the Bubble Bros. Thank you for always being there to check in on each other, listen to my bad puns, and enrich my grad school experience. In particular, thank you, Ben, Etienne, Rosie, Giancarlo, Mike, Giulia, Carolyn, Brian, Jinyong, Zach, Monica, and Ralph.

Thank you to the administrative staff in MechE, CCSE, and NSE for helping me out at various points during my Ph.D. In particular, I want to thank Leslie Regan from the MechE administration in particular for always being available, watching out for me, and finding ways to help me in times of need.

I was involved with the Society of Industrial and Applied Mathematics (SIAM) student chapter and the Communication Lab during my Ph.D., which contributed towards a well-rounded graduate school experience. I want to thank the directors of the CCSE and SIAM officers for supporting me to offer an IAP course on “Practical Computer Science for Computational Scientists” for three years

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in a row. I immensely benefited from the training and mentoring I received during my service as a communication lab fellow. I am incredibly lucky to have the opportunity to work with Thomas Mazzocco and Pasquale Conte from the University of Pisa, on their master theses, as part of the MIT-UNIPI alliance. I thank Prof. Walter Ambrosini from the University of Pisa for these successful collaborations.

The summer internship I did at the Center for Atomic Energy and Alternative Energy (CEA) in Paris was a pivotal moment that shaped my research experience. I learned a lot by working in close quarters with Dr. Guillaume Bois, my internship advisor, and running buddy. The different perspectives I picked up during my time at CEA has helped me in my Ph.D. research. Thank you, Guillaume, for all your guidance and help.

I am very fortunate to have developed great and lasting friendships over the years from all the places I have lived in. You have stuck by me during the highs and the lows cheered me up when I was stressed, checked me when I was going astray, and always gave me something to look forward to. Thank you to my friends from school (Harish, Sree Lekha), college (G.K.), Germany (Atlanter), U.S.A. (Arko, Rohit, Corbin, David, Simon). I want to thank the Pasha family from Baku, who have jumped in to help during uncertain times at the start of my program. Special thanks are due to my girlfriend, Kelly, for her unwavering love and support during these grueling last few months.

I am truly grateful for the unconditional love, guidance, and support of my family. Thank you to my mom and dad for giving me the freedom to pursue my passions, and instilling in me the discipline and determination to accomplish them. You have always led by example and inspired me to keep improving. Thank you to my dear uncles, Seenu, Ramu, and Shyam, who have been my mentors and role models since I was a child, and always had a watchful eye and a helping hand for me. Last, but certainly not the least, I want to thank my grandmother, Lalithamma, who raised me as a kid and was my biggest cheerleader. She would have been the happiest person on earth to see this work come to fruition.

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Bibliography 113 Appendices 121 A Image processing algorithm to measure the bubble departure diameters 121 B Point-averaged implementation of the heat flux partitioning approach in MATLAB 123 C Boiling curves of all experiments used for the model assessment 125 C.1 Kennel Experiment . . . 125

C.2 Bucci Experiment . . . 133

C.3 Jens-Lottes Experiment . . . 139

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List of Figures

1.1 Effective heat transfer coefficients measured in literature for various server cooling technologies obtained from [6] . . . 22 1.2 Comparison of pool boiling and subcooled flow boiling data for a stainless steel tube

showing the extremely efficient heat transfer in subcooled flow boiling. Data from Bergles and Rohsenow [7], reproduced from [8] . . . 23 1.3 Photographic study of subcooled flow boiling from Kennel [9]. (Left) Onset of

nucleate boiling. (Center) Fully developed nucleate flow boiling. (Right) Critical heat flux . . . 24 1.4 Nukiyama Boiling curve. Various regimes and regime transitions are labeled. . . 25 1.5 Trends of boiling curve with variation in pressure, velocity, and subcooling. Data

from Kennel [9]. Flow conditions: (Left) Velocity = 0.3m/s, Subcooling = 22.2K. (Center) Pressure = 4bars, Subcooling = 55.5K. (Right) Pressure = 2bars, Velocity = 0.3m/s . . . 26 1.6 Effect of dissolved air and surface oxidization on the boiling curves. Data from

Kennel [9]. Flow conditions: (Left) Pressure = 6bars, Velocity = 0.3m/s, Subcooling = 22.2K. (Right) Pressure = 2bars, Velocity = 3.6m/s, Subcooling = 11.1K . . . 26 1.7 Dependence of bubble departure diameter with pressure. Data from Semeria [19] . . 27 1.8 High-speed video (top) and temperature of heater surface from infrared camera

(bottom) of a heater surface with a sliding bubble from [22]. The thermal footprint in the wake of the sliding bubble indicates transient conduction heat transfer. . . 28 1.9 Correlations and experimental data of Jens and Lottes [31] and Thom [32]. These

correlations have been calibrated over a limited range of wall heat flux and cannot match the boiling curve from the other experiment . . . 29 1.10 Illustration of the RPI mechanistic heat flux partitioning approach. Heat flux at the

wall is partitioned into forced convection, quenching, and evaporation components 31 1.11 Mechanisms modeled in Gilman’s heat flux partitioning framework (from [24]) . . . 32 1.12 Mindmap of the heat flux partitions and mechanisms in Gilman’s boiling framework

(from [24]) . . . 33 1.13 Heat flux partitions and boiling curve predicted by Gilman’s framework when all

inputs from closure models are replaced with experimental measurements from Richenderfer [25]. Flow conditions: Pressure: 1 bar, Velocity: 1 m/s, Subcooling: 10 K 35

2.1 Illustration of the bubble departure and liftoff events in flow boiling . . . 37 2.2 Assessment of the predictions of Tolubinsky Kostanchuk [39] and Kocamustafaogullari

[41] bubble departure diameter correlations with the experimental data of Sugrue [51] 38 2.3 Illustration of the different forces acting on a bubble growing on a nucleation site . . 40 2.4 Illustration of the heat transfer mechanisms during bubble growth. Microlayer

evaporation fuels bubble growth and the subcooled liquid condenses the bubble cap 41 2.5 Bubble departure diameter predictions of the reassessed mechanistic force balance

approach. Note: Right plot shows area within the red dotted lines in left plot . . . . 43 2.6 Measurements of the bubble sliding velocity from the experiments of Phillips [22].

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2.7 Bubble liftoff diameter predictions of the reassessed mechanistic force balance ap-proach. Note: Right plot shows area within the orange dotted lines in left plot . . . . 44 2.8 Arrangement of the infrared and high-speed video cameras in the experiment by

Bucci [25] . . . 45 2.9 Algorithm used to measure the bubble departure diameter. Left: The time signals

of temperature and heat flux at the nucleation site are used to detect the departure event. Right: A bubble is measured in the location and time corresponding to the departure event detected . . . 46 2.10 Measurements of the bubble departure diameter obtained from the image processing

algorithm using the experimental data of Richenderfer [25]. The diameters are plotted against the average wall superheat (left) and average wall temperature (right). The predictions of the mechanistic force balance approach presented in Section 2.1 are shown in solid lines. . . 46 2.11 Experimental trends of bubble departure diameter with variation in wall superheat

(left), flow subcooling (middle), and velocity (right). Data from experiments listed in Table 2.1 . . . 48 2.12 Bubble departure diameters predicted by the reduced correlation for the experiments

listed in Table 2.1 . . . 49

3.1 Illustration of the Cole [40] bubble departure frequency model. The frequency is modeled as a scaling of the bubble diameter to the terminal rise velocity . . . 52 3.2 Measurements of bubble departure diameter (left) and frequency (right) from the

experiment of Richenderfer [25] show and increasing trend with the wall heat flux . 52 3.3 Assessment of various bubble departure frequency models by Yoo et. al. [66] shows

that each model is accurate only in a range of flow conditions . . . 54 3.4 Illustration of the Hsu’s criterion [70] for nucleation. (Left) Illustration of the vapor

embryo and the thermal boundary layer prior to nucleation. (Right) Evolution of the temperature profile in the thermal boundary layer with time. Nucleation occurs when the temperature of the liquid one cavity radius away reaches the saturation temperature of the vapor embryo. . . 54 3.5 Distribution of cavity sizes on a stainless steel heater surface from the work of Hibiki

[75]. The distribution of cavity sizes that are active for two superheats T2> T1are

shown. . . 55 3.6 Illustration of the energy limit of the thermal boundary layer. Nucleation occurs

when the growing thermal boundary layer can no longer sustain the energy balance on the heater surface. . . 56 3.7 Bubble departure frequency predictions of the model developed in this work and

the correlation by Cole. The nucleation criterion based on the energy limit of the thermal boundary layer captures the correct trends and shows significantly improved predictions at all frequency ranges . . . 57

4.1 Images from high-speed video of flow boiling at 1, 2, 5, 10 bars shows the rapidly evolving physics with pressure. Adapted from experiment of Richendefer [77] . . . 59 4.2 Illustration of the various heat transfer and micro-scale mechanisms in the MITB

heat flux partitioning approach . . . 60 4.3 Assumed shape of the microlayer under the bubble . . . 63

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List of Figures

4.4 Compilation of the initial microlayer thickness measurements by various researchers. Microlayer thickness of 4µm is typical away from the nucleation site. Reproduced from [86] . . . 65 4.5 Illustration of static interaction on the heater surface resulting in suppression of

nucleation sites . . . 66 4.6 Approximation of the Lambert’s W-function in Equation 4.30 compared to the exact

solution . . . 67 4.7 Approximation of the Lambert’s W-function in Equation 4.30 compared to the exact

solution, obtained using the inbuilt implementation in MATLAB R2019b . . . 68

5.1 Flow loop in the experiment of Kennel [9] . . . 71 5.2 Test section consisting of a flow through an annulus with a rod heater in the

experi-ment of Kennel [9] . . . 72 5.3 Representative boiling curve for the Kennel Experiment, corresponding to 4 bars

pressure, 0.3 m/s velocity, and 27.8 K subcooling. Boiling curves for all cases listed in Table 5.2 are provided in Appendix C . . . 73 5.4 Comparison of boiling curve predictions for increasing pressure. The MITB model

captures the correct trends with pressure, which is quite challenging due to large changes in fluid properties and bubble dynamics . . . 74 5.5 Predictions of the bubble departure diameter (Top) and frequency (Bottom) for the

three mechanistic heat flux partitioning approaches. The trend with fluid properties and flow conditions are adequately captured by the closures in the MITB model . . . 75 5.6 Comparison of boiling curve predictions for increasing flow subcooling in the Kennel

experiment. Subcooling has an effect of increasing the relevance of single-phase heat transfer mechanisms during boiling. . . 76 5.7 Heat flux partitioned to single-phase heat transfer for the case with a subcooling of

83.3 K. The MITB model captures the enhancement of single-phase heat transfer at high subcooling. Note: RPI and CASL curves overlap . . . 76 5.8 Comparison of boiling curve predictions for increasing flow velocity in the Kennel

experiment. Flow velocity enhances the single-phases heat transfer mechanisms due to increased turbulence and bubble sliding on the heater surface . . . 77 5.9 Heat flux partitioned to single-phase heat transfer for the case with a velocity of

3.6 m/s. The MITB model captures the enhancement of single-phase heat transfer at high velocity. Note: RPI and CASL curves overlap. . . 77 5.10 Scatter plot of the measured and predicted wall superheat for the Kennel experiment.

The MITB model captures the trends with flow conditions and fluid properties, resulting in a low root-mean-squared error of 3.2 K. Note: dotted lines show±20% error bounds . . . 78 5.11 Plot of the cumulative distribution function of the error in predicting the wall

superheat for the Kennel experiment. The MITB model predicts over 80 % of the measured points within an error of 3.7 K . . . 79 5.12 Schematic of the flow loop (left) and the flow boiling test section (right) in the

experiment by Bucci [25] . . . 80 5.13 Distribution of cavity sizes in a stainless steel (left) and ITO heater (right). Adapted

from [25, 75] . . . 82 5.14 Comparison of nucleation site densities for the ZR and ITO heaters . . . 82 5.15 Comparison of boiling curves for the ZR and ITO heaters . . . 83

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5.16 Comparison of boiling curve predictions for increasing flow subcooling in the Bucci experiment. Subcooling has an effect of increasing the relevance of single-phase heat transfer mechanisms during boiling. . . 83 5.17 Heat flux partitioned to single-phase heat transfer for the case with a subcooling of

25 K. The MITB model captures the enhancement of single-phase heat transfer at high subcooling. Note: RPI and CASL curves overlap . . . 84 5.18 Comparison of boiling curve predictions for increasing flow velocity in the Bucci

experiment. Flow velocity enhances the single-phases heat transfer mechanisms due to increased turbulence and bubble sliding on the heater surface . . . 85 5.19 Heat flux partitioned to single-phase heat transfer for the case with a velocity of

1.5 m/s. The MITB model captures the enhancement of single-phase heat transfer at high velocity. Note: RPI and CASL curves overlap. . . 85 5.20 Scatter plot of the measured and predicted wall superheat for the Bucci experiment.

superheat for the Bucci experiment. The MITB model predicts over 80 % of the measured points within an error of 3.2 K . . . 86 5.22 Comparison of boiling curve predictions for increasing flow subcooling in the MIT

experiment reported in the work of Jens and Lottes . . . 88 5.23 Comparison of boiling curve predictions for increasing flow subcooling in the UCLA

experiment reported in the work of Jens and Lottes . . . 89 5.24 Scatter plot of the measured and predicted wall superheat for the Jens-Lottes

experi-ment. The MITB model captures the trends with flow conditions and fluid properties, resulting in a low root-mean-squared error of 1.5 K. Note: dotted lines show±20% error bounds . . . 90 5.25 Plot of the cumulative distribution function of the error in predicting the wall

superheat for the Jens-Lottes experiment. The MITB model predicts over 80 % of the measured points within an error of 2 K . . . 90 5.26 Flow loop used in the experiment of Thom [32] . . . 91 5.27 Comparison of boiling curve predictions for increasing flow subcooling in the Thom

experiment . . . 92 5.28 Scatter plot of the measured and predicted wall superheat for the Thom experiment.

superheat for the Thom experiment. The MITB model predicts over 80 % of the measured points within an error of 1.1 K . . . 94 5.30 Comparison of boiling curve predictions for increasing flow subcooling reported in

the work of Jens and Lottes (top) and Thom (bottom) . . . 95 5.31 Scatter plot of the measured and predicted wall superheat for the different

mecha-nistic heat flux partitioning approaches and correlations across all experiments. The MITB model captures the trends with flow conditions and fluid properties, resulting in a low root-mean-squared error of 3 K. Note: dotted lines show±20% error bounds 96

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List of Figures

5.32 Plot of the cumulative distribution function of the relative error in predicting the wall superheat across all experiments. The MITB model predicts over 80 % of the

measured points within 20% error . . . 96

6.1 Illustration of ensemble averaging in the two-fluid approach. The volume fraction of vapor αgin the averaged approach reduces from left to right as the bubbles move away from the heated wall and condense in the subcooled liquid. . . 100

6.2 Two dimensional axi-symmetric setup used in the CFD simulation . . . 103

6.3 Results of the CFD simulation of the Bartolomei experiment using the MITB model. liquid temperature (left) and volume fraction of vapor(right) fields in the test section 104 6.4 Comparison of the axial profiles of liquid temperature (left) and volume fraction of vapor(right) along the heated length predicted by various heat flux partitioning approaches . . . 104

6.5 Comparision of axial profiles of fraction of heat flux partitioned to evaporation by various heat flux partitioning approaches . . . 105

A.1 Image processing . . . 122

C.1 Boiling curve for Kennel experiment case 3 from Table 5.2 . . . 125

C.16 Boiling curve for Bucci experiment case 1 from Table 5.3 . . . 133

C.28 Boiling curve for Jens-Lottes MIT experiment case 1 from Table 5.4 . . . 139

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C.33 Boiling curve for Jens-Lottes UCLA experiment case 6 from Table 5.4 . . . 141

C.43 Boiling curve for Thom experiment case 1 from Table 5.5 . . . 147

C.44 Boiling curve for Thom experiment case 2 from Table 5.5 . . . 148

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List of Tables

1.1 Typical convective heat transfer coefficients(W/m2K)of various heat transfer

con-figurations from [2] . . . 21

1.2 Closure models used in the RPI and CASL heat flux partitioning approaches . . . 32

2.1 Experimental data used to develop the reduced correlation for predicting the bubble departure diameter . . . 48

3.1 Experimental data used to develop the bubble nucleation criterion based on the thermal boundary layer energy limit . . . 57

5.1 Experiments used for the assessment of correlations and mechanistic heat flux partitioning approaches . . . 69

5.2 Flow conditions explored in the experiment of Kennel [9] . . . 73

5.3 Flow conditions explored in the experiment of Bucci [25, 96] . . . 81

5.4 Flow conditions reported in the work of Jens and Lottes [31] . . . 88

5.5 Flow conditions explored in the Thom experiment . . . 92

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Nomenclature

Latin Symbols

Notation Description Unit

A Area [m2] Ca Capillary Number [-] D Diameter [m] cp Specific Heat [J kg−1K−1] f Frequency [Hz] F Force [N]

g Standard Gravity / Gravitational Acceleration on Earth [9.81m s−2_]

G Mass Flux [kg m−2s−1]

hf g Latent Heat [J kg−1]

h Enthalpy [J kg−1]

Ja Jacob Number [-]

k Thermal Conductivity [W m−1K−1]

K Bubble growth constant [m/√sec]

l (Sliding) Length [m]

L (Heated) Length [m]

Nu Nusselt Number [-]

N” Nucleation Site Density [m−2]

P Pressure [Pa]

P Probability [-]

Pr Prandlt Number [-]

R Radius [m]

Re Reynolds Number [-]

S_dry Dry area fraction coefficient [-]

t Time [s]

T Temperature [K]

u Velocity [m s−1]

U_b Bubble Growth Rate [m s−1_]

V Volume [m3]

x Thermodynamic Quality [-]

Greek Symbols

Notation Description Unit

α Void fraction, volume fraction of gas phase; α≡ αg [-]

α∗ Wetted fraction [-]

δ (Micro Layer) Thickness [m]

∆ Difference [-]

e Surface Roughness [m]

η Thermal Diffusivity [J m−3K−1]

π 3.14159265359... [-]

φ Heat Flux [W m−2]

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Nomenclature – Continued from previous page

µ Dynamic Viscosity [Pa s]

ρ Density [kg m−3]

σ Surface Tension [N m−1]

θ Contact Angle [circ]

χ Temperature Ratio [-]

Subscripts

Subscript Description

b Bubble

B Buyoyancy Force

CP Contact Pressure Force

d Departure

D Drag Force

dry Dry Area/Spot

e Evaporation f Fluid Phase f c Forced Convection g Gas Phase g Growth (time) g Gravity Force H Hydrodynamic Force h Heater L Lift Force lo Lift-Off ml Micro Layer nb Nucleate Boiling q Quenching s Surface

Sx/Sy Surface Tension Force

sat Saturation sl Sliding sl Shear Lift sc Sliding Conduction sub Subcooling sup Superheat

wait Wait (time)

w/wall Wall

wet Wetted Area

Acronyms

Acronym Description

CASL Consortium for Advanced Simulation of Light Water Reactors CFD Computational Fluid Dynamics

CHF Critical Heat Flux

CRUD Chalk River Unidentified Deposits CSR Complete Spatial Randomness theory

ITO Indium Tin Oxide

MITB MIT Boiling model

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Nomenclature

Nomenclature – Continued from previous page RPI Rensselaer Polytechnic Institute

ONB Onset of Nucleate Boiling PWR Pressurized Water Reactor

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Chapter 1

Introduction

1.1 Motivation

Heat transfer is ubiquitous in domestic and industrial applications, accounting for over half of the global energy consumption and 40% of the global carbon dioxide emissions [1]. Increasing the efficiency of heat transfer is an essential step towards enhancing energy efficiency and reducing carbon emissions. Boiling can enhance the efficiency of heat transfer up to 20 times compared to similar configurations without boiling. Table 1.1 from Rohsenow and Choi [2] summarizes the typical convective heat transfer coefficients for various configurations and shows the extremely efficient heat transfer observed in boiling.

Table 1.1:Typical convective heat transfer coefficients(W/m2_K₎_{of various heat transfer}

configurations from [2]

Gases, natural convection 5-30 Gases, forced convection 10-250 Liquids, natural convection 15-1000 Liquids, forced convection 150-5000

Condensation 2500-25,000

Boiling liquids 1000-250,000

Boiling heat transfer has been successfully leveraged as the heat removal mechanism in large power plants —including nuclear reactors, and is being adopted more recently in electronics cooling applications. Data centers worldwide account for roughly 1% of the global electricity use [3], contributing about 0.3% to overall carbon emissions, and is expected to grow to 8% of the projected total electricity demand by 2030 [4]. Roughly 40% of the total energy consumed by a typical data center goes into cooling and ventilation systems [5]. A review of existing strategies to cool server electronics by Kheirabadi and Groulx [6] comparing the effective heat transfer coefficients obtained for different methods is presented in Figure 1.1. A dramatic increase in the heat transfer rate is observed in two-phase cooling, which employ boiling, compared to conventional air cooling technology. While active cooling technologies such as spray cooling and jet impingement can offer higher heat transfer coefficients, the pumping power requirements and emergency access protocols limit their applicability.

Nuclear reactors utilize boiling heat transfer to exchange the heat generated in the fuel rods to water at high pressure, flowing in the space around. In particular, Pressurized Water Reactors (PWRs) utilize subcooled flow boiling to boost the efficiency of heat transfer. The fundamental role of subcooled flow boiling to successful operation of nuclear reactors was recognized early in their development, and has motivated the advancement of its understanding and adoption.

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Figure 1.1:Effective heat transfer coefficients measured in literature for various server cooling technologies obtained from [6]

Figure 1.2 shows experimental data from Roshenow and Bergles [7] comparing the heat transfer rates observed in pool boiling and subcooled flow boiling for the same heater surface, and shows the significant enhancement of the heat transfer in subcooled flow boiling. A photographic study of subcooled flow boiling heat transfer at the onset of nucleate boiling, fully developed nucleate boiling, and the critical heat flux, is shown in Figure 1.3. The flow conditions correspond to a velocity of 3.6 m/s at 2 bars pressure and 27.7 K subcooling.

The efficiency of heat transfer in nuclear reactors can be enhanced by leveraging predictive simulation and optimization in the design process. Nuclear power accounts for 20% of the elec-tricity produced in the United States with 99 reactors in operation. Increasing the efficiency of these nuclear reactors is crucial for nuclear power to remain economically competitive [10], failing which, many of these reactors face the prospect of shutting down in the next decade. To address these concerns, the Department of Energy (DoE) set up the Consortium for Advanced Simulation of Light Water Reactors (CASL) as the first Energy Innovation Hub, to develop simulation tools to improve the predictive performance of existing and next-generation commercial nuclear reactors. Computational Fluid Dynamics (CFD) can provide insights about the effect of flow conditions and geometrical features on heat transfer that can improve the design and operational efficiency of nuclear reactors [11].

In order to run these simulations in CFD, we require models that represent the physics of boiling heat transfer at the heater surface. These models must be capable of predicting the heat transfer accounting for the effects of oxidization and fouling over time, which in nuclear fuel takes the unique form of CRUD (Chalk River Unidentified Deposits). A uniquely valuable characteristic would be the ability to support the modeling of new "accident tolerant fuels" [12, 13] such as SiC claddings, and heater surfaces deposited with micro and nano-scale structures [14, 15] to enhance the critical heat flux.

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CHAPTER 1. INTRODUCTION

Figure 1.2:Comparison of pool boiling and subcooled flow boiling data for a stainless steel tube

showing the extremely efficient heat transfer in subcooled flow boiling. Data from Bergles and Rohsenow [7], reproduced from [8]

Since most of the understanding of boiling heat transfer derives from empirical observations, existing models do not incorporate the ability to account for the surface effects in new heater designs. The lack of predictive simulation capability has forced the industry to rely on expensive experimental campaigns to inform design and regulation. Developing physics-based models that can instead reliably and more generally predict boiling heat transfer can bring the power of predic-tive simulation to design and operate more efficient nuclear reactors, and forms the motivation for this thesis.

An introduction to boiling heat transfer and physical insights into the process are provided in Section 1.2. An overview of the existing approaches to model boiling heat transfer is presented in Section 1.3. A second-generation mechanistic heat flux partitioning framework that accounts for physical phenomenon relevant to flow boiling is described in Section 1.4. The objectives of this thesis are presented in Section 1.5, and the thesis structure is outlined in Section 1.6.

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Figure 1.3:Photographic study of subcooled flow boiling from Kennel [9]. (Left) Onset of nucleate boiling. (Center) Fully developed nucleate flow boiling. (Right) Critical heat flux

1.2 Physical insights into boiling heat transfer

Boiling heat transfer has been an area of focused research for several decades, and our first understanding of boiling comes from the experimental work of Nukiyama [16] in 1934. The experiment consisted of a Nichrome wire immersed in a pool of saturated water. The nichrome wire functioned as a joule heater and a resistance thermometer. Water was boiled by applying a power input to the wire, and the relationship between the applied heat flux(φ)and the wall

superheat∆Tsup = Twall−Tsat was plotted. The plot, shown in Figure 1.4, is referred to as the

Nukiyama boiling curve, although researchers before Nukiyama made measurements of the applied heat flux and the resultant wall superheat [17]. The points labeled in Nukiyama boiling curve identify several regimes:

A:Forced convection heat transfer. No bubble nucleation is observed.

B:Onset of Nucleate Boiling (ONB). The heat flux at which bubbles start to form and addi-tional heat transfer due to boiling is observed.

C:Fully developed Nucleate boiling. Bubbles are produced at a constant rate at the heater surface, and liquid quenches the voids left by departing bubbles. This region is characterized by extremely efficient heat transfer signified by the steep slope of the boiling curve.

D:Departure from Nucleate Boiling (DNB). A limiting heat flux, also referred to as the Critical Heat Flux (CHF), beyond which liquid can no longer quench the heater surface and nucleate boiling cannot be sustained.

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E:Transition boiling: Region only attainable if the wall temperature is the control parameter. An increase in wall temperature results in a decrease in the heat flux that can be transferred from the wall.

F:Leidenfrost Point: The temperature beyond which the liquid is no longer in physical contact with the heater surface and hovers over the surface separated by an insulating layer of steam.

G: Film boiling: Heater surface is fully in contact with steam, and extremely high wall temperatures characterize this regime. The boiling curve transitions from the critical heat flux (D) directly to film boiling (G) when heat flux is the control parameter.

Figure 1.4:Nukiyama Boiling curve. Various regimes and regime transitions are labeled.

Most of the research on modeling boiling heat transfer is focused on predicting the different regimes in the boiling curve and the criteria for various regime transitions within. In particular, we focus on modeling nucleate boiling heat transfer in regimes B-D in this work. Modeling the critical heat flux (point D in the Nukiyama boiling curve) is a research topic of its own, with several decades of focused research. A review of existing approaches to model the critical heat flux and a new micro-hydrodynamics approach based on the dry area fraction on the heater surface is presented in [18].

Insights from integral experiments: Early experimental work was focused on performing integral experiments that study the overall trends of boiling curves with flow conditions. Thermo-couples were used to measure the temperature of the heater surface for an imposed wall heat flux. Integral experiments by Kennel [9] studied the trends of the boiling curve for variation in pressure, velocity, and flow subcooling, shown in Figure 1.5. An increase in the efficiency of boiling heat transfer is observed with pressure, with the boiling curve shifting towards a lower wall superheat and a higher wall heat flux. An increase in velocity and subcooling enhances the forced convection heat transfer, resulting in an increase in the onset of nucleate boiling and the critical heat flux, while not altering the boiling curve.

The effect of dissolved air and surface oxidization was also studied in the experiments of Kennel and the resultant trends are shown in Figure 1.6. Boiling experiments were performed on surfaces that were progressively oxidized by steam at an elevated temperature and pressure for 0, 30, and

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Figure 1.5:Trends of boiling curve with variation in pressure, velocity, and subcooling. Data from Kennel [9]. Flow conditions: (Left) Velocity = 0.3m/s, Subcooling = 22.2K. (Center) Pressure = 4bars,

Subcooling = 55.5K. (Right) Pressure = 2bars, Velocity = 0.3m/s

90 minutes. A shift in the boiling curves towards a lower superheat is observed for the oxidized surfaces, indicating more efficient boiling heat transfer. Early-onset of nucleate boiling is observed in the presence of dissolved air, resulting in more efficient boiling at lower superheats due to enhanced nucleation. This gain in efficiency disappears at higher superheats and has an adverse effect of lowering the critical heat flux.

Figure 1.6:Effect of dissolved air and surface oxidization on the boiling curves. Data from Kennel [9].

Flow conditions: (Left) Pressure = 6bars, Velocity = 0.3m/s, Subcooling = 22.2K. (Right) Pressure = 2bars, Velocity = 3.6m/s, Subcooling = 11.1K

Integral experiments were used to develop correlations to predict boiling heat transfer described in Section 1.3.1. Integral experiments allow studying the trends of boiling curves with flow parame-ters, but fail to offer any insights into the physics that drives these trends.

Insights from individual-effect experiments:Integral experiments were supplemented with individual-effect experiments aimed at understanding the underlying physics, such as the bubble departure diameters and frequencies, and the number of active cavities on the heater surface that can support nucleation. For example, experimental measurements of the bubble departure diameter by Semeria [19] showing a large variation of the bubble departure diameter with pressure, is shown in Figure 1.7. Quantifying the diameters, frequencies, and the number of bubbles on the heater

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surface allows for estimating the amount of vapor generation and the additional heat transfer due to evaporation. Assembling models for these mechanisms allow us to represent boiling as a sum of different heat transfer mechanisms occurring at the wall. These models are referred to as mechanistic heat flux partitioning approaches and are described in Section 1.3.2.

Figure 1.7:Dependence of bubble departure diameter with pressure. Data from Semeria [19]

Individual-effect experiments relied on optical measurements to study the physics of boiling heat transfer, severely limiting their scope. These experiments were often restricted to heat fluxes close to the onset of nucleate boiling, beyond which the heater surface becomes crowded by bubbles, making optical measurements infeasible. For instance, the highest heat flux at which a bubble departure diameter measurement could be made was limited to 200kW/m2in the experiment of Situ [20], while nucleate boiling can extend up to several MW/m2. Crowding of the surface with an increase in heat flux can be observed in Figure 1.3. Moreover, most of these experiments were conducted in pool boiling conditions to avoid the complexities that arise from flow boiling. The first photographic experiment that studied the physics of flow boiling was performed by Gunther [21] in 1951 to measure the rate of bubble growth. These limitations resulted in the individual-effect experiments being performed separately from the integral experiments and often could not repli-cate the range of flow conditions explored in integral experiments. Consequently, early mechanistic heat flux partitioning approaches suffered from a lack of sufficient description of the underlying physics, limiting their predictive capability and general applicability.

Insights from next-generation experiments:The gap between individual-effect and integral experiments can be bridged by leveraging new advances in infrared thermometry, that allow to study the instantaneous distributions of temperature and heat flux on the heater surface. These experiments record the infrared radiation emitted from the heater’s backside using an infrared camera, and post-process the intensity of radiation to obtain the instantaneous temperature and heat flux distributions. Phillips [22] observed a large thermal footprint in the wake of a sliding bubble, which indicates an efficient transient conduction heat transfer due to the reestablishment of the thermal boundary layer. The high-speed video and temperature distribution of the heater in the presence of a sliding bubble from the experiment of Phillips [22] is shown in Figure 1.8. Yoo [23] studied the trajectory of sliding bubbles and the frequency of bubble formation on a heater surface that was fabricated to have a single nucleation site. The experiments of Phillips [22] also

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evidenced a significant heat transfer contribution due to the evaporation of the microlayer, which is a thin layer of liquid trapped underneath the bubble. These experiments provide extremely valuable new insights which have led to the development of a second-generation mechanistic heat flux partitioning framework by Gilman [24], described in Section 1.4.

Figure 1.8:High-speed video (top) and temperature of heater surface from infrared camera (bottom)

of a heater surface with a sliding bubble from [22]. The thermal footprint in the wake of the sliding bubble indicates transient conduction heat transfer.

Recent subcooled flow boiling experiments by Richenderfer [25] further adopted infrared thermometry to provide advanced diagnostics into the physics of boiling heat transfer including:

1. Indirect measurements of the bubble departure diameter and frequency, nucleation site density, and dry area fraction up to the critical heat flux

2. Heat flux partitioning to different heat transfer mechanisms 3. Boling curves and critical heat flux

These experiments represent the latest advancement in the field. Insights from these experiments are used to inform the development of closure models for bubble departure diameter and frequency in this work.

1.3 Existing approaches to modeling boiling heat transfer

A review of the existing approaches to modeling boiling heat transfer was conducted by Dhir and Warrier [26, 27]. An abridged review focusing on subcooled flow boiling heat transfer is presented in this Section. Modeling approaches can be categorized into correlations based on experimental data of boiling curves, and mechanistic heat flux partitioning approaches that aim to describe the heat transfer mechanisms at the wall. A review of the existing correlations is presented in Section 1.3.1, and the existing mechanistic heat flux partitioning approaches are discussed in Section 1.3.2.

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1.3.1 Correlations

Early work in modeling boiling heat transfer consisted of correlations that were developed using experimental data of boiling curves. Rohsenow [28] proposed one of the earliest correlations (in 1952) to predict pool boiling heat transfer based on a non-dimensionalization of the wall superheat and the wall heat flux. The first correlation to predict saturated flow boiling heat transfer, based on a nucleate pool boiling equation, was proposed by Chen [29] in 1966. Liu and Winterton [30] in 1991 introduced corrections to the Chen correlation to account for flow subcooling. The most popular correlations to model subcooled flow boiling heat transfer for water at high pressures, which is of interest for nuclear reactor applications, are the Jens and Lottes [31] and Thom [32] correlations.

1.3.1.1 Jens-Lottes and Thom Correlations

The Jens-Lottes and Thom correlations were developed using in-house experimental data for subcooled flow boiling heat transfer in a heated pipe. Both correlations share a similar formulation, predicting the wall superheat as a function of the applied heat flux φwall, and the system pressure

P, as shown in Equation 1.1.

∆Tsup=

A(φ_wall)B

exp(P/C) (1.1)

where A, B, and C are calibration constants, which differ for the Jens-Lottes and Thom correlations. The experiments of Jens-Lottes and Thom span complementary ranges of flow conditions as shown in Table 5.1. The Jens-Lottes experiments are run at higher flow velocity, subcooling, and wall heat flux, than the Thom experiment. Figure 1.9 shows the boiling curves predicted by the Jens-Lottes and Thom correlations plotted with the data from both experiments. It is evident that neither correlation can match the boiling curves from the other experiment.

Figure 1.9:Correlations and experimental data of Jens and Lottes [31] and Thom [32]. These

correlations have been calibrated over a limited range of wall heat flux and cannot match the boiling curve from the other experiment

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1.3.1.2 Limitations of correlation approach

Empirical correlations have been developed using experimental data from specific heater/coolant combinations for simple geometries, which limits their application range. The functional form of these correlations, shown in Equation 1.1, captures a dependency only on the system pressure and shows no dependence on flow conditions. Correlations cannot be consistently applied in CFD as the interfacial energy transfer, described in Section 6.1, cannot be prescribed without introducing ad-hoc assumptions. The applicability of correlations cannot be extended to new heater surfaces by providing the missing information as an additional closure model. These correlations fail to provide reliable predictions when applied to complex geometries for varying flow conditions, due to the lack of sufficient description of the underlying physics.

1.3.2 Mechanistic heat flux partitioning approaches

Mechanistic heat flux partitioning approaches are suitable for application to CFD as they model the heat transfer based on the local flow conditions at the wall, therefore seamlessly coupling to the 3-D local representation of CFD. Among the numerous heat flux partitioning approaches, the classic RPI formulation by Podowski [33], together with a recent optimization of the RPI formulation by Feng [34], are presented in this Section.

1.3.2.1 RPI partitioning approach

The RPI approach was adapted by Kurul and Podowski [33] based on the work of Judd and Hwang [35] to develop a heat flux partitioning for nucleate pool boiling. The RPI approach is widely used in CFD applications due to its implementation suitability and ease of calibration. The heat flux applied at the wall is partitioned to three heat transfer mechanisms:

1. Forced convection: The heat removed by single-phase convection heat transfer.

φf c =hf c(Twall−Tliquid) (1.2)

2. Quenching: The enhancement of heat transfer as the heater surface is quenched by cold liquid following the departure of a bubble.

φquench =hquench(Twall−Tliquid) (1.3)

The Del Valle and Kenning model [36] is used to determine the heat transfer coefficient of quenching(hquench)as:

hquench=2Kquenchf

r

ρfcplkltw

π (1.4)

The wall influence area fraction (Kquench)is defined in the original model formulation of

Kurul and Podowski as:

Kquench= FA

πD2_d

4 N

” _(1.5)

where FArepresents the area of the wall influenced by the bubble and is set equal to 2.0.

3. Evaporation: The latent heat required to form the vapor phase.

φe=ρghf g

π

6D

3

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Figure 1.10:Illustration of the RPI mechanistic heat flux partitioning approach. Heat flux at the wall

is partitioned into forced convection, quenching, and evaporation components

The wall heat flux is computed as a sum of the three heat transfer mechanisms, illustrated in Figure 1.10.

φ_wall =φ_{f c}+φ_quench+φe (1.7)

Closure models for the nucleation site density (N”), bubble departure diameter (D_d), and bubble departure frequency(f)are prescribed based on the work of Podowski [37]. The nucleation site density is modeled based on the work of Lemmart and Chawla [38], while the bubble departure diameter is modeled by a correlation from Tolubinsky and Kostanchuk [39]. The bubble departure frequency is approximated as a scaling of the bubble diameter to the terminal rise velocity based on the work of Cole [40].

1.3.2.2 CASL formulation

Closure models for the bubble departure diameter and the nucleation site density in the RPI model have been developed mostly based on experimental observations at low pressure. Feng [34] optimized the RPI formulation to improve its applicability at high pressures through better repre-sentation of closures, which resulted in the CASL formulation. The Tolubinsky and Kostanchuk correlation [39] used in the RPI model has a dependency only on the flow subcooling. Experimental measurements of the bubble departure diameter by Semeria [19] show a large variation of the bubble departure diameter with pressure, as shown in Figure 1.7. This dependence on pressure is modeled in the correlation developed by Kocamustafaogullari [41] and was chosen as the closure model for bubble departure diameter in the CASL formulation. The nucleation site density model from Lemmert and Chawla [38] shows a dependence only on the wall superheat. Experimental data from Borishanskii [42] shows a significant increase in the active nucleation sites with an increase in pressure. These observations were taken into account by Li et al. [43] to develop a new model to predict the nucleation site density as a nested power function that has a dependency on the pressure and the wall superheat. A summary of the closure models for the RPI and CASL model formulations is given in Table 1.2.

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Table 1.2:Closure models used in the RPI and CASL heat flux partitioning approaches

Closure RPI CASL

Bubble departure diameter Tolubinsky and Kostanchuk [39] Kocamustafaogullari [41]

Bubble departure frequency Cole [40] Cole [40]

Nucleation site density Lemmert and Chawla [38] Li et al. [43]

1.4 Second-generation heat flux partitioning framework

Gilman [24, 44] used insights from the experiments of Phillips [22] to develop a second-generation mechanistic heat flux partitioning framework. The framework aims at consistently including all physical mechanisms relevant to subcooled flow boiling, including bubbles sliding on the heater surface, microlayer evaporation, and bubble interaction. The mechanisms in Gilman’s framework are illustrated in Figure 1.11. Bubbles sliding on the heater surface enhance the heat transfer in their wake due to the reestablishment of the thermal boundary layer. The evaporation of a thin layer of liquid trapped under the bubble (microlayer) fuels the bubble growth and con-tributes to the evaporative heat transfer. Static interaction of neighboring nucleation sites results in a suppression of the number of sites that can support nucleation.

Figure 1.11:Mechanisms modeled in Gilman’s heat flux partitioning framework (from [24])

The heat flux at the wall is partitioned into four components, shown in the form of a mindmap in Figure 1.12. The inputs and outputs, in addition to the mechanisms accounted for in the framework, are shown in the mindmap. The framework tracks the complete bubble ebullition cycle, including the bubble nucleation and growth, bubble departure and liftoff, and the bubble sliding area. A description of the different heat flux partitions follows:

φwall =φf c+φsc+φsq+φe (1.8)

1. Forced convection: The heat removed by single-phase convection heat transfer.

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Figure 1.12:Mindmap of the heat flux partitions and mechanisms in Gilman’s boiling framework

(from [24])

2. Sliding Conduction: The enhancement of heat transfer in the wake of a sliding bubble due to the reestablishment of the thermal boundary layer.

φsc =

2kf(Twall−Tliquid)

p

πη_ft∗ t

∗_f _(1.10)

where t∗ is the time to reestablish the thermal boundary layer given by:

t∗ = k 2 f hf c 1 πηf ! (1.11)

3. Solid Quenching: The solid quenching component was introduced in the Gilman framework and accounts for the heat transfer due to the change in temperature of the heater substrate.

φsq =ρ_hc_ph2 3π Ddry 2 2 f N”∆T_h (1.12)

where∆This the average change in temperature of the heater substrate over the bubble period.

Experimental observations in pool boiling from Gerardi [45] are used to set∆Th=2.0K.

4. Evaporation: The evaporation heat flux is computed as a combination of the volume of vapor produced during the inception of the bubble and the volume of microlayer under the bubble that undergoes evaporation.

φe= hf gN”f ρg π 6 Dd 2 3 +ρ_f2 3π Dd 4 2 δ_ml ! (1.13)

where δ_ml is the maximum thickness of the microlayer and is set to δ_ml =2µm based on the work of Guion [46].

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Gilman leveraged closure models available in the literature to describe the bubble ebullition cycle. A mechanistic force balance model based on Sugrue [47] was used to predict the bubble departure diameter. The bubble departure frequency model of Cole [40] and the Hibiki-Ishii model, which provides improved predictions of the nucleation site densities, were adopted. A statistical approach based on complete spatial randomness theory [48] was used to model a static interaction of neighboring nucleation sites. The framework with existing closure models shows improved capabilities in predicting the boiling curves, compared to the RPI model.

The capability of the second-generation mechanistic heat flux partitioning approach to accu-rately represent the physics of subcooled flow boiling was demonstrated as part of this work [49], by replacing all closure models that served as inputs to the framework with experimental measurements. This approach allowed to remove the errors introduced by the closure models and evaluate the heat flux partitioning approach in isolation. The experimental data of Richenderfer [25] provided measurements of the boiling closures, heat flux partitions, and the boiling curves up to CHF. The heat flux partitions and the boiling curves predicted by the framework are shown in Figure 1.13. The heat flux partitions for two points on the boiling curve representing: i) a moderate heat flux in fully developed flow boiling and ii) a high heat flux close to CHF, are shown. It is notable that at moderate heat flux, the framework partitions approximately 80% of the energy applied at the wall to single-phase heat transfer, which is in agreement with the experimental measurements. This finding contradicts the classic understanding in the field that most of the heat flux in boiling heat transfer is partitioned to evaporation [50]. The increase in evaporation at high heat flux and the boiling curves are also correctly predicted, indicating a faithful representation of the underlying physics. This study demonstrates the potential of Gilman’s framework to accurately predict subcooled flow boiling heat transfer when equipped with correct closure models.

1.5 Objectives of the Thesis

The improved representation of the physical mechanisms relevant to subcooled flow boiling in Gilman’s framework is an important step towards developing a general model formulation. However, Gilman [44] observes that improvements to closure models for the bubble departure diameter and frequency need to be made to achieve generality in predictions. Gilman also notes that the inception diameter of the bubble and microlayer thickness have been prescribed based on limited experimental observations, and need to be revisited. Moreover, the closure models employed in the framework have been validated only at low pressures, restricting the framework’s applicability at high pressures.

Recent advancements in experimental capabilities provide advanced diagnostics into the physics of subcooled flow boiling, including measurements of the closures, heat flux partitions, and the boiling curves up to CHF. These insights can be used to develop new closure models and extend Gilman’s framework into a closed model formulation that accurately captures the underlying physics and demonstrates wide applicability. The objectives of this thesis are listed below:

– Review existing approaches to boiling closures modeling

– Leverage insights from experiments and direct numerical simulations to inform closure model development

– Develop post-processing techniques to perform indirect measurements of boiling closures from infrared and high-speed videos of the heater surface

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Figure 1.13:Heat flux partitions and boiling curve predicted by Gilman’s framework when all inputs

from closure models are replaced with experimental measurements from Richenderfer [25]. Flow conditions: Pressure: 1 bar, Velocity: 1 m/s, Subcooling: 10 K

– Propose closure models that more accurately predict bubble departure diameter and fre-quency

– Assemble a closed model formulation that can be consistently applied at all pressures

– Assess the predictive capabilities of the model on experimental data of boiling curves for varying flow conditions

– Demonstrate the extendability of the model to novel heater surfaces

– Provide a robust implementation of the model in a commercial CFD software and demonstrate the predictive capabilities of the model

1.6 Structure of the Thesis

The work performed towards improving modeling capabilities of the bubble departure diameter is presented in Chapter 2. A new criterion to model bubble nucleation based on the energy limit of the thermal boundary layer is developed in Chapter 3, and a model to predict the bubble departure diameter is proposed. The complete model formulation, applicable at all pressures is presented in Chapter 4. An assessment of the model capabilities against existing approaches in predicting the boiling curves and the trends with flow conditions is provided in Chapter 5. A

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robust implementation of the model is provided in a commercial CFD software (STAR-CCM+), and the results of a CFD simulation of the Bartolomei experiment are presented in Chapter 6. The conclusions of this thesis and scope for future work are listed in Chapter 7.

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Chapter 2

Bubble Departure Diameter

The diameter at which a growing bubble departs the heater surface has been referred to in literature as the bubble departure diameter. This definition is sufficient to describe a simple configuration of pool boiling on an upward-facing heater, wherein the bubble departs from the nucleation site and lifts off the heater surface at the same instant. However, in the case of flow boiling, a distinction between departure and liftoff needs to be made to account for a variety of scenarios, including:

1. Departure to bulk: A bubble departing from the nucleation site and lifting off the heater surface

2. Departure by sliding: A bubble departing from the nucleation site and sliding on the heater surface

3. Liftoff: A sliding bubble lifting off the heater surface

These scenarios are illustrated in Figure 2.1. The diameter of the bubble when it departs the nucleation site either by sliding or by departing into the bulk, is defined as the bubble departure diameter. The diameter of a sliding bubble at the instant it lifts off the heater surface is defined as the bubble liftoff diameter.

Figure 2.1:Illustration of the bubble departure and liftoff events in flow boiling

Accurately predicting bubble departure diameters is crucial for the overall accuracy of the mechanistic heat flux partitioning approach, owing to the evaporation heat flux’s sensitivity to the bubble departure diameter. The two approaches to modeling the bubble departure diameter are correlations and mechanistic force balance models. Correlations estimate the bubble departure diameter based on the flow conditions while mechanistic force balance models track the evolution

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of forces on the bubble to predict departure. Most commonly, heat flux partitioning approaches use either the Tolubinsky-Kostanchuk [39] or the Kocamustafaogullari [41] correlation to model the bubble departure diameter. The Tolubinsky Kostanchuk correlation models the dependency of the bubble departure diameter with liquid subcooling, as shown in Equation 2.1.

Dd =d0exp −∆Tsub ∆T0 (2.1)

where d0is the reference diameter with a default value of 0.0015 m,∆Tsubis the flow subcooling and

∆T0is the reference subcooling with a default value of 45 K. The default values of the diameter and

subcooling were calibrated from subcooled flow boiling experiments at atmospheric pressure. The correlation does not capture a dependency with the wall superheat and pressure. Moreover, the exponential dependence on subcooling results in un-physically low values of the bubble departure diameter at high subcooling.

The Kocamustafaogullari correlation models the dependence of bubble departure diameter on pressure using experimental data from pool boiling, as shown in Equation 2.2.

Dd=2.64×10−5 σ g∆ρ 0.5∆ρ ρg 0.9 (2.2)

where σ is the surface tension of the liquid, g is the acceleration due to gravity,∆ρ is the difference in densities of liquid and gas, and ρg is the density of the gas. While the correlation models a

dependency of the bubble departure diameter due to pressure, it shows no dependency on the applied heat flux and flow conditions.

An assessment of these correlations for variation in pressure and mass flux is shown in Figure 2.2 with experimental data of Sugrue [51]. The Tolubinsky-Kostanchuk correlation, which only depends on the flow subcooling, predicts a constant value for all conditions. While the Kocamustafaogullari correlation shows a dependency on pressure, the predictions vastly overestimate the measured departure diameters.

Figure 2.2:Assessment of the predictions of Tolubinsky Kostanchuk [39] and Kocamustafaogullari