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Thermodynamic analysis of two-phase fluids stored in stationary and slowly

moving tanks Dibakar Rakshit

This thesis is presented for the degree of

Doctor of Philosophy At

The University of Western Australia

School of Mechanical and Chemical Engineering 2012

Dedicated to

My Parents

ii

Abstract The present work deals with heat and mass transfer study of stationary and slowly moving,

partially filled containers. The focus is on an increase in the mass transfer of the system due

to vibration. For the stationary tank, the aim is to propose an optimum condition for (i)

maximum mass transfer and (ii) minimum entropy generation rate. The study commenced

with analysis of mass transfer and entropy generation of a storage container (asymptotic

limit) for two cases: (a) open and (b) closed top systems and proceeded to the mass transfer

study of a slowly moving container. For the moving tank the aim is to determine the effect

of vibration on the mass transfer.

A thermodynamic analysis of the two-phase physics involving a liquid-vapor combination

had been studied under the regime of conjugate heat and mass transfer phenomena. An

experiment had been designed and performed to estimate the interfacial mass transfer

characteristics of a liquid-vapor system by varying the liquid temperature. The

experimental set-up consisted of an instrumented stationary tank partially filled with water

and maintained at different temperatures. The evaporation of liquid from the interface and

the gaseous condensation had been quantified by calculating the interfacial mass transfer

rate for both covered and uncovered tanks. The dependence of interfacial mass transfer

rate on the liquid-vapor interfacial temperature, fractional concentration of the evaporating

liquid, the surface area of the liquid vapor interface and the fill level of the liquid had been

established through the present experimental study. An estimation of the overall mass

transfer rate from the interface due to a concentration gradient showed an analogy with the

multiphase heat transfer that took place across the interface due to the temperature

gradient. It was seen that at low fill levels and with a temperature difference of about 30 K

between the liquid and the ullage, the mass transfer rate of a closed system was nearly

doubled when compared to its open system counterpart.

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An experiment was also designed and performed to estimate the interfacial mass transfer

characteristics of a slowly moving tank. The tank was swayed at varying frequencies and

constant amplitude using 6 degrees of freedom motion actuator. The experiments were

conducted for a range of liquid temperatures and filling levels. The experimental set-up

consisted of a swaying tank partially filled with water at different temperatures. The

experiments were conducted for a frequency range of 0.7-1.6 Hz with constant amplitude

of 0.025 m. The evaporation of liquid from the interface and the gaseous condensation

was quantified by calculating the instantaneous interfacial mass transfer rate of the slow

moving tank. The dependence of interfacial mass transfer rate on the liquid-vapor

interfacial temperature, the fractional concentration of the evaporating liquid, the surface

area of the liquid vapor interface and the filling level of the liquid was established.

For optimizing the stationary container a “holistic” approach was utilized to thermally

optimize a storage container that is partially filled with liquid and vapor in its ullage. This

optimization was attempted by considering two symbiotic physical parameters influencing

the thermal performance of the storage container. The two physical quantities viz.

interfacial mass transfer and the entropy generated by the system were simultaneously

considered to arrive at a holistic optimum. The entropy of the system was estimated by

considering the entropy of the diabatic saturated liquid and the ullage vapor separately.

The impact of temperature increase influencing the entropy generation was studied keeping

in mind the various heat transfer characteristics of the system. This was followed by the

determination of individual optima through a synergistically considered composite

objective function based on the penalty involved in deviation from the optima. Thus, by

way of this, a “holistic” optimum was arrived at. It was seen that stored liquids showed a

better Second law efficiency at higher fill levels as compared to its lower fill level

counterparts due to various influencing factors that existed at high fill levels because of

roof condensation.

KEY WORDS: Concentration gradient; evaporation; condensation; conjugate heat and

mass transfer; filling level; holistic optimization; entropy generation

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Table of Contents ABSTRACT ................................................................................................................II

TABLE OF CONTENTS.........................................................................................IV

ACKNOWLEDGEMENTS .................................................................................... VII

STATEMENT OF CANDIDATE CONTRIBUTION............................................ X

LIST OF FIGURES ...............................................................................................XIII

LIST OF TABLES ...................................................................................................XV

CHAPTER 1 ................................................................................................................ 1

INTRODUCTION..................................................................................................................................1

1.1 BACKGROUND .............................................................................................................1

1.2 ASPECTS OF TWO-PHASE FLOW STUDIES OF STORED LIQUIDS..............................2

1.3 THERMODYNAMIC STUDIES OF LNG.......................................................................6

1.4 ORGANISATION OF THE THESIS ................................................................................8

1.5 CLOSURE.......................................................................................................................9

1.6 REFERENCES................................................................................................................9

CHAPTER 2...............................................................................................................13

ON THE CHARACTERIZATION OF INTERFACIAL MASS TRANSFER RATE OF STORED LIQUIDS

........................................................................................................................................................... 13

ABSTRACT ............................................................................................................................... 13

2.1 INTRODUCTION........................................................................................................ 14

2.2 THEORETICAL BACKGROUND................................................................................ 16

2.3 EXPERIMENTAL SETUP ............................................................................................ 19

2.4 UNCERTAINTY ANALYSIS........................................................................................ 24

2.5 VALIDATION ............................................................................................................. 24

v

2.6 RESULTS AND DISCUSSION...................................................................................... 25

2.6.1 INFLUENCE OF TANK FILL LEVELS ON MASS TRANSFER...................................... 27

2.7 CONCLUSIONS ........................................................................................................... 30

2.8 NOMENCLATURE ...................................................................................................... 30

2.9 ACKNOWLEDGEMENT ............................................................................................. 31

2.10 REFERENCES ............................................................................................................. 32

CHAPTER 3 ............................................................................................................. 34

ESTIMATION OF ENTROPY GENERATION BY HEAT TRANSFER FROM A LIQUID IN AN

ENCLOSURE TO AMBIENT ............................................................................................................... 34

ABSTRACT................................................................................................................................ 34

3.1 INTRODUCTION ........................................................................................................ 35

3.2 MATHEMATICAL FORMULATION............................................................................ 36

3.2.1 MATHEMATICAL FORMULATION FOR ENTROPY CALCULATION OF TWO-PHASE

ULLAGE REGION..................................................................................................................... 37

3.2.2 MATHEMATICAL FORMULATION FOR ENTROPY CALCULATION OF DIABATIC

SATURATED TWO PHASE LIQUID .......................................................................................... 38

3.3 EXPERIMENTAL SETUP............................................................................................. 40

3.4 VALIDATION.............................................................................................................. 41

3.5 RESULTS AND DISCUSSION...................................................................................... 42

3.6 CONCLUSIONS ........................................................................................................... 45

3.7 NOMENCLATURE ...................................................................................................... 46

3.8 ACKNOWLEDGEMENT ............................................................................................. 47

3.9 REFERENCES ............................................................................................................. 47

CHAPTER 4 ............................................................................................................. 49

THERMODYNAMIC OPTIMIZATION OF STORED LIQUID-A HOLISTIC APPROACH.................... 49

ABSTRACT................................................................................................................................ 49

4.1 INTRODUCTION ........................................................................................................ 50

4.2 THEORETICAL BACKGROUND AND MATHEMATICAL FORMULATION ............. 52

4.2.1 INTERFACIAL MASS TRANSFER RATE CALCULATION ............................................ 52

4.2.2 ENTROPY ESTIMATION OF STORED LIQUID.......................................................... 54

4.2.2.1 MATHEMATICAL FORMULATION FOR ENTROPY CALCULATION OF TWO-PHASE

ULLAGE REGION..................................................................................................................... 54

4.2.2.2 MATHEMATICAL FORMULATION FOR ENTROPY CALCULATION OF DIABATIC

SATURATED TWO PHASE LIQUID .......................................................................................... 55

4.3 EXPERIMENTAL SETUP............................................................................................. 58

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4.4 VALIDATION ............................................................................................................. 60

4.5 RESULTS AND DISCUSSION ..................................................................................... 61

4.5.1 NON DIMENSIONAL ENTROPY GENERATION: .................................................... 64

4.6 HOLISTIC OPTIMIZATION........................................................................................ 65

4.7 CONCLUSIONS........................................................................................................... 67

4.8 NOMENCLATURE...................................................................................................... 68

4.9 REFERENCES............................................................................................................. 70

CHAPTER 5.............................................................................................................. 73

EFFECT OF MILD SWAY ON THE INTERFACIAL MASS TRANSFER RATE OF STORED LIQUIDS. 73

ABSTRACT ............................................................................................................................... 73

5.1 INTRODUCTION........................................................................................................ 74

5.2 THEORETICAL BACKGROUND................................................................................ 77

5.3 EXPERIMENTAL SETUP ............................................................................................ 82

5.4 UNCERTAINTY ANALYSIS........................................................................................ 86

5.5 VALIDATION ............................................................................................................. 87

5.6 RESULTS AND DISCUSSION ..................................................................................... 88

5.6.1 INFLUENCE OF SWAY FREQUENCY ON MASS TRANSFER ..................................... 88

5.6.2 INFLUENCE OF TANK FILLING LEVELS ON MASS TRANSFER.............................. 91

5.6.3 INFLUENCE OF LIQUID TEMPERATURE ON MASS TRANSFER .............................. 92

5.7 CONCLUSIONS........................................................................................................... 95

5.8 NOMENCLATURE...................................................................................................... 97

5.9 ACKNOWLEDGEMENT............................................................................................. 98

5.10 REFERENCES............................................................................................................. 98

CHAPTER 6.............................................................................................................102

CONCLUSIONS AND SUGGESTIONS FOR FUTURE WORK .......................................................... 102

6.1 INTRODUCTION...................................................................................................... 102

6.2 BROAD CONCLUSIONS OF THE PRESENT STUDY ............................................... 102

6.3 SUGGESTIONS FOR FUTURE WORK ...................................................................... 105

APPENDIX-1 .............................................................................................................. I

APPENDIX-2............................................................................................................XI

APPENDIX-3...........................................................................................................XX

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Acknowledgements In life certain matters are too precious to be evaluated, especially when it is something as

important as consummation of a degree from an institute of international importance. In

such situations I feel that the word acknowledgement is very meagre to be used at. Anyway

as a part of routine curriculum this should also be placed in its correct location of

formalities. But whatever is informal and affectionate has already become an indelible

chapter of my life and will always serve as lighthouse for the harsher portions of the life to

come and will help me to overcome those situations.

At first I would like to convey my heartiest gratitude to my principal supervisor Prof. Krish

Thiagarajan for undertaking the long journey of my research along with me for over four

years. The innumerous ups and downs of my professional and personal life would not have

been easy to overcome without his support, for which I sincerely thank him. With a

wistful of hopes when I started my research it was Prof. Thiagarajan’s patience and faith on

me, which helped me to channelize my work in the right direction. His sharp analytical

skills and proper guidance encouraged me to think more precisely on my research problem.

His guidance with theory and instruction in data analysis was invaluable. I would like to

thank him for patiently listening when I felt exacerbated by my experiment, and would

spend countless hours helping me to figure things out. Whatever problem I may have

encountered Prof. Thiagarajan would have a solution on hand. His knowledge and ability

to see the “big picture” has greatly influenced my work. I show my sincere thankfulness

for the person who has demonstrated to me what it takes to be a successful researcher.

I also express my sincere indebtedness and profound sense of gratitude to my co-

supervisor Dr. Ramesh Narayanaswamy whose unstinted views and gracious

encouragement helped me to bring perfection in my work. He was always willing to

discuss my technical problems and suggest the most insightful solutions. I would like to

thank him for sharing all his expertise in Experimental Heat Transfer while looking at

viii

countless approaches to achieve optimum solution for my research. He inspired my

interest in the subject while taking time to help me identify my strengths and implement it

in my research for which I am deeply thankful. Dr. Narayanaswamy’s thoughtful

suggestions and dynamic personality always induced activeness and freshness in my work.

He has provided me with the knowledge and motivation to successfully complete my

research. Dr. Narayanaswamy not only paid special attention on my research problem but

also helped me to evolve more prominently for my future endeavors.

Thanks must also go to all the students and staffs of the Floating Structures and Naval

Architecture Group of The School of Mechanical Engineering for providing such a

stimulating and enjoyable environment to work. The scientific discussions with Dr.

Fabrizio Pistani and his assistance in setting up initial experiments helped me to conduct

my experiments smoothly. Dr. Pistani was always willing to consider my technical

problems and suggest the most insightful avenues for investigation. Suggestions regarding

Design of Experiments with Dr. Tam Truong were helpful in resolving some of the initial

setbacks. Mr. Gerald Wright’s assistance with electronics and Mr. Mike’s support with

workshop for the experimental setup has been indispensable. I owe a profound debt of

gratitude to Prof. Hong Yang who took over the reins from Prof. Thiagarajan when he

moved from The UWA to The University of Maine. Prof. Hong Yang’s ability to

coordinate my final thesis submission made all my administrative obligations easy to

handle. I also thank Prof. Des Hill from School of Mathematics for helping me out with

the mathematical integration of my research problem.

I would like to thank “The UWA Student Services” for their excellent support and

guidance. I could not imagine seeing the valediction to my thesis without help and

coordination of Dr. Krystyna Haq. The quality time spent with her in bringing the thesis

to required level of perfection is invaluable. Her indelible patience to go through my thesis

and suggest corrective measures has been rewarding in writing my thesis. I am thankful to

her for all her help and support.

The opportunity to do this research was made possible by an International Postgraduate

Research Scholarship and University Postgraduate Awards. I would like to acknowledge

The University of Western Australia for providing financial support for this research. I

show sincere gratitude to The Graduate Research School in particular, Dr. Sato Juniper, for

her unconditional support towards the completion of my thesis.

ix

On a more personal note I thank my research buddies Hema, Nitin, Ashkan, Asamudin,

Jagadeesh and Kamarudin with whom I spent the most relaxing times over the last four

years. I must also acknowledge and give praise to my closest buddy “Patience” and the

most valuable thought “One baby step at a time”. It has been with me in every step to

reach my thesis completion. I would also like to thank Ms. Christine Shervington from

The UWA Oceans Institute for her measures to conduct seminars and trainings at the

Institute to provide a holistic research environment for budding researchers. The biggest

acknowledgement must go to my friends who have been with me every step of the way

from the lows to the joyful highs. Nothing makes me feel better at the end of the day,

seeing you all smile (Rupali, Shruti, Carol, Mani, Pooja, Karen, Anisha, Sonia, Paritosh,

Tejas, Thejaswi, Sameer, Surendra, Abishek, Ramji, Vishesh and Saumya). I must

acknowledge Ms. Sushama Wagh for her continuous encouragement to retain my morale

and the spirit to keep fighting without giving up. And finally I would like to sincerely

thank my parents for their unconditional love and support. They are always there for me,

never judgmental irrespective of which path I choose to follow. Thank You Ma Baba.

Perth Dibakar Rakshit August 2011

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Statement of candidate contribution In accordance with The University of Western Australia’s regulations regarding Research

Higher Degrees, this thesis is presented as a series of journal papers which are-

• published

• under review or

• prepared for submission.

The contribution of the candidate and co-authors for the papers comprising chapters 2, 3,

4, 5 along with Appendix 1, Appendix 2 and Appendix 3 are hereby set forth.

Paper 1

The paper presented in Chapter 2 is first-authored by the candidate and co-authored by Dr.

Ramesh Narayanaswamy and Prof. Krish Thiagarajan, and is under review as-

Dibakar Rakshit, R. Narayanaswamy and K. P. Thiagarajan. On the characterization of

interfacial mass transfer rate of stored liquids. Experimental Fluid and Thermal

Science.

The candidate planned and carried out the experimental curriculum under the supervision

of Prof. Krish Thiagarajan and Dr. Ramesh Narayanaswamy. The experimental apparatus

was developed by the candidate. The candidate analyzed the experimental results and

wrote the paper under supervision of the co-authors.

Paper 2

The paper presented in Chapter 3 is first-authored by the candidate and co-authored by Dr.

Ramesh Narayanaswamy and Prof. Krish Thiagarajan and is published as-

D. Rakshit, R. Narayanaswamy, K. P. Thiagarajan. Estimation of entropy generation

by heat transfer from a liquid in an enclosure to ambient. ANZIAM Journal, 2011,

Vol.51, p.852-873. ISSN-1446-8735

The mathematical formulation for entropy generation calculation in partially filled liquid

tanks introduced in this paper has been developed by the candidate. The candidate

planned and carried out the experimental curriculum under the supervision of Prof. Krish

xi

Thiagarajan and Dr. Ramesh Narayanaswamy. The candidate analyzed the experimental

results and wrote the paper under supervision of the co-authors.

Paper 3

The paper presented in Chapter 4 is first-authored by the candidate and co-authored by Dr.

Ramesh Narayanaswamy and Prof. Krish Thiagarajan and is prepared for publication as-

D. Rakshit, R. Narayanaswamy, K. P. Thiagarajan. Thermodynamic optimization of

stored liquid-a holistic approach. Journal of Energy Research.

The concept of holistic optimization in partially filled liquid tanks dissipating heat to the

ambient has been introduced in this paper by the candidate. The candidate planned and

carried out the experimental curriculum under the supervision of Prof. Krish Thiagarajan

and Dr. Ramesh Narayanaswamy. The candidate analyzed the experimental results and

wrote the paper under supervision of the co-authors.

Paper 4

The paper presented in Chapter 5 is first-authored by the candidate and co-authored by

Prof. Krish Thiagarajan and Dr. Ramesh Narayanaswamy and is prepared for publication

as-

D. Rakshit, K. P. Thiagarajan, R. Narayanaswamy, Effect of mild sway on the

interfacial mass transfer rate of stored liquids. International Journal of Thermal

Sciences.

The concept of interfacial mass transfer estimation in partially filled slow moving tanks

under different conditions has been studied by the candidate. The candidate planned and

carried out the experimental curriculum under the supervision of Prof. Krish Thiagarajan

and Dr. Ramesh Narayanaswamy. The candidate analyzed the experimental results and

wrote the paper under supervision of the co-authors.

Paper 5

The paper presented in Appendix 1 is first-authored by the candidate and co-authored by

Dr. Ramesh Narayanaswamy, Dr. Tam Truong, Prof. Krish P Thiagarajan and is published

as-

Dibakar Rakshit, Ramesh Narayanaswamy, Tam Truong, Krish P Thiagarajan, An

experimental study on the interface mass transfer governing thermodynamics of

stored liquids. Proceedings of the 20th National and 9th International ISHMT-ASME

Heat and Mass Transfer Conference, January 4-6, 2010. ISBN- 978-981-08-3813-3

The candidate planned and carried out the experimental curriculum under the supervision

of Prof. Krish Thiagarajan and Dr. Ramesh Narayanaswamy. Dr. Tam Truong helped with

initial Design of Experiments. The experimental apparatus was developed by the

xii

candidate. The candidate analyzed the experimental results and wrote the paper under

supervision of the co-authors.

Paper 6

The paper presented in Appendix 2 is first-authored by the candidate and co-authored by

Nitin Repalle, Jagadeesh Putta and Krish P Thiagarajan and is published as-

Dibakar Rakshit, Nitin Repalle, Jagadeesh Putta and Krish P Thiagarajan, A numerical

study on the impact of filling level on liquid sloshing pressures, Proceedings of the

27th International Conference on Offshore Mechanics and Arctic Engineering

OMAE2008, June 15-20, Estoril, Portugal. ISBN- 0-7918-3821-8

The candidate carried out the CFD simulations presented in this paper based on the

requirements suggested by Prof. Krish Thiagarajan and Dr. Jagadeesh Putta. Mr. Nitin

Repalle helped with post processing of the data. The candidate analyzed the experimental

results and wrote the paper except Numerical Approach Basics (written by Dr. Jagadeesh

Putta) under supervision of Prof. Krish Thiagarajan.

Paper 7

The paper presented in Appendix 3 is first-authored by the candidate and co-authored by

Dr. S Dev, and Prof. K. P. Thiagarajan and is published as-

D Rakshit, S Dev, and K P Thiagarajan, Application of nano-paints for hydrodynamic

pressure visualization, Proceedings of the Fifth Australian Conference on Laser

Diagnostics in Fluid Mechanics and Combustion, The University of Western Australia

3-4 December 2008. ISBN-978-1-74052-177-2

The candidate carried out the detailed study of the pressure measurement techniques

presented in this paper based on the requirements suggested by Prof. Krish Thiagarajan.

Dr. Selvi Dev helped with collecting literature. The candidate performed the basic 3-

Dimensional CFD analysis and wrote the paper under supervision of Prof. Krish

Thiagarajan.

I certify that, except where specific reference is made in the text to the work of others, the

contents of this thesis are original and have not been submitted to any other university.

(Dibakar Rakshit)

(Prof. Hong Yang)

xiii

List of Figures Figure 1. 1 Tank configurations...................................................................................................... 4

Figure 1. 2 Two-Phase free surface comparisons of CFD simulations with high speed

imaging................................................................................................................................................ 5

Figure 2. 1 Diffusion of water vapor through air between two planes x1 and x2 .................. 16

Figure 2. 2 Schematic set-up of the test rig ................................................................................. 19

Figure 2. 3 Moisture sensing device used to measure the relative humidity along tank height

........................................................................................................................................................... 20

Figure 2. 4 Layout of the test rig................................................................................................... 20

Figure 2. 5 Schematic of the experimental setup with position of moisture sensing probe 23

Figure 2. 6 Experimental and analytical relative humidity variation across tank height ....... 26

Figure 2. 7 Transient temperature history for 20% filling level ............................................... 26

Figure 2. 8 Interface mass transfer variation with temperature ............................................... 27

Figure 2. 9 Model of the heat and mass transfer in the tank with characteristics zone........ 28

Figure 2. 10 Mass transfer variations with vapor pressure of ullage ....................................... 29

Figure 3. 1 Schematic of two-phase liquid and ullage with ambient conditions.................... 37

Figure 3. 2 Schematic of diabatic saturated two-phase liquid in stored container ................ 38

Figure 3. 3 Schematic set-up of the test rig ................................................................................. 41

Figure 3. 4 Transient temperature history for 20% filling level with step increase of

temperature ...................................................................................................................................... 43

Figure 3. 5 Variation of heat transfer coefficient across the front wall with temperature ...44

Figure 3. 6 Variation of heat transfer coefficient across the side wall with temperature ..... 44

Figure 3. 7 Variation of total entropy per unit volume with temperature .............................. 45

Figure 3. 8 Variation of entropy per unit volume in ullage with temperature ....................... 45

Figure 4. 1 Diffusion of water vapor through air between two planes x1 and x2 .................. 53

Figure 4. 2 Schematic of two-phase liquid and ullage with ambient conditions.................... 54

Figure 4. 3 Schematic of diabatic saturated two-phase liquid in stored container ................ 55

Figure 4. 4 Schematic set-up of the test rig ................................................................................. 58

xiv

Figure 4. 5 Layout of the test rig .................................................................................................. 59

Figure 4. 6 Mass transfer variation with filling levels for an open container......................... 61

Figure 4. 7 Mass transfer variation with filling levels for a closed container......................... 61

Figure 4. 8.Variation of total entropy per unit volume with temperature.............................. 63

Figure 4. 9 Variation of entropy per unit volume in ullage with temperature ....................... 63

Figure 4. 10 Variation of non-dimensional entropy generated with temperature................. 64

Figure 4. 11 Variation of NQ and 1/NS with temperature for open and closed tanks....... 66

Figure 5. 1 Diffusion of water vapor through air between two planes y1 and y2 .................. 78

Figure 5. 2 Schematic set-up of instrumented tank ................................................................... 82

Figure 5. 3 Schematic of the experimental setup with position of moisture sensing probe 83

Figure 5. 4 Layout of the experimental set up............................................................................ 85

Figure 5. 5 Standing wave generation by swaying liquid at 30 ° C with Amplitude 0.025m,

Frequency 0.7 Hz............................................................................................................................ 88

Figure 5. 6 Interfacial mass transfer variation at 10% filling level and 30° C for different

sway frequencies ............................................................................................................................. 89

Figure 5. 7 Ullage height variation for10% filling level tank swaying at 0.7 Hz frequency . 90

Figure 5. 8 Interfacial mass transfer variation with filling levels at 30° C and 0.7 Hz sway

frequency.......................................................................................................................................... 93

Figure 5. 9 Interfacial mass transfer variation with temperature for 10% filling level and 0.7

Hz sway frequency.......................................................................................................................... 94

Figure 5. 10 Model of the heat and mass transfer in the tank with characteristic zones ..... 95

xv

List of tables Table 2. 1 Range of Parameters .................................................................................................... 23

Table 2. 2 Uncertainty in the physical quantities of present experimental study................... 24

Table 3. 1 Range of parameters..................................................................................................... 41

Table 3. 2 Comparison of the heat transfer coefficients ........................................................... 42

Table 4. 1 Range of Parameters .................................................................................................... 60

Table 4. 2 Overall penalties in mass transfer and entropy generation for holistic

optimization of tank with optimum temperature....................................................................... 67

Table 5. 1 Natural sloshing periods of tank for different filling levels ................................... 84

Table 5. 2 Hexapod Operating Conditions (Input/Output Matrix) ....................................... 85

Table 5. 3 Range of Parameters .................................................................................................... 86

Table 5. 4 Uncertainty in the physical quantities of present experimental study................... 87

Table 5. 5 Mean Interfacial mass transfer comparison between stationary and vibrating

tanks .................................................................................................................................................. 90

Table 5. 6 Effect of filling level on mean interfacial mass transfer ......................................... 91

1

Chapter 1 Introduction

1.1 Background

The study of two-phase physics is a major step forward in the quest for understanding fluid

behavior as they metamorphose through change of phase, properties and structure. The

actual challenge in the study of two-phase physics lies in the vastness of scale where it can

be encountered. Two-phase physics finds utility in a broad spectrum of applications,

ranging from next generation nuclear machinery and space engines to pharmaceutical

manufacturing, food technology, energy and environmental remediation. Some of the

applications in which two-phase flow studies find strong relevance are: power systems, heat

transfer systems, process systems, transport systems, lubrication systems, environmental

control and geo-meteorological phenomenon.

The design of engineering systems and the ability to predict their performance depends

upon both the availability of experimental data and of conceptual mathematical models that

can be used to describe the physical processes with the required degree of accuracy. The

theoretical discourse and experimental proposition of two-phase flow are closely linked

with each other. The synthesis that arises from this correlation generates immense

technological abilities for measurements informing and validating dynamic models and vice

versa. Therefore the conceptual models of two-phase systems should be formulated in

terms of the appropriate field and relevant applications that can bestow the assumptions

made for a particular scenario. The derivation of such equations for two-phase flow is

considerably more complicated than single-phase flow. The complex nature of two-phase

flow initiates from the existence of multiple, deformable and moving interfaces. These

2

interfaces create significant discontinuities of fluid properties resulting in complicated flow

fields. The present study of two-phase physics deals with the heat and mass transfer

characterizations of liquids stored in containers.

1.2 Aspects of two-phase flow studies of stored liquids

The two-phase scenario involving liquid storage has a wide range of applications. From

cryogenic fuel storage in spacecrafts to transportation of liquid fuels from its source to

destination, two-phase studies of stored liquids has become inevitable. As mentioned

earlier the interfacial physics of two-phase systems becomes complicated when the liquid

starts moving resulting in moving interfaces. This free surface motion of a liquid inside its

containment is defined as sloshing. The type of motion to which the free surface is

subjected mainly depends on the type of disturbance and container shape. These motions

include simple planar, non planar, rotational, irregular beating, symmetric, asymmetric,

quasi-periodic and chaotic [1]. Because of these motions, the free surface of the liquid

takes some wavy structure in accordance to the response of the elasticity of the

containment material which contains the fluid.

The study of liquid sloshing dates back to 19th century when Sir Horace Lamb [2]

attempted to explain the fluid dynamics of sloshing phenomenon. He analyzed the

problem in a two dimensional matrix, under the wave propagation theory explaining the

effect of a travelling pressure disturbance having similarities with sloshing. Later with

revolutionary space science advances, the criticality of sloshing was much felt in aerospace

applications and rocket technologies. The cryogenic effects of the liquid propellant made

the study of the two-phase sloshing inevitable for space applications. Engineers in the

1960s and 1970s were involved in computing and predicting the thermo-hydraulic behavior

of the propellants (liquid hydrogen as fuel and liquid oxygen as oxidizer) in their respective

tanks. The various conclusions which were drawn from the research of sloshing were

mainly concentrated with space technologies. Besides space applications one contemporary

field where sloshing tanks have become a matter of concern is transportation of fuel from

its source to its destination.

In recent years, economic prosperity has led to a tremendous growth in energy

consumption. This has created increased utilization of various non-renewable resources of

3

energy. Transportation of various forms of energy from its source to destination is one of

the important factors, which require proper attention and understanding. In this context

Liquefied Natural Gas (LNG) has become a significant energy resource. LNG is obtained

from Natural Gas (NG) which is thermodynamically processed to attain its liquid state by

cooling to a temperature of −163 °C. For LNG the thermodynamic process of

condensation is carried out at a pressure close to atmospheric pressure by cooling it to

approximately −163 °C. The maximum pressure that is set for LNG transportation is

around 125 kPa.

When vaporized, LNG is combustible in stoichiometric concentrations of 5% to 15%

when mixed with air. When the concentration is less than 5% there is insufficient natural

gas to burn, whereas when the concentration is in excess of 15% there is not enough

oxygen to stimulate combustion. Besides these when cold LNG comes in contact with

warmer air, it starts changing its phase and becomes a visible vapor cloud. Finally if the

vapor cloud reaches a concentration of 5% in the air mixture then the mixture becomes

combustible. In addition to these LNG when spilled on water at a very fast rate, a Rapid

Phase Transition (RPT) occurs. In this process heat transfer takes place from the water to

the LNG at a very high rate. This causes instantaneous conversion of LNG from its liquid

phase to its gaseous phase. This type of rapid transition between phases releases large

amount of energy without any combustion and can sometimes result in catastrophic

physical explosions.

LNG is transported in double-hulled ships specifically designed to handle the cryogenic

properties. These carriers are insulated to limit the amount of LNG that boils off or

evaporates. LNG carriers are up to 305 meters long, and require a minimum water depth

of about 12 meters when fully loaded. MOSS design of LNG tanks (Figure 1.1a) is the

traditional form of design in which spherical vessels are used to transport LNG. The Moss

LNG carrier with spherical LNG tanks represents the safest and most reliable LNG

containment system on the market. Card and Lee [3] in their innovative approach to

design LNG tanks to meet various multi-dimensional requirements came up with some

noble design modifications. Their strength assessment procedure involved impact strength

of insulation. An example of this new membrane type of vessels is shown in Figure 1.1b.

4

Figure 1. 1 Tank configurations

a. MOSS Type System b. GTT No 96 System (Courtesy www.mossww.com) (Janssens and Lee, 2005)

To fulfil the spot market requirement partially filled LNG tanks when transported from

one place to another get subjected to loading, unloading and sloshing which has been

studied in details for the first time by Bass etal [4]. Because of this continuous loading,

unloading and sloshing LNG cannot maintain its thermodynamic stability, and thus in long

run the LNG properties get modified. As far as hydrodynamic aspects of sloshing is

considered, the key issue is to find the hydrodynamic pressure distribution, forces,

moments and the natural frequency of oscillation which sometimes becomes vulnerable for

the containment. A review of literature shows that hydrodynamics studies of liquid

sloshing in storage containers have been done for the following parameters [5]:

• Impact Pressure

• Impact Velocity

• Total forces and moments acting on the tank

• Temporal behavior of Pressure

• Temperature change of the fluid system

The impact pressure measurement on sloshing tanks has a significant contribution in

understanding the sloshing hydrodynamics. The impact pressure measurement is the

measurement of the pressure that the fluid exerts in terms of force per unit area on the

walls of its container. The local fluid pressure may be dependent upon many variables viz.

elevation, flow velocity, fluid density and temperature. The exact location of peak impact

pressure cannot be obtained with “traditional” instrumentation such as pressure

5

transducers that provide local, single-point measurements. Moreover, the time period for

which the impact pressure measurement exists is of the order of milliseconds.

Therefore there is a need for sophisticated measurement techniques which report the

globalized whole-field values of pressure at each location of the wall. Pressure Sensitive

Paints (PSPs) is one such non intrusive pressure measurements technique which gives

globalized values of pressure. The viabilities of using PSP for hydrodynamic study have

been explored in details in Appendix 3. With advancement in computational efficiency and

sophisticated instrumentations the ability to predict sloshing impact took a new path. One

of the basic studies detailing the parameters influencing tank design subjected to sloshing

has been explained in Appendix 2. A most recent study in the field of liquid sloshing

containers is done by Thiagarajan et al. [6]. With Computational Fluid Dynamics (CFD)

and High Speed Imaging as predictive tools the process of depicting impact pressure has

become experimentally less cumbersome. Figure 1.2 shows the comparison of a CFD

simulated sloshing free surface with high speed images obtained from experiments.

Figure 1. 2 Two-Phase free surface comparisons of CFD simulations with high speed imaging (Sway Frequency- 1.1 Hz, Amplitude- 0.025 m)

(50% filling level) (10% filling level)

6

Besides the resonating frequency of the sloshing fluid, the other key parameter which

contributes to the structural instability of the sloshing tank is ullage pressure [1]. Ullage

pressure in a closed container is the pressure above the free surface of the liquid, in the

vapor region. The ullage pressure plays an important role in the boil off phenomenon of

thermally insulated LNG tanks. In the process of filling LNG inside the tank the region

above the LNG becomes a stratified zone with vapor of LNG as a part of that zone.

Further when new LNG is added to already existing LNG in partially filled tank, the

density variation plays an important role in stratifying the ullage zone. It occasionally results

in mixing accompanied by a sudden release of large amounts of boil off gas, which causes

rapid pressure rise [7]. Traditionally, LNG tankers have utilized this boil off as fuel for the

ship propulsion system.

The ullage pressure above the liquid in the container plays a critical role in making a proper

selection of the various fill levels, and temperature with which LNG can be transported

from one place to another. To date, understanding of the effect of sloshing has been

based solely on hydrodynamic considerations. In order to understand the underlying

principles and the physical mechanisms of LNG sloshing, both fluid mechanics and

thermodynamics must be studied.

1.3 Thermodynamic studies of LNG

A review of literature shows that the heat transfer and thermodynamic studies of LNG

have mainly been done in the area of LNG tank design. The main considerations in these

studies were the insulation design of the tank to maintain it at sub-zero temperatures.

Kobayashi and Tsutsumi [8] studied the hybrid thermal insulation system of supporting

skirt of MOSS type LNG carrier by inserting a non-metallic material of low thermal

conductivity in between the aluminum alloy and mild steel. Bates and Morrison [9] studied

the characteristics of LNG by considering two principal phases through which the liquid

passes when roll over phenomenon takes place. Studies by Chen et al. [10], Boukeffa et al.

[11] and Khemis et al. [12] attempted to estimate the heat transfer coefficient of cryogenic

containers under various thermal conditions.

In the periphery of two-phase study of sloshing fluids, related to cryogenics the mechanism

of evaporation of a liquid immersed in another liquid was the main area of study. Gradon

and Selecki [13], Raina and Wanchoo [14] in their studies on direct contact heat transfer

with phase change provided expressions for the instantaneous velocity of a two-phase

7

bubble. Raina and Grover [15] introduced sloshing effect into the direct contact heat

transfer modeling of a multiphase system. Fang and Ward [16], Fedorov and Luk’yanova

[17], Kozyrev and Sitnikov [18], Scurlock [19], Krahl and Adamo [20] studied interfacial

multiphase heat transfer of cryogenic fluids. These studies were based on the methods of

predicting the interfacial temperature and vapor pressure with various factors affecting their

measurement. The multiphase study of sloshing physics in the vicinity of miscible fluid

combination viz. water-vapor-water system was first addressed by Dent [21]. Rose [22]

then studied the heat transfer mechanisms involved in a drop undergoing condensation.

Similar studies of forced convection boiling were undertaken by Tolubinskiy et al. [23].

Antonenko et al. [24] identified various boiling modes and showed that heat transfer is

enhanced when a vibration is induced into the system. A detailed study of condensation

heat transfer performance involving various gyroscopic motions was undertaken by Katsuo

et al. [25].

From these studies it can be concluded that the thermodynamics of the phase change

phenomenon makes a significant contribution to the proper conditioning, transportation

and storage of LNG. Hence the study of thermodynamics becomes critical when studying

the sloshing physics. The two-phase interaction of the liquid with ullage vapor in LNG

containers mainly depends on the interfacial physics of the evaporation and the

condensation of the fluid in the container. The multiphase characteristics of LNG may be

well understood by examining any miscible fluid combination like water-vapor and water

system at its equilibrium. Similar to all other non-reactive liquid-gas combinations, the

liquid phase of natural gas always maintains a dynamic equilibrium with its gaseous phase

under all storage conditions [26]. The two-phase interaction of the liquid with ullage vapor

in partially filled containers mainly depends on the interfacial physics of evaporation and

condensation of the fluids in the container irrespective of the type of fluid.

The literature review shows that published work involving the phase change mechanism of

a liquid to its gaseous phase during its storage is very scarce. In most of these studies, the

regime of evaporation and condensation of the interface was investigated without

consideration of the effects of temperature variation and fill levels at which the liquid was

being stored. The present work is an attempt to estimate the overall mass transfer from a

water vapor - water interface taking into consideration the dynamic equilibrium of the air-

water system in which the two participating phases undergo a phase change phenomenon.

The interfacial mass transfer of water vapor – water system has been explored in this thesis

because of its analogous behavior with non-reactive NG –LNG combination. The

8

dependence of interfacial mass transfer rate on the liquid-vapor interfacial temperature,

fractional concentration of the evaporating liquid, the surface area of the liquid vapor

interface and the fill level of the liquid has been established through the present study. The

study described in this thesis aims to fill this gap by estimating the overall mass transfer

from a stationary and slowly moving water vapor - water interface, taking into

consideration the dynamic equilibrium of the water vapor - water system.

1.4 Organisation of the thesis

This thesis is divided into six chapters. The first chapter gives a brief introduction regarding

the needs and scope of two-phase study of liquid stored in a container. A brief synopsized

form of the literature survey is also presented in this chapter to make it more informative

and convenient for its later use. This is followed by chapter 2, 3, 4 and 5 which are

independent publications either published or in the process of getting published. Chapter 2

describes a thermodynamic analysis of the two-phase physics involving a liquid-vapor

combination under the regime of conjugate heat and mass transfer phenomena. This

chapter also describes the basic assumptions postulated for modeling the system without

any loss of generality.

The overall success of a system depends on the design efficient feature of that product.

Hence the analysis of a system should be multi-dimensional that addresses every

requirement of the designing. It may no longer be adequate just to develop a system that

performs the desired task to satisfy a recognized need. There is much more to be looked

upon to make the system utilized in its best possible means. Hence, the technologies of the

day are directed towards producing optimized design with conceptual understanding of the

physics. Keeping in view all these factors the two-phase thermodynamic analysis of stored

liquids is analyzed further to estimate the amount of entropy generated by the system.

Chapter 3 mainly deals with the mathematical formulation, used for estimating the entropy

generated by liquids stored in a container. This study outlines the estimation of entropy

generation of a stored liquid-vapor combination. A salient feature of the present study is

the incorporation of a separated flow model for the calculation of entropy generated by a

diabatic two-phase system.

Chapter 4 deals with the impact of entropy generated by the system on the interfacial mass

transfer. This chapter deals with best possible way to design a thermal system by

holistically optimizing the systems. This is done by simultaneously considering the two

9

competing criteria of maximum interfacial mass transfer and minimum entropy generation

rate.

This is followed by Chapter 5 in which an exploratory study of two-phase physics was

undertaken in a slow moving tank containing liquid. An experiment was designed and

performed to estimate the interfacial mass transfer characteristics of a slowly moving tank.

The interfacial mass transfer rate estimated for the swaying tank was then compared with

the interfacial mass transfer rate of stationary tank estimated in Chapter 2. This is followed

by conclusions and appendices. Appendices deals with the explanation of initial

mathematical formulation for determining interfacial mass transfer, CFD techniques used

to determine impact pressures in sloshing tanks and utility of PSP for measuring pressures

in two-phase hydrodynamic system.

1.5 Closure

This chapter gave a brief introduction to the main theme of the present study. A broad

outline of the problems to be considered in the study has also been given. The organization

of the thesis highlighting the salient features of each chapter has been presented.

1.6 References

[1] Raouf A., Ibrahim, 2005. “Liquid Sloshing Dynamics Theory and Applications”

Cambridge Press University Sir Horace Lamb, Hydrodynamics, Cambridge

Mathematical Library, Sixth Edition 1932.

[2] Card, J., and Lee, H., 2005. “Leading Technology for Next Generation of LNG

Carriers” International Offshore and Polar Engineering Conference Seoul, Korea

p. 83-89.

[3] Faltinsen, O. M., Olsen, H. A., Abramson, H. N., and Bass, R. L., 1974. “Liquid

sloshing in LNG carriers”, DNV Classification and Registry of Shipping, 85

[4] Bass, R. L., Bowles, E. B., Trudell, R.W., Navickas, J., Peck, J. C., Yoshimura, N.,

Endo, S., Pots, B. F. M., 1985. “Modelling Criteria for Scaled LNG Sloshing

Experiments”, Transaction of the ASME, 107: p. 272-280.

10

[5] Thiagarajan, K. P., Rakshit, D., Repalle, N., 2011. “The air-water sloshing problem:

fundamental analysis and parametric studies on excitation and fill levels”,

International Journal of Ocean Engineering, 38, 498-508.

[6] Koyamada, K., Tamura, S., and Ono, O., 2006. “CFD Simulation on LNG Storage

Tank to Improve Safety and Reduce Cost”, Systems Modelling and Simulation

Theory and Applications, Asia Simulation Conference p. 39-43

[7] Kobayashi, Y., and Tsutsumi, T., 2000. “Heat Transfer Characteristic of the

Spherical Tank Skirt of LNG and LH2 Having a Hybrid Construction”,

Transactions of the West Japan Society of Naval Architects. 99, 395-402.

[8] Bates, S., and Morrison, D. S., 1997. “Modelling the behavior of stratified liquid

natural gas in storage tanks: a study of the rollover phenomenon”, Int. J. Heat Mass

Transfer 40(8) 1875-1884.

[9] Chen, Q. S., Wegrzyn, J., Prasad, V., 2004. “Analysis of temperature and pressure

changes in liquefied natural gas (LNG) cryogenic tanks”, Cryogenics. 44 701–709.

[10] Boukeffa, D., Boumaza, M., Francois, M., X., and Pellerin, S., 2001. “Experimental

and Numerical Analysis of heat losses in a liquid Nitrogen Cryostat”, Applied

Thermal Engineering. 21, 967-975.

[11] Khemis, O., Boumaza, M., Ali, A. Francois, M., X., 2003. “Experimental analysis of

heat transfers in a cryogenic tank without lateral insulation”, Applied Thermal

Engineering, 23, 2107–2117.

[12] Gradon, L., and Selecki, A., 1977. “Evaporation of a liquid drop immersed in

another immiscible liquid. The Case of σc < σd”, International Journal of Heat

and Mass Transfer, 20, 459-466

[13] Raina, G. K., and Wanchoo, R. K., 1984. “Direct Contact Heat Transfer with Phase

Change: Theoretical Expression for Instantaneous Velocity of a Two-Phase

Bubble”, International Communication in Heat and Mass Transfer, 11, 227-237.

11

[14] Raina, G. K., and Grover, P. D., 1985. “Direct Contact Heat Transfer with Phase

Change: Theoretical Model Incorporating Sloshing Effects”, AIChE Journal, 31

507-510.

[15] Fang, C., Ward, A., 1999. “Temperature measured close to the interface of an

evaporating liquid”, Phys. Rev. 59, 417–428.

[16] Fedorov, V. I., Luk’yanova, E. A., 2000, “Filling and storage of cryogenic

propellant components cooled below boiling point in rocket tanks at atmospheric

pressure”, Chem. Pet. Eng. 36, 584–587.

[17] Kozyrev, A. V., Sitnikov, A. G., 2001. “Evaporation of a spherical droplet in a

moderate pressure gas”, Phys. Usp. 44, 725–733.

[18] Scurlock, R., 2001. “Low loss Dewars and tanks: liquid evaporation mechanisms

and instabilities”, Coldfacts 7–18.

[19] Krahl, R., Adamo, M., 2004. “A model for two phase flow with evaporation”,

Weierstrass-Institut for Angewandte Analysis and Stochastic, Berlin 899.

[20] Dent, J. C., (1969-1970) “Effect of Vibration on condensation heat transfer to a

Horizontal Tube”, Proceedings Institution of Mechanical Engineers, 184.

[21] Rose, J. W., 1981. “Dropwise Condensation Theory”, International Journal of Heat

and Mass Transfer, 24.

[22] Tolubinskiy, V. I., Kichigin, A. M., Povsten, S. G., and Moskalenko, A. A., 1980.

“Investigation of Forced-Convection Boiling by the Acoustical Method”, Heat

Transfer- Soviet Research, 12.

[23] Antonenko, V. A., Chistyakov, Y. G. and Kudritskiy, G. R., 1992. “Vibration-Aided

boiling heat transfer”, Heat Transfer- Soviet Research, 24.

[24] Nishiyama, K., Murata, A., and Kajikawa, T., 1986. “The Study of Condensation

Heat Transfer Performance Due to Periodic Motion of a Vertical Tube”. Heat

Transfer: Japanese Research, 15.

12

[25] Bird, R. B., Stewart, W. E., Lightfoot, E. N., 2007. “Transport Phenomena”, 2nd

Edition, John Wiley and Sons, Inc.

.

13

Chapter 2 On the characterization of interfacial mass transfer rate of stored liquids

Abstract

A thermodynamic analysis of the two-phase physics involving a liquid-vapor combination

has been studied under the regime of conjugate heat and mass transfer phenomena. An

experiment has been designed and performed to estimate the interfacial mass transfer

characteristics of a liquid-vapor system by varying the liquid temperature. The experimental

set-up consists of an instrumented tank partially filled with water and maintained at

different temperatures. The evaporation of liquid from the interface and the gaseous

condensation has been quantified by calculating the interfacial mass transfer rate for both

covered and uncovered tanks. The dependence of interfacial mass transfer rate on the

liquid-vapor interfacial temperature, fractional concentration of the evaporating liquid, the

surface area of the liquid vapor interface and the fill level of the liquid has been established

through the present experimental study. An estimation of the overall mass transfer rate

from the interface due to a concentration gradient shows an analogy with the multiphase

heat transfer that takes place across the interface due to temperature gradient. It was seen

that at low fill levels and with a temperature difference of about 30 K between liquid and

ullage, the mass transfer rate of a closed system was nearly doubled when compared to its

open system counterpart.

Keywords: concentration gradient, evaporation, condensation, conjugate heat and mass transfer, filling level

14

2.1 Introduction

Most engineering operations related to process industries encounter the transfer of heat,

mass and momentum between different phases of a working fluid. It is, in fact, almost

impossible to think of any dynamic process in which one or more of these phenomena are

not involved. The transportation of liquid fuels such as Liquefied Natural Gas (LNG) from

their source to destination is one area where the liquid continuously interacts with its vapor

phase through interfacial heat, mass and momentum transfers. Most importantly safe

transportation of LNG calls for extreme care and knowledge of the interfacial liquid-vapor

systems. The diverse aspects of LNG include handling, storage and transportation, which

can be related to loading, offloading and sloshing, whose characteristics are not yet

completely understood.

Natural gas is thermodynamically processed to attain its liquid state by cooling to a

temperature of −163 °C. In order to maintain LNG at such a low temperature, proper

thermally insulated storage is required. The principle objective of any thermal insulation

system is to maintain the operating condition at some constant temperature over a long

time period. To maintain this thermal equilibrium of LNG, the ullage pressure above the

liquid containment and the fill level of the LNG also plays a critical role.

Let us assume that there exists equilibrium in terms of temperature and species for any two

diffusive participants of a heat and mass transfer phenomenon. Then multiphase

characteristics of LNG may be well understood by examining any miscible fluid

combination e.g. water-vapor and water system at its equilibrium. It is worth mentioning

here that similar to all other non-reactive liquid-gas combinations, the liquid phase of

natural gas (NG) always maintains a dynamic equilibrium with its gaseous phase under all

storage conditions [1]. The two-phase interaction of the liquid with ullage vapor in LNG

containers mainly depends on the interfacial physics of the evaporation and the

condensation of the fluid in the container. The present work is an attempt to estimate the

overall mass transfer from a water vapor - water interface taking into consideration the

dynamic equilibrium of the air-water system in which the two participating phases undergo

a phase change phenomenon.

A review of literature shows that the heat transfer and thermodynamic studies of LNG have

mainly been done in the area of LNG tank design. The main considerations in these

studies were insulation design of the tank to maintain it at sub-zero temperatures.

15

Kobayashi and Tsutsumi [2] studied the hybrid thermal insulation system of supporting

skirt of MOSS type LNG carrier by inserting a non-metalic material of low thermal

conductivity in between the aluminum alloy and mild steel. Bates and Morrison [3] studied

the characteristics of liquid natural gas by considering two principal phases through which

the liquid passes when roll over phenomenon takes place. Studies by Chen et al. [4],

Boukeffa et al. [5] and Khemis et al. [6] attempted to estimate the heat transfer coefficient

of cryogenic containers under various thermal conditions. In the periphery of two-phase

studies of fluids, the mechanism of evaporation of a liquid immersed in another liquid was

first studied by Gradon and Selecki [7]. Raina and Wanchoo [8] in their studies on direct

contact heat transfer with phase change provided expressions for the instantaneous velocity

of a two-phase bubble. Later, Raina and Grover [9] introduced sloshing effect into the

direct contact heat transfer modeling of a multiphase system. This theoretical model was

later validated through experiments by Raina and Grover [10].

Fang and Ward [11], Fedorov and Luk’yanova [12], Kozyrev and Sitnikov [13], Scurlock

[14], Krahl and Adamo [15] studied interfacial multiphase heat transfer of cryogenic fluids.

These studies were based on the methods of predicting the interfacial temperature and

vapor pressure with various factors affecting their measurement. Besides the above

mentioned studies, some studies were made to quantify the stratified region at the interface

which mainly governs the heat and mass transfer phenomenon. These studies conclude

that although the thickness of this layer is of the order of a few nanometers, the presence of

this non-condensable gas layer makes the temperature of the interfacial liquid undergoing

the process of evaporation significantly higher than the bulk liquid.

The literature review relating to the present work shows that published work involving the

phase change mechanism of a liquid to its gaseous phase during its storage is very scarce. In

most of these studies, the regime of evaporation and condensation of the interface was

investigated without consideration of the effects of temperature variation and fill levels at

which the liquid was being stored. The study described in this paper is aimed to fill this

gap. A consolidated experimental study has been conducted to understand the interface

physics of evaporation and condensation of stored liquids keeping in view the fill level

effects. These experiments were conducted on a 1:150 scaled down model of an actual

LNG tank at the hydrodynamics laboratory of the University of Western Australia [A2].

The experiment makes use of a range of measuring instruments to characterize the entire

physics of the interface mass transfer.

16

2.2 Theoretical Background

The analytical prediction of mass transfer is based on the principle of conservation of mass,

momentum and energy. A multi-component system with variations in concentration and

temperature continuously transfers mass across their interface. In this system of two

phases, the concentration of water molecules is higher just above the liquid surface

compared to that in the main portion of the air stream, thereby leading to diffusion. Thus

the concentration of liquid water molecules and its vapor mainly governs the mass transfer

which takes place across the interface. The assumptions that govern the diffusion of water

vapor through air for the cases considered in the present work are the following:

1 The system is under quasi steady-state conditions.

2 Both fluids (air and water vapor) behave as perfect gases.

3 Air that flows across the free surface has negligible solubility in water.

Figure 2. 1 Diffusion of water vapor through air between two planes x1 and x2 As water evaporates and diffuses, the mass diffusion can be expressed by Fick’s law of

diffusion. Under steady state conditions, the upward water evaporation into air is balanced

by downward diffusion of air so that concentration at any location from the water surface

remains constant [16]. Figure 2.1 explains the mass diffusion of water vapor taking place

through air between planes x1 and x2. The diffusion rates of the two species, air and water

vapor, participating in this interface phenomenon can be expressed as:

Water vapor and air 2 Plane x2

x Plane x1 Fill level

Top end of tank

1

Water

17

⎟⎠⎞

⎜⎝⎛−==

RTaM

apdxdD

Aam

aN (2.1)

⎟⎠⎞

⎜⎝⎛−==

RTwM

wpdxdD

Awm

wN (2.2)

Rearrangement of equation (2.1) gives the mass transfer rate of air as-

dxadp

RTaAMDam −= (2.3)

Symbols are defined in the nomenclature in section 2.8. Since there can be no net mass

movement of air downward at the surface of water, the bulk mass movement upward with a

velocity is large enough to compensate for the mass diffusion of air downward. The bulk

mass transfer of air is equal to

AVRT

aMapAVaρam −=−= (2.4)

By combining Equations (2.3) and (2.4), the bulk mass velocity of air can be expressed as:

dxadp

apDV = (2.5)

The mass transfer of water from the liquid phase to the vapor phase is due to upward mass

diffusion of water into the gaseous phase. Thus the overall interfacial mass transfer of

water vapor is

VART

wMwpdx

wdpwM

RTADtotal)w(m +−= (2.6)

18

Substituting the value of bulk mass velocity V from Equation 2.5 in Equation 2.6 following

relation is obtained:

Adx

adp

apD

RTwMwp

dxwdp

wMRTADtotal)w(m +−= (2.7)

From Dalton’s law of partial pressures,

wpaptp += (2.8)

Since the ullage pressure of the system remains constant, we have

dxwdp

dxadp

−= (2.9)

Equation (2.9) when substituted in Equation (2.7) gives the following equation-

⎥⎥⎦

⎤

⎢⎢⎣

⎡+−=

apwp

1dx

wdpwM

RTDA

total)w(m (2.10)

Equation (2.10) when integrated between the planes x1 and x2 can be rewritten as following-

(2.11)

The derivation of Equation (2.11) has been explained in detail in Appendix 1. The two

planes x1 and x2 through which the phenomenon of mass transfer takes place will have

different pressures which can be calculated thermodynamically. The partial pressure of

water vapor near the surface is assumed to be the saturated pressure corresponding to a wet

bulb temperature. The wet bulb temperature is the same as the dry bulb temperature for

saturated water at the interface. The partial pressure of water vapor through the region of

mass transfer is determined by using the relation between partial pressure and relative

humidity as stated by the psychrometric principle [1].

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−

−

−=

1wptp2wptp

elog)1x2(x

tpwMRTDA

total)w(m

19

Figure 2. 2 Schematic set-up of the test rig

(1.Thermocouples, 2. Conductivity probe at various test fill levels and top, 3. Humidity sensing probe, 4. Pressure transducer for measuring ullage pressure, 5.Cartridge heaters.)

2.3 Experimental setup

The experiments were carried out in a thermodynamically controlled system, which is

schematically shown in Figure 2.2. The set-up consisted of a Plexiglass tank of dimensions

260 mm × 180 mm × 50 mm. The bottom plate was an aluminum block embedded with

five 250 W cylindrical cartridge heaters each of 0.9 cm diameter. A neoprene rubber

insulation layer below the heater ensured that all heat was transferred upwards to the liquid.

The temperature was controlled by a variable thermostat with an operating temperature of -

10 ºC to 55 ºC and operating humidity of 25% to 85%. The accuracy of this controller was

± 0.5% of the indicated value. The accuracy with which the temperature controller

maintained the preset temperature was ± 0.2%. The instrumentation schematic used for

the present experiment is also shown in Figure 2.2. One side of the tank was instrumented

with nine equally spaced K-Type thermocouples having a measurement accuracy of ± 0.2%.

The ullage pressure was measured using a 2 mm diameter Kulite XCL8M 100 series

pressure transducer (Figure 2.2) with a measurement accuracy of ± 1%. The pressure

transducer was powered with 5 volts DC, with a gain of 100 for signal amplification. The

relative humidity of the ullage was measured using an auto-timer data logging humidity

meter. The range of this humidity meter was 10% to 95%, and its resolution was 0.1% of

the full scale reading with a response time less than 60 seconds. The sampling rate of the

humidity meter was set at 2 Hz. In order to obtain the relative humidity variation along the

tank height from the free surface, a set of moisture sensing probes were used in

combination with the relative humidity meter. These moisture sensing probes work on the

principle of DC capacitance using Astable Multivibrator circuitry [17]. In order to obtain

1

Temperature controller

2

34

Data acquisition

system

5

Plexiglass tank

20

the overall mass transfer characteristics of the system these probes were placed at 30%,

60%, 80% and 100% filling level (roof top) as shown in Figure 2.3. These moisture sensing

probes thus give the overall concentration profile of the system with a decaying value of

moisture from the free surface to the top of the tank. The layout of the test rig along with

various measuring devices is shown in Figure 2.4.

Figure 2. 3 Moisture sensing device used to measure the relative humidity along tank height

Figure 2. 4 Layout of the test rig

1. Constant temperature bath (container), 2. K-Type thermocouples series, 3. Conductivity Probe, 4. Pressure Transducer, 5. Humidity meter, 6. Data Acquisition System, 7. Temperature controller, 8.

Heater assembly, 9. Humidity meter Astable Multivibrator The moisture sensing probes are considered analogous to a heat dissipating fin referred to

as a ‘standard situation’ of extended surface heat transfer. It is worth mentioning here that

Top end of tank

30%

100%

60%

80%

21

in contrast to augmentation of heat transfer, a system can also be designed to quantify the

mass transfer enhancement. The present experiment uses these moisture sensing probes to

quantify the relative humidity variation across the tank height. These probes sense the

variable concentration of moist air in a conduit involving heat and mass transfer. The

principle used for modeling the concentration variation across the height is analogous to

the one-dimensional fin modeling [18] and is given by-

(2.12)

Since Ac is constant, the second term of Equation (2.12) is 0. The moisture sensing probes

are modeled as the nodal points of a concentration measuring device. At a particular state

of thermodynamic equilibrium these probes report the moisture content of the system at

their respective positions. It is worth mentioning that the equation explaining the diffusive

action of humidity probe is a steady state equation without any perturbations. This

equation states that at a particular instant of time (t = t1) the diffusive mass transfer through

probe at the tip is equal to the convective mass transfer in the ullage.

Equation (2.12) is used to calculate the concentration of the water vapor at the nodal

points. Further simplification of equation 2.12 may occur through transformation of

variable ‘c’ given by-

c(x) = C(x)-C∞ (2.13)

For a constant C∞ and Ac Equation (2.12) becomes-

(2.14)

where

The solution of the linear, homogenous, second-order differential equation is of the form

mxe2Cmxe1Cc −+= (2.15)

The constants C1 and C2 are obtained by applying appropriate boundary conditions. The

convective mass transfer from the top end (x = l) is assumed to be equal to the diffusive

mass transfer, which leads to the following boundary condition:

( ) 0CCdx

sdA

Dmh

cA1

dxdT

dxcdA

cA1

2dx

C2d=∞−−+

0c2m2dxc2d

=−

cDAPmhm =

22

(2.16)

At the base

c = c0 (2.17)

Applying these boundary conditions to Equation (2.14) gives the following constants.

(2.18)

(2.19)

Substituting the values of C1 and C2 in Equation (2.14) gives

(2.20)

This is elegantly written as-

(2.21)

( )mlemleDm

mhmlemle

x)m(lex)m(leDm

mhx)m(lex)m(le

0cc

−−+−+

⎟⎠⎞⎜

⎝⎛ −−−−+−−+−

=

)C(Cmhlxx

CD ∞−==

⎟⎠⎞

⎜⎝⎛∂∂

−

( ) ( )mlemleDm

mhmlemle

mleDm

mh10c

2C−−+−+

⎟⎠⎞

⎜⎝⎛ +

=

( ) ( )mlemleDm

mhmlemle

mleDm

mh10c

1C−−+−+

−⎟⎠⎞

⎜⎝⎛ −

=

( ) ( )( ) ( )mlsinhmhmlDmcosh

x)m(lsinhmhx)m(lDmcosh

0cc

+

−+−=

23

Figure 2. 5 Schematic of the experimental setup with position of moisture sensing probe

Equation (2.21) gives the value of the relative humidity with respect to its base value across

the conduit height. This experimental comparison of the data against the analytical results

obtained for the system is discussed in the results and discussion. Details of the position of

the moisture sensing device are explained in Figure 2.5.

Table 2. 1 Range of Parameters

At the commencement of the experiments, the ullage was kept at atmospheric pressure.

However as the temperature of the stored fluid was increased, the ullage pressure also

changed. The temperature of the liquid was raised from 19 ºC to 50 ºC in steps of 5 ˚C. At

each temperature, the experiments were carried out for a range of filling levels as shown in

Table 2.1. This was followed by the characterization of the heat and mass transfer across

the interface.

Parameters Range Fluid Water-Air system

Temperature 19-60 º C Fill level of total height 10%-50%

Tank condition Stationary

C∞, h mass transfer

Mass diffusion

Moisture sensing probes with uniform cross section

Cb

24

2.4 Uncertainty Analysis

The interfacial mass transfer rate is a function of interfacial temperature, relative humidity

and ullage pressure. These variables in turn are dependent on other thermodynamic

parameters like saturation pressure and saturation temperature. Hence the accuracy of

measurements of these parameters has an impact on the overall uncertainty with which

interfacial mass transfer is estimated. The uncertainty in the interfacial mass transfer was

computed by perturbing the values of the independent variables between their possible

extremes as dictated by the respective uncertainties in their measurement [19]. The

uncertainty in estimation of the mass transfer as obtained for the extremities was within ±

3%. The uncertainties in measurements and calculated parameters are shown in Table 2.2.

Table 2. 2 Uncertainty in the physical quantities of present experimental study

Parameters Range

Temperature ± 0.2 % Pressure ± 1%

Relative Humidity ± 3% Interfacial Mass Transfer rate ± 3%

2.5 Validation

The validation of the present experimental setup was performed for different regimes of

boiling through which the system has passed. The set-up was validated for various boiling

regimes through an asymptotic mode of validation [20]. When a liquid is heated in a

controlled environment, the first phase is free convection boiling within the liquid followed

by nucleation, resulting in vapor bubbles that form at certain favorable spots. To validate

the present system, water was heated over its saturation temperature in steps. The heat flux

obtained from the present system was compared with the classical correlation of Rohsenow

and Jakob for nucleate boiling [22] as given in the following equations-

( ) 3

1.7

0.5

fPrfghsfC

∆tpfc

σgρfρg

fghfµAQ

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡⎥⎦

⎤⎢⎣

⎡ −= (2.22)

( ) degW/m∆T5.58AQ1.54h 23

0.75=⎟

⎠⎞

⎜⎝⎛= (2.23)

The surface fluid constant to be used between the polished aluminum surface and water in

the correlation by Rohsenow was obtained from Pioro [21]. The thermophysical properties

25

of the fluid were obtained from Liley [22]. The heat transfer coefficients obtained from the

present experimental setup were found to be comparable with those obtained using

correlations available in the literature as detailed in Chapter 3.

2.6 Results and Discussion

Figure 2.6 shows the variation of relative humidity across the tank height at 10% filling level

for different temperatures. The thermophysical property of diffusive mass transfer

coefficients as obtained from the diffusion coefficient chart [16] were utilized in the

calculation. In this graph, series of experiments performed to calculate the moisture

content of the tank was also compared with the analytical solution of the system. The

relative humidity as recorded by the sensors when compared with Equation (2.20) shows

good agreement. The relative humidity error bars are within 1 standard deviation range of

± 5% of the total deviation. This variation can be explained due to the stratified zone

formation in the mid region of the tank creating a significant convection vapor movement

from the side wall along with a non-linear vapor gradient.

Figure 2.7 shows the transient temperature history for the bulk, interface and ullage region

of the tank at 20% filling level. The liquid and the interphase after attaining a constant

temperature show some undulations with a periodicity of about 0.002 cycles per second.

These undulations are due to thermostat functioning within temperature limit of ± 0.5 °C.

The bold vertical line depicts the time at which the power was switched ‘OFF’ thus allowing

the water to come to equilibrium with the ambient temperature. The slopes of the

undulations before and after the vertical lines are similar. This decay refers to the bulk

capacitance heat loss of the water at temperature above ambient [18]. The variation was

consistent for all the temperature levels tested and even beyond 5 hours.

There was uniform thermal decay for the liquid and the ullage. It was found that even after

a period of 8.5 hours there was a temperature difference of 1°C between bulk and the ullage

for each temperature level. Thus, it can be inferred that even if the bulk and the ullage were

allowed to be in equilibrium over a long period of time, a temperature difference can exist

and lead to heat and mass transfer from and to the bulk liquid throughout the storage time.

It was also seen that water, which undergoes the highest degree of heating, cools more

rapidly as compared to air.

26

15161718192021222324252627282930313233343536373839404142

0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5 2.75 3 3.25 3.5 3.75 4 4.25 4.5 4.75 5Time (Hours)

Tem

pera

ture

(C)

Bulk

Interface

50% fill level ullage

80% fill level ullage

90

91

92

93

94

95

96

97

98

99

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

% of total height of tank

Rel

ativ

e H

umid

ity %

Analytical 10% filling level 30 CExperimental 10% filling level 30 CAnalytical 10% filling level 25 CExperimental 10% filling level 25 C

Analytical 10% filling level 20 CExperimental 10% filling level 20 C

Figure 2. 6 Experimental and analytical relative humidity variation across tank height

Figure 2. 7 Transient temperature history for 20% filling level

27

0

0.000125

0.00025

0.000375

0.0005

0.000625

0.00075

0.000875

0.001

0.001125

0.00125

0.001375

290 292.5 295 297.5 300 302.5 305 307.5 310 312.5 315 317.5 320

Temperature K

Mas

s tra

nsfe

r kg

/hr

20 percent fill level open container20 percent fill level closed container50 percent fill level open container50 percent fill level closed container10 percent fill level open container10 percent fill level closed container30 percent fill level open container30 percent fill level closed container

Figure 2. 8 Interface mass transfer variation with temperature

2.6.1 Influence of tank fill levels on mass transfer

The mass transfer at the interface was determined using the expression described by

Equation 2.11. This mass transfer was estimated when the system reaches equilibrium and

the bulk temperature of water reaches a constant value. The mass diffusivity of air through

water was calculated using the following empirical correlation suggested by Martínez [23]-

constantTDp

2.72 = (2.24)

The datum coefficients were obtained from the experimental results mentioned by

Rohsenow and Hartnett [24]. Figure 2.8 shows the mass transfer taking place across the

interface at various temperatures for the fill levels under consideration, with and without

the top cover of the tank. It can be seen in Figure 2.8 that at lower temperatures, the mass

transfer across the interface was the same for the open and closed container experiments

undertaken. In contrast at a high temperature the mass transfer increased by about twice its

initial value.

28

Figure 2. 9 Model of the heat and mass transfer in the tank with characteristics zone In terms of the two-phase physics, this increase of mass transfer in the closed container is

due to the Stefan phenomenon taking place at the core of the ullage region, with convection

playing a vital role near the walls. Stefan phenomenon is a very common interfacial mass

transfer phenomenon. It is the driver of natural circulation of fluids due to difference in

their densities. It actually deals with movement of chemical species by their production or

removal at the interface. Unlike diffusion it only occurs at the interface where addition of

species takes place and results in an altercation of the mean flow. Therefore the total

transport phenomenon of interfacial mass transfer involves Stefan phenomenon along with

diffusion. The density of the air-vapor mixture in the immediate vicinity of the wall surface

and further away from it determines the convection pattern in the system. Density driven

convective current in the present experiment is significant because of the difference in

densities of water and air-vapor mixture (nearly four times). As temperature of the tank

walls rise, the thermal boundary layer being nearest to the heated wall heat up faster, as

compared to the central tank axis. This creates a density difference between the two

regions. The buoyancy acting on the heated air-vapor volume of the thermal boundary

layer lifts it upward through a distance of the order of thermal boundary layer thickness.

This bulk movement of the air-vapor mixture through a distance stops, when the density of

the lifted mixture becomes equal to that of the mixture away from the wall.

In another situation the movement of the mixture continues till the density becomes equal.

On the basis of this vapor-air mixture movement in the heated tank Malyshev and Zlobin

[25] has described three zones of mass transfer (Figure 2.9) as follows-

1. Zone I immediately above the interface, having negligible buoyancy currents and

characterized by appreciable vapor concentration.

x

I

III

II

29

0

0.000125

0.00025

0.000375

0.0005

0.000625

0.00075

0.000875

0.001

0.001125

0.00125

0.001375

1000 1500 2000 2500 3000 3500Water vapor saturation pressure (Pa)

Mas

s tr

ansf

er (k

g/hr

)

20 percent fill level open container

20 percent fill level closed container

50 percent fill level open container

50 percent fill level closed container

10 percent fill level open container

10 percent fill level closed container

30 percent fill level open container

30 percent fill level closed container

2. Zone II above Zone I mainly governed by convection at the wall with very small

vapor concentration gradient along the tank height.

3. Zone III above Zone II extending to the top wall characterized by minimum vapor

concentration and governed by conductive heat and mass transfer phenomenon.

The height of the central Zone II as measured using scale in the present experiments varied

from 8.5 cm to 5 cm above the stratified layer (Zone I) for 10 % filling level and 50 %

filling level respectively. Zone II is sandwiched between the stratified layer (Zone I)

influenced by diffusive mass transfer (density and temperature difference) and Zone III,

having entirely different physics. It is significant to mention that this zone mass transfer

takes place only because of the density gradient with little condensation effect. The width

of this zone depends on the concentration gradient, the heating rate, and the ratio of the

molecular weights of water vapor and air.

Figure 2. 10 Mass transfer variations with vapor pressure of ullage These zonal regimes can also be justified by an increase in the saturation pressure of the

water vapor in the ullage region for a closed container in comparison to an open container.

Figure 2.10 gives the variation of the interface mass transfer with the saturation pressure of

water vapor in the ullage.

30

2.7 Conclusions

An experimental study of a tank partially filled with water has been carried out with

different combinations of filling levels and water temperatures. The thermodynamic

variables of the filling fluids were mapped over a wide range of parameters. The following

conclusions make the present study unique and contribute new knowledge in the area of

heat and mass transfer of stored liquids-

1. The interface mass transfer is primarily governed by the concentration gradient

across the liquid and gaseous phases, irrespective of the container being open or

closed.

2. The central zone of mass transfer prevails for a region with maximum wall

convection and minimum concentration gradient for a height of 8.5 cm and 5 cm

for a 10 % filling level and a 50 % filling level respectively.

3. The interface mass transfer in a closed container was almost 50 % more than its

open counterpart.

4. The validity of the use of humidity sensors was established by comparing their

performance with the derived analytical solution based on heat and mass transfer

analogy.

2.8 Nomenclature

A base area of the tank, mPP

2

ABBBc cross section area, mPP

2

ABBBs surface area, mPPP

2P

c concentration difference, kg-mol/mP

3P

C molar concentration, kg-mol/mP

3P

CBBBp specific heat of water, J/kg/K

CBsf surface fluid constant

D diffusion coefficient mPPP

2P/s

Gr Grashof number, 2µ

2ρ3Tlg ∆β

g acceleration due to gravity, m/sP

2P

h convective heat transfer coefficient, W/mP

2PK

hBBBm convective mass transfer coefficient, m/s

hBfg latent heat of vaporization J/kg

l characteristic length of tank, m

31

M molecular weight of species, kg

m mass flow rate, kg/s (also fin parameter, eq 13)

N diffusion rate, kg-mol/s-mPPP

2P

P Perimeter, m

Pr Prandtl number of fluid, α

ν

p partial pressure, N/mPPP

2

R universal gas constant, J KP

−1PmolP

−1P

R’ characteristic gas constant of air, J kgP

-1PKP

−1PmolP

−1P

T temperature, K

∆T excess temperature over saturation, K

V bulk mass velocity upward, m/s

x co-ordinate perpendicular to free surface of the liquid

Greek Symbols

α thermal diffusivity, mP

2P/s

β isobaric cubic expansivity of the fluid, pT

ρ

ρ

1⎟⎠⎞

⎜⎝⎛∂

∂− , KP

-1

µ dynamic viscosity of water, kg/ms

ν kinematic viscosity, mP

2P/s

ρ Bmass density of water, kg/m3

σ surface tension, N/m

∞ ambient

Subscripts

a air

d dry

s saturation

t total

v vapor

w water

0 base (x=0)

2.9 Acknowledgement

The study was funded by an Australian Research Council Discovery Grant-DP0881447.

32

2.10 References

[1] Bird, R. B., Stewart, W. E., Lightfoot, E. N., 2007. “Transport Phenomena”, 2nd

Edition, John Wiley and Sons, Inc.

[2] Kobayashi, Y., and Tsutsumi, T., 2000. “Heat Transfer Characteristic of the

Spherical Tank Skirt of LNG and LH2 Having a Hybrid Construction”,

Transactions of the West Japan Society of Naval Architects. 99, 395-402.

[3] Bates, S., and Morrison, D. S., 1997, “Modelling the behavior of stratified liquid

natural gas in storage tanks: a study of the rollover phenomenon”, Int. J. Heat Mass

Transfer 40(8), 1875-1884.

[4] Chen, Q. S., Wegrzyn, J., Prasad, V., 2004. “Analysis of temperature and pressure

changes in liquefied natural gas (LNG) cryogenic tanks”, Cryogenics. 44, 701–709.

[5] Boukeffa, D., Boumaza, M., Francois, M. X., and Pellerin, S., 2001. “Experimental

and Numerical Analysis of heat losses in a liquid Nitrogen Cryostat”, Applied

Thermal Engineering. 21, 967-975.

[6] Khemis, O. M., Boumaza, M. Ali, A., Francois, M. X., 2003. “Experimental analysis

of heat transfers in a cryogenic tank without lateral insulation”, Applied Thermal

Engineering 23, 2107–2117.

[7] Gradon, L., and Selecki, A., 1977, “Evaporation of a liquid drop immersed in

another immiscible liquid. The Case of σc < σd”, International Journal of Heat

and Mass Transfer 20, 459-466

[8] Raina, G. K., and R. K., Wanchoo, 1984. “Direct Contact Heat Transfer with Phase

Change: Theoretical Expression for Instantaneous Velocity of a Two-Phase

Bubble”, International Communication in Heat and Mass Transfer 11, 227-237.

[9] Raina, G. K., and Grover, P. D., 1985. “Direct Contact Heat Transfer with Phase

Change: Theoretical Model Incorporating Sloshing Effects”, AIChE Journal, 31

507-510.

[10] Raina, G. K., and Grover, P. D., 1988. “Direct Contact Heat Transfer with Change

of Phase: Experimental Technique”, AIChE Journal 34, 1376-1380.

[11] Fang, C. A., Ward, 1999. “Temperature measured close to the interface of an

evaporating liquid”, Phys. Rev. 59, 417–428.

[12] Fedorov, V. I., Luk’yanova, E., A., 2000. “Filling and storage of cryogenic

propellant components cooled below boiling point in rocket tanks at atmospheric

pressure”, Chem. Pet. Eng. 36, 584–587.

[13] Kozyrev, A. V., Sitnikov, A. G., 2001. “Evaporation of a spherical droplet in a

moderate pressure gas”, Phys. Usp. 44, 725–733.

33

[14] Scurlock, R., 2001. “Low loss Dewars and tanks: liquid evaporation mechanisms

and instabilities”, Coldfacts 7–18.

[15] Krahl, R., and Adamo, M., 2004. “A model for two phase flow with evaporation”,

Weierstrass-Institut for Angewandte Analysis and Stochastic, Berlin 899.

[16] Lienhard, John, H. IV John, H., Lienhard, V, 2005. “Heat Transfer Text Book”.

Cambridge Press.

[17] Parameswaram, M., Baltes, H. P., Brett, M. J., Fraser, D. E., and Robinson, A. M.,

1988. “A Capacitive Humidity Sensor based on CMOS Technology with adsorbing

film”, Sensors and Actuators, 15, 325-335

[18] Incropera, F. P., Dewitt, D. P., Bergmen, T. L., Lavine, A. S., 2005. “Introduction

to Heat Transfer”, 5th ed, John Willey and Sons.

[19] Coleman, H. W., and Steele, W. G., 1989. “Experimentation and uncertainty

analysis for engineers”, 2nd edition John Willey and Sons.

[20] Balaji, C., Hölling, M., Herwig H., 2007. “Entropy generation minimization in

turbulent mixed convection flows”, International Communications in Heat and

Mass Transfer 34, p. 544–552.

[21] Pioro, L., 1999. “Experimental evaluation of constants for the Rohsenow pool

boiling correlation”, International Journal of Heat and Mass Transfer 31, p. 2003-

2013

[22] Liley, P. E., 1984. Steam Tables in SI Units, private communication. School of

Mechanical Engineering, Purdue University, West Lafayette, IN

[23] Martínez, H., and Isidoro, H., Unpublished, Mass diffusivity data by Polytechnic

University of Madrid See also URL http://webserver.dmt.upm.es/~isidoro/

[24] Rohsenow, W. M., Hartnett, J. P., Cho, Y. I., 1988. “Handbook of Heat Transfer”.

3rd ed, McGraw-Hill New York.

[25] Malyshev, V. V., and Zlobin, E. P., 1972. “Evaporation of liquid hydrocarbons in

heated closed containers”, Translated from Inzheneruo-Fizicheskii Zhurnal, Vol.

23, No. 4, p. 701-708.

34

Chapter 3 Estimation of entropy generation by heat transfer from a liquid in an enclosure to ambient

Abstract

This study outlines the estimation of entropy generation of a stored liquid-vapor

combination. A salient feature of the present study is the incorporation of a separated flow

model for the calculation of entropy generated by a diabatic two-phase system. Equations

of state corresponding to different thermodynamic equilibria are used for these calculations.

Two distinct expressions have been proposed for the determination of the total entropy

generated by the diabatic saturated two-phase system that comprises of a liquid and an

ullage region. The dissipative energy losses in various liquid-ullage systems can be

quantified through the derived mathematical relations obtained for the overall entropy

generated by the system. These expressions form the basis of an experimental

determination of the entropy generation rate. A parametric study on temperature

influencing entropy generation has been performed in the context of different thermal

transport mechanisms. The influence of fill levels on entropy generation was also analyzed

by considering a low and an intermediate filling ratio. It is concluded that the prevailing

central zone of thermodynamic equilibrium with maximum wall convection and minimum

concentration gradient influences the overall entropy generated by the system.

Keywords: two-phase flow, Entropy generation, Separated flow model

35

3.1 Introduction

Mass transfer augmentation techniques provide a simple and economical method of altering

the overall thermal performance of any two-phase heat exchanging device. In recent years

mass transfer enhancement concepts have advanced significantly. From complex boiling

phenomena to storage of highly thermodynamically unstable liquids, studies of mass

transfer mechanism of a stored liquid in conjunction with its thermodynamic features have

become inevitable. The basic objective of any mass transfer augmentation technique is to

enhance the thermal performance of the system by increasing the surface mass transfer

coefficient relative to the mass transfer coefficient that characterizes the base mass transfer.

The impediment to attaining this objective is associated with the thermodynamic

irreversibility of the system. Furthermore, even when the highest possible mass transfer

from the system is achieved, the thermodynamic efficiency of the system may remain less

than optimal. The main factor which deals with thermodynamic irreversibility of a system

is the entropy generated by a system when augmentation of mass transfer takes place.

Though the foundations of knowledge of entropy production goes back to Clasius’ and

Kelvin’s studies on the irreversible aspects of the second law of thermodynamics, the

approach to incorporate its viability in the study of heat and mass transfer is relatively new.

The two prime factors which make the entropy analysis of a system significant are (i) heat

transfer across a finite temperature gradient and (ii) the viscous effects of the fluid flow. A

disadvantage of the entropy generated by the system is that there is a reduction in the

available energy of the system.

Bejan [1] proposed to deal with the heat transfer and entropy generated by a system

unilaterally, by considering fundamental convective heat transfer situation. The four basic

flow situations which were analyzed under this study were: pipe flow, boundary layer over

plate, single cylinder in cross flow and flow in the entrance region of a rectangular duct.

Later, Bejan [2] proposed an optimum Reynolds number for pipe flow leading to the

minimum entropy generation rate under fixed heat duty and flow rate. Sarangi and

Chowdhury [3] analyzed entropy generation in a counter flow heat exchanger and derived

expressions in terms of non-dimensional parameters. The study derived an expression for

entropy production of an ideal heat exchanger with a balanced capacity rate that matched

closely with exact calculations. Nag and Mukherjee [4] studied the effect of various duty

parameters on the thermodynamic behavior of a convecting duct. From this analysis, it

became clear that generation of entropy destroys the available work of a system and

therefore it made good engineering sense to focus on the irreversibility of heat. Bejan [5]

36

extensively reviewed the entropy generation minimization technique of various flow

geometries. He observed that design improvements are possible if imperfections in flow are

spread through out the system. Zimparov [6] estimated the entropy generated from

enhanced surfaces taking into consideration the irreversibility associated with heat transfer

and fluid friction characteristics. This general evaluation criterion added new information

to Bejan’s entropy generation minimization technique. Recently Abbassi [7] performed a

second law analysis of a uniformly heated micro channel heat sink using Darcy’s equation

for porous medium flow. This is an innovative approach where second law optimization

was utilized in a modern fluid transport setting.

These studies mainly highlight the estimation of entropy generation in a single-phase flow

with no attention to the local entropy generation of two-phase flow. A theory of two-phase

flow entropy generation was attempted by Collado [8] through the development of an

expression for entropy based boiling curve for subcooled flow boiling. This work was

based on the classical 1-D entropy formulation for a steady boiling flow. Although this was

a novel attempt to incorporate two-phase flow in an entropy minimization study, it did not

consider the entropy generated by pressure drop. Besides, the specific contributions by

heat transfer, pressure drop and phase change to entropy generation have not been

explicitly expressed. In contrast to this Vargas and Bejan [9] estimated the irreversibility

generated by two streams of fluids passing through a heat exchanger giving an optimal ratio

between the stream mass flow rates.

So it can be seen that in continuation to the previous study of interfacial mass transfer study

it is legitimate to explore the consequences of increased mass transfer coefficient on the

second law efficiency of the system. Also from the literature presented here, it is evident

that most studies on thermodynamic optimization are based on entropy minimization of

single-phase flow systems. No significant contribution has been made for the local entropy

generation of diabatic saturated two-phase flow. Besides, for a liquid stored in a container,

the concept of entropy generation needs to be studied in conjunction with its two-phase

ullage region. The present study is an attempt in this direction, in which a partially filled

rectangular container subjected to external natural convection is studied to estimate the

overall entropy generated with both an open lid and a closed lid system.

3.2 Mathematical Formulation

The system consists of a rectangular storage tank partially filled with water, subject to

natural convection of air in and around the system. The overall estimation of entropy is

37

made by aggregating the entropy generated by the diabatic saturated two-phase liquid

(water) and the entropy generated by the two-phase ullage region at different temperatures

for an open top and closed top system. Figure 3.1 shows the schematic of the liquid storage

system explaining the physical phenomenon of two-phase diabatic saturated liquid along

with two-phase ullage.

Figure 3. 1 Schematic of two-phase liquid and ullage with ambient conditions

3.2.1 Mathematical formulation for entropy calculation of two-phase ullage region

The energy and mass conservation of a diabatic saturated two-phase flow can be dealt with

in a simple but realistic manner using a separated flow model approach which assumes that

both liquid and vapor phases travel at the same velocity. From classical thermodynamics,

the vapor quality (x), when represented in terms of mass flow rate ( ) along with two-

phase enthalpy (htp) and entropy (stp) generated by the system, can be described as follows-

(3.1)

(3.2)

(3.3)

where subscripts v and l refer to vapor and liquid phases respectively.

Equations (3.1-3.3) form the basis of the calculation of entropy generated by the two-phase

ullage region.

20% and 50%

Fill levels

h∞, T∞h∞, T∞

Water Diabatic

saturated two phase liquid

Air

Interphase

Qflux

lmvmvmx&&

&

+=

lx)h(1vxhtph −+=

lx)s(1vxstps −+=

m&

38

3.2.2 Mathematical formulation for entropy calculation of diabatic saturated two phase liquid

The estimation of entropy generation by the diabatic saturated flow is achieved by applying

the second law of thermodynamics to a control volume of length dz as shown in Figure 3.2.

Figure 3. 2 Schematic of diabatic saturated two-phase liquid in stored container

(3.4)

Where .

dS' is the entropy generation rate per unit length that corresponds to the heat rate

applied to the control volume and Tw is the wall temperature which is assumed to be the

temperature of the system boundary. Steam is a pure substance undergoing an infinitesimal

reversible process of pressure change at constant volume (vtp). Invoking TdS equation [10]

for steam-

(3.5)

Since the entropy of steam (two-phase) is an extensive property, its rate of change can be

expressed as the sum of the rates of change of entropy of the vapor and liquid phases that

constitute the steam-

(3.6)

Substituting the values of equations (3.5) and (3.6) in equation (3.4) we obtain-

(3.7)

Liquid Vapor

Heat flux

dz

wTQδ)lslmvsvmd(dz'sd&

&&& −+=

dptpvtpdssatTtpdh +=

)lslmvsvmd(tpdstm &&& +=

wTQδdp

satTtpvtm

tpdhsatT

tmdz'sd

&&&& −−=

39

Assuming the air water system to be at constant pressure at a particular temperature with

negligible kinetic energy and gravity terms (≈0) the first law of thermodynamics, when

applied to liquid and ullage region, gives-

(3.8)

Substituting equation (3.8) in (3.7) we have

(3.9)

This expression can be used to estimate the overall entropy generated by a system subjected

to any thermodynamic change. As our study deals with the estimation of diabatic saturated

flow entropy subject to phase change, the wall temperature of the container can be

expressed in terms of liquid saturation temperature (∆T=Tw-Tsat). This, when incorporated

in equation 3.9, gives the following relation-

(3.10)

The ratio can be neglected as the degree of superheat of the walls will be much

less compared to the saturation temperature of the liquid (373 K for water at one

atmosphere pressure).

The heat input along with the convective heat losses taking place, can be used to estimate

the overall increase in the heat content of the system. Assuming a constant perimeter P of

the tank with respect to z (as shown in figure 3.2), the following relation holds for the heat

content of the system:

(3.11)

Substituting equation 3.11 into equation 3.10 and neglecting the ratio of ∆T/Tsat, the

entropy of the diabatic saturated flow can be expressed as –

(3.12)

lx)dh(1vxdhdxlvh)tpd(htm

Qδ−++==

&

&

dpsatT

tpvtm

wTQδ

satTQδdzsd

&&&& −−=′

⎟⎠⎞

⎜⎝⎛−+

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

+=

dzdp

satTtpvtm

)sat∆T/T(12satT

∆TdzQδ'sd

&&&

satT∆T

dzPQδThq&

=∆=

⎟⎠⎞

⎜⎝⎛−+=′

dzdp

satTtpvtm

2sathT

P2q S d

&&

40

The first and second terms of the Right Hand Side of equation 3.12 correspond to the

entropy generated due to heat transfer and due to the pressure drop respectively.

Here h is the wall heat transfer coefficient calculated using the vertical wall Nusselt number

(Nu = hl/k) correlation suggested by Rohsenow et al. [11].

(3.13)

where Gr is the Grashof number which quantifies the natural convection heat loss from the

system, and Pr is the Prandtl number of the fluid.

Since the pressure drop contribution of the generated entropy is negligible in our case with

stationary fluid in the tank, the total entropy generated by the system is the sum of the

entropy generated by the diabatic saturated flow and the entropy generated by the two-

phase ullage. This is obtained by combining equations (3.3) and (3.12)

(3.14)

3.3 Experimental setup

The experiments were carried out in a thermodynamically controlled system as shown in

figure 3.3. The system consisted of a Plexiglass tank of dimension 260 mm × 180 mm × 50

mm with heating arrangements at the bottom. The system was heated in a controlled

environment with a temperature controller. To measure the temperature of the fluid,

interphase and the ullage contained in the tank, nine equally spaced k-type thermocouples

having a measurement accuracy of ±0.2 % were used. To estimate the convective heat

losses through the walls, four equally spaced thermocouples were placed on the outer side

of each wall of the tank. These thermocouples monitored the elevated temperatures of the

wall with respect to the ambient temperature and in turn were used to estimate the heat

losses. The top plexiglass consisted of two slots having arrangements to measure the ullage

pressure and relative humidity of the ullage.

0.250.59(GrPr)Nu =

sat2hT

P2qlx)s(1vxss +−+=

41

Figure 3. 3 Schematic set-up of the test rig (1. Thermocouples, 2. Conductivity probe at various test fill levels and top, 3. Humidity sensing probe, 4. Pressure transducer for measuring ullage pressure, 5.Cartridge heaters)

Table 3. 1 Range of parameters

The details of the experimental setup are provided by Rakshit et al. [Appendix 1]. Entropy

estimation was carried out for a range of temperatures and two filling levels as shown in

Table 3.1.

3.4 Validation

The experiment was designed such that the heat and mass transfer phenomena of the two

phases could be studied by changing the temperature. The present experimental setup was

validated for various boiling regimes through an asymptotic mode of validation [12]. In this

experiment water was heated above room temperature in steps, first to reach its saturated

temperature and further beyond its saturation. The heat flux provided for heating was

utilized to calculate the heat transfer coefficients of various boiling regimes. This was done

using the classical correlation of Rohsenow and Jakob for nucleate boiling [13] as given by

the following equations-

(3.15)

S.No Parameters Range

1 Fluid Water-Air system

2 Water Temperature 20-45 ºC

3 Ambient Temperature 18-19 ºC

3 Fill level of total height 20%, 50%

1

Temperature controller

2

34

Data acquisition

system

5

Plexiglass tank

( )3

1.7fPrfghsfC

∆tpfc0.5

σgρfρg

fghfµAQ

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎦

⎤⎢⎣

⎡ −=

42

(3.16)

All thermophysical properties of the fluids were calculated from steam tables of Liley [15].

The heat transfer coefficients as obtained from the present experimental setup are tabulated

in Table 3.2. The values of the heat transfer coefficients are comparable with those

obtained through the correlations present in the literature within the error limits of a boiling

experiment. The detailed validation procedure of the experimental setup is described by

Rakshit et al. [Appendix 1].

Table 3. 2 Comparison of the heat transfer coefficients

3.5 Results and Discussion

Figure 3.4 shows the transient temperature history for the bulk, interphase and ullage region

of tank at 20% filling level for both closed and open systems. Irregular time dependent

periodic spikes were observed for all the measured temperatures at all fill levels. After

reaching a constant level the temperature of the liquid showed some undulations with a

periodicity of about 7 cycles per hour. Unlike the ullage temperature this wavy pattern of

the liquid temperature was found to be repeatable for all 10 sets of experiments. The slopes

of the spikes were similar through the entire process of heating. The temperature rise refers

to the bulk capacitance heat gain of the heated water and the temperature drop corresponds

to the switch ‘OFF’ of the thermostat [13]. The variation was consistent for all the

temperature levels and even for the entire duration of the experiment. The difference in

temperature response of the liquid and the ullage can be explained in terms of difference of

their thermal inertias. This difference in the temperature levels of liquid and vapor phase

also forms the basis of the difference in entropy generation rate of the two phases. The

entropy of the system was calculated at each step of system equilibrium with the

environment.

S.No Temperature differential

hcalculated

(W/m2K) hliterature

(W/m2K)

1 10 9615.38 13236.22

2 11 8741.25 7426.98

3 12 8012.82 9642.24

4 14 6868.13 8409.02

( ) K2W/m3∆T 5.580.75

AQ1.54 h =⎟⎠⎞

⎜⎝⎛=

43

15

17.5

20

22.5

25

27.5

30

32.5

35

37.5

40

42.5

45

0 5000 10000 15000 20000

Time (s)

Tem

pera

ture

(Deg

ree

Cen

tigra

de)

bulk closed

bulk open

interphase closed

interphase open

70 percent ullage closed

70 percent ullage open

40 percent ullage closed

40 percent ullage open

Figure 3. 4 Transient temperature history for 20% filling level with step increase of temperature

The entropy generated by the system was determined using equation (3.14). The datum

thermophysical properties of water and steam as described above were taken from Liley

[14] with the intermediate temperature properties being calculated through the empirical

correlations obtained by regression analysis of the best fit data. Figures 3.5 and 3.6 show

the variation of heat transfer coefficient of the tank walls with temperature by considering

the heat loss across the front and side walls. These heat transfer coefficients were

calculated using the correlation suggested for vertical wall heat transfer coefficient by

equation (3.13). At lower temperature the heat transfer coefficient for both open and

closed system increased rapidly with an increase in temperature, but after attaining a

temperature of 30ºC the slope tended to change with the heat transfer coefficient following

an asymptotic trend.

Figure 3.7 shows the variation of total entropy generated per unit volume by the saturated

diabatic liquid and the two-phase ullage with temperature at two different fill levels with

and without the top cover. The entropy generated per unit volume at higher temperature is

appreciably higher than its lower fill level counterpart. This considerable hike in the

entropy generation occurs because of the entropy contributed by the ullage region. This

becomes clear from Figure 3.8 which shows the variation of the entropy generated per unit

volume of the ullage with temperature for closed systems. In terms of the two-phase

physics, this predominant increase of entropy generation in the ullage region can be

44

2

3

4

5

6

7

8

9

15 20 25 30 35 40 45 50

Temperature C

Hea

t tra

nsfe

r co

eefic

ient

w/m

2 K

Heat transfer coefficient 20 % fill

Heat transfer coefficient 50 % fill

Heat transfer coefficient 20% fill open

Heat transfer coefficient 50% fill open

2

3

4

5

6

7

8

9

15 20 25 30 35 40 45 50

Temperature C

Hea

t tra

nsfe

r co

eefic

ient

w/m

2 K

Heat transfer coefficient 20 % fill

Heat transfer coefficient 50 % fill

Heat transfer coefficient 20% fill open

Heat transfer coefficient 50% fill open

explained by the Stefan phenomenon that takes place at the core of the ullage region with

convective heat and mass transfer playing a vital role near the walls. The height of this zone

is about 7 cm and 5 cm above the stratified layer for 20% filling level and 50% filling level

respectively. This zone is coupled between the stratified nano layer zone and the top wall

zone where heat and mass transfer is predominantly by conduction. This proposition is in

good agreement with findings of Malyshev and Zlobin [15].

Figure 3. 5 Variation of heat transfer coefficient across the front wall with temperature

Figure 3. 6 Variation of heat transfer coefficient across the side wall with temperature

45

0

0.1

0.2

0.3

0.4

0.5

0.6

15 20 25 30 35 40 45 50

Temperature C

Ent

ropy

per

uni

t vol

ume

w/K

m3

Entropy 20 percent closed Entropy 20 percent open

Entropy 50 percent closed Entropy 50 percent open

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

15 20 25 30 35 40 45 50Temperature C

Ent

ropy

ulla

ge p

er u

nit v

olum

e w

/Km

3

entropy 20 percent ullage total per unit volume closed

entropy 50 percent ullage total per unit volume closed

Figure 3. 7 Variation of total entropy per unit volume with temperature

Figure 3. 8 Variation of entropy per unit volume in ullage with temperature

3.6 Conclusions

The present study considered the mathematical modeling and the experimental

determination of total entropy generated by a system consisting of a saturated diabatic two-

phase liquid along with two-phase ullage using the first principle of separated flow model.

The entropy is estimated at two different fill levels and varying fluid temperatures. The

46

heat transfer coefficient affecting the overall entropy generated by the system has also been

estimated. The unique findings of this study which will give further insight and new

knowledge to the field of Second Law analysis of stored liquids are-

1. Heat transfer coefficients of the panel losing heat to ambient are experimentally

determined by calculating the overall surface temperature.

2. Entropy generation of the ullage is greater than the saturated liquid.

3. Ullage entropy which is governed by a region of mist with high humidity (dryness

fraction) contributes to the maximum part of the overall entropy.

4. The central zone of mass transfer prevails for a region with maximum wall

convection and minimum concentration gradient for an approximate height of 7 cm

and 5 cm for a 20 % filling level and a 50% filling level respectively.

3.7 Nomenclature

A heat transfer surface area, m2

Csf surface fluid constant

Gr Grashof number, 2µ

2ρ3Tl∆gβ

g acceleration due to gravity, m/s2

h convective heat transfer coefficient, W/m2K

htp two-phase enthalpy, W

hfg latent heat of vaporization J/kg

l characteristic length of tank, m

k thermal conductivity of fluid, W/mK .

m mass flow rate kg/sec

p fluid pressure, Pa

Pr Prandtl number of fluid α

ν

P perimeter, m .

Qδ heat flux applied W

S entropy generated by system w/K

entropy generation rate per unit length W/Km

T temperature, K

∆T excess temperature over saturation

'.S

47

z co-ordinate parallel to free surface of the liquid

Greek Symbols

α thermal diffusivity, m2/s

β isobaric cubic expansivity of the fluid, K-1

µ dynamic viscosity of water, kg/ms

ν kinematic viscosity, m2/s

ρ mass density of water, kg/m3

σ surface tension, N/m

Subscripts

∞ ambient

g gas

f fluid

v vapor

l liquid

tp two-phase

v vapor

sat saturation

3.8 Acknowledgement

The study was funded by an Australian Research Council Discovery Grant-DP0881447

3.9 References

[1] Bejan, A., 1979. “A study of Entropy Generation in Fundamental convective heat

transfers”. ASME J of Heat Transfer, 101 718-725.

[2] Bejan, A., 1980. “Second Law Analysis in Heat Transfer”. Energy 5721-732.

[3] Sarangi, S., and Chowdhury, K., 1982. “On the generation of entropy in a counter

flow heat exchanger”. Cryogenics, 2263-65.

[4] Nag, P. K., and P., Mukherjee, 1987. “Thermodynamic optimization of convective

heat transfer through a duct with constant wall temperature”. Int J of Heat and

Mass Transfer, 30 401-405

pT

ρ

ρ

1⎟⎠⎞

⎜⎝⎛∂

∂−

48

[5] Bejan, A., 2001. “A thermodynamic optimization of geometry in engineering flow

systems”. Exergy Int. Journal 4 269-277.

[6] Zimparov, V. D., 2001. “Extended performance evaluation criteria for enhanced

heat transfer surfaces: heat transfer through ducts with constant heat flux”. Int J

Heat and Mass Transfer 44 169-180.

[7] Abbassi, H., 2007. “Entropy generation analysis in a uniformly heated micro

channel heat sink”. Energy 32, 1932-1947.

[8] Collado, F. J., 2005. “The law of stable equilibrium and the entropy-based boiling

curve for flow boiling”. Energy 30, 807-819.

[9] Vargas, J. V. C., Bejan, A., 2000. “Thermodynamic optimization of the match

between two streams with phase change”. Energy 25 15-33.

[10] P. K., Nag “Engineering Thermodynamics”, 1996. Tata McGraw-Hill Publishing

Company Limited

[11] Rohsenow, W. M., Hartnett, J. P., Cho, Y., I., 1988 “Hand book of Heat

Transfer” 3rd ed, McGraw-Hill New York,

[12] Balaji, C., Hölling, M., Herwig, H., 2007. “Entropy generation minimization in

turbulent mixed convection flows”. International Communications in Heat and

Mass Transfer, 34, 544–552

[13] Incropera, F. P., Dewitt, D. P., Bergmen, T. L., Lavine, A. S., 2005. “Introduction

to Heat Transfer”. 5th ed, John Willey and Sons.

[14] Liley, P. E., May, 1984. “Steam Tables in SI Units, private communication”. School

of Mechanical Engineering, Purdue University, West Lafayette, IN.

[15] Malyshev, V. V., and Zlobin, E. P., 1972. “Evaporation of liquid hydrocarbons in

heated closed containers”. Translated from Inzheneruo-Fizicheskii Zhurnal, 23, 4

701-708.

49

CHAPTER 4 Thermodynamic optimization of stored liquid-a holistic approach

Abstract

A holistic approach was adopted to thermally optimize a storage container that was partially

filled with liquid (water). The ullage region above the liquid was composed of water vapor

and air. In this optimization two symbiotic physical parameters that influence the thermal

performance of the storage container, namely interfacial mass transfer and the entropy

generated by the system, were simultaneously considered. The optimization proposed a set

of temperatures for two different filling levels that would maximize the interfacial mass

transfer by keeping the entropy generated by the system at its minimum possible level. The

mass transfer estimation involved the determination of (i) liquid-vapor interfacial

temperature, (ii) fractional concentration of the evaporating liquid present in the gaseous

state, and (iii) surface area of the liquid-vapor interface. The entropy of the system was

estimated separately by considering the entropy of the diabatic saturated liquid and the

ullage vapor. This was followed by the composition of a synergistically considered

objective function based on the penalty involved in deviation from the individual optima.

Thus, a holistic optimum was determined. It was seen that stored liquids had a better

optimum for open containers as compared to its closed counterpart when maximization of

mass transfer is sought for in combination with minimum entropy generation. The main

influencing factor that contributes to this type of optimum is the roof condensation that

occurred for closed containers at 50% filling level.

Keywords: Second Law analysis, Holistic Optimization, Two-phase flow, Entropy generation, conjugate

heat and mass transfer, filling level

50

4.1 Introduction

Professional engineering activities often involves design of physical systems that accomplish

specific tasks. Design and optimization are essential elements of all practical engineering

applications. The optimization of thermal systems has become important with their prolific

usage in industries related to energy conversion, aerospace, chemical processing and

metallurgical operations. The functioning of various thermal systems can be explained by

using the principles of heat and mass transfer. Heat and mass transfer augmentation

techniques provide a simple and economical method of improving the overall thermal

performance of any heat/mass exchanging device. In recent years mass transfer

enhancement concepts have advanced significantly. From complex boiling phenomena to

storage of highly thermodynamically unstable liquids, the studies of mass transfer

mechanisms in conjunction with their thermodynamic features have become critical.

The basic objective of any mass transfer augmentation technique is to enhance the mass

transfer of a system by increasing the surface mass transfer coefficient relative to the

coefficient that characterizes the base mass transfer. The impediment to attaining this

objective is associated with the thermodynamic irreversibility of the system. The main

factor which deals with thermodynamic irreversibility of a system is the entropy generated

by a system when an augmentation of mass transfer takes place. Even when the highest

possible mass transfer from the system is achieved, the thermodynamic efficiency of the

system may remain less than optimal because of the increased entropy generation.

Although the foundations of knowledge of entropy production go back to Clasius’ and

Kelvin’s studies on the irreversible aspects of the Second Law of thermodynamics, its

application in heat and mass transfer study is relatively new. The two prime factors which

make the entropy analysis of a system significant are (i) heat transfer across a finite

temperature gradient and (ii) the viscous effects of the fluid flow. Bejan [1] proposed

dealing with the heat transfer and entropy generated by a system unilaterally, by considering

fundamental convective heat transfer situation. The four basic flow situations which were

analyzed under this study were: pipe flow, boundary layer over plate, single cylinder in cross

flow and flow in the entrance region of the rectangular duct. Later, Bejan [2] proposed an

optimum Reynolds number for pipe flow leading to the minimum entropy generation rate

51

under fixed heat duty and flow rate. Entropy generation in a counter flow heat exchanger

was analyzed by Sarangi and Chowdhury [3]. They derived an expression for entropy

generation in terms of non-dimensional parameters. The effect of various duty parameters

on the thermodynamic behavior of a convecting duct was studied by Nag and Mukherjee

[4]. Bejan [5] extensively reviewed the entropy generation minimization technique of

various flow geometries. He observed that design improvements are possible if

imperfections in flow are spread through out the system. The entropy generated from

enhanced surfaces taking into consideration the irreversibility associated with heat transfer

and fluid friction characteristics was estimated by Zimparov [6]. This general evaluation

criterion added new information to Bejan’s entropy generation minimization technique. A

second law analysis was recently performed by Abbassi [7] for a uniformly heated micro

channel heat sink using Darcy’s equation for porous medium flow. This is an innovative

approach where second law optimization was utilized in a modern fluid transport setting.

These studies highlight the estimation of entropy generation in a single-phase flow. In

these studies of local entropy generation, two-phase fluids were not the focus. A theory of

two-phase flow entropy generation was proposed by Collado [8] through the development

of an expression for entropy based boiling curve for subcooled flow boiling. This work was

based on the classical 1-D entropy formulation for a steady boiling flow. Although this was

a novel attempt to incorporate two-phase flow in an entropy minimization study, it did not

consider the entropy generated due to pressure drop. However, the specific contributions

due to heat transfer, pressure drop and phase change to the entropy generation were not

explicitly expressed. In contrast to this, Vargas and Bejan [9] estimated the irreversibility

generated by two streams of fluids passing through a heat exchanger giving an optimal ratio

between the stream mass flow rates. From these studies, it became clear that entropy

generation associated with various heat transfer processes cannot always be neglected. It

therefore makes good engineering sense to focus on the irreversibility of the system during

designing.

The above studies were based on the effect of thermodynamic irreversibility of the system

without giving consideration to mass transfer analysis. Indeed, there are many studies that

analyze the mass transfer behavior of the system. In the periphery of two-phase studies of

fluids, the mechanism of evaporation of a liquid immersed in another liquid was first

studied by Gradon and Selecki [10]. Raina and Wanchoo [11] in their studies on direct

contact heat transfer with phase change provided expressions for the instantaneous velocity

52

of a two-phase bubble. Later, Raina and Grover [12] introduced sloshing effect into the

direct contact heat transfer modeling of a multiphase system. This theoretical model was

later validated through experiments by Raina and Grover [13].

Fang and Ward [14], Fedorov and Luk’yanova [15], Kozyrev and Sitnikov [16], Scurlock

[17], Krahl and Adamo [18] studied interfacial multiphase heat transfer of cryogenic fluids.

These studies were based on the methods of predicting the interfacial temperature and

vapor pressure with various factors affecting their measurement. Besides the above

mentioned studies, some studies were made to quantify the stratified region at the interface

which mainly governs the heat and mass transfer phenomena. These studies conclude that

although the thickness of this layer is of the order of a few nanometers, the presence of this

non-condensable gas layer makes the temperature of the interfacial liquid undergoing the

process of evaporation significantly higher than the bulk liquid.

The literature review related to the present work shows that published work involving

interfacial mass transfer in conjunction with the entropy generated by the system is not

available. It is clear that designing of a thermal system by considering the mass transfer as

well as the entropy generated by the system, could make a significant improvement to the

Second Law efficiency of the system. The purpose of this study was to attempt to deal with

both of these competing criteria simultaneously and to search for a holistic optimum. In

this study the aim is to amalgamate the concept of entropy generated by the system with the

mass transfer in its most original form. This is achieved by determining the entropy and the

interfacial mass transfer of a storage container partially filled with liquid (water) losing heat

to the environment.

4.2 Theoretical Background and Mathematical Formulation

4.2.1 Interfacial mass transfer rate calculation

The analytical prediction of mass transfer was based on the principle of conservation of

mass, momentum and energy. A multi-component system with variations in concentration

and temperature continuously transferred mass across its interface. A holistic approach was

adopted to thermally optimize a storage container that was partially filled with liquid

(water). The ullage region above the liquid was composed of water vapor and air. In this

system of two phases, the concentration of water molecules was higher just above the liquid

surface compared to that in the main portion of the air, thereby leading to diffusion. The

53

concentrations of liquid water molecules and water vapor mainly governed the mass

transfer across the interface.

The mass transfer of water from the liquid phase to the vapor phase was due to upward

mass diffusion of water into the gaseous phase. Thus the overall interfacial mass transfer of

water vapor is given by-

⎥⎥⎦

⎤

⎢⎢⎣

⎡+−=

apwp

1dx

wdpwM

RTDA

total)w(m (4.1)

Equation (4.1) when integrated between the top of water surface (plane x1) and top end of

the tank (plane x2) as shown in Figure 4.1, can be rewritten as following-

(4.2)

Figure 4. 1 Diffusion of water vapor through air between two planes x1 and x2 The derivation of Equation (4.2) has been explained in detail in Chapter 2. The two planes

x1 and x2 through which the phenomenon of mass transfer took place had different

pressures which could be calculated thermodynamically. The partial pressure of water

vapor near the surface was assumed to be saturated pressure corresponding to the wet bulb

temperature. The wet bulb temperature was same as the dry bulb temperature for saturated

water at the interface. The partial pressure of water vapor through the region of mass

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−

−

−=

1wptp2wptp

elog)1x2(x

tpwMRTDA

total)w(m

Water vapor and air 2 Plane x2

x Plane x1 Fill level

Top end of tank

1

Water

54

transfer was determined by using the relation between partial pressure and relative humidity

as stated by the psychrometric principle [19].

4.2.2 Entropy estimation of stored liquid

In this study the overall estimation of entropy was made by aggregating the entropy

generated by the diabatic saturated two-phase liquid (water) and the entropy generated by

the two- phase ullage region at different temperatures for an open top and closed top

system. Figure 4.2 shows the schematic of the storage system explaining the physical

phenomenon of two-phase diabatic saturated liquid along with two-phase ullage.

Figure 4. 2 Schematic of two-phase liquid and ullage with ambient conditions

4.2.2.1 Mathematical formulation for entropy calculation of two-phase ullage region

The energy and mass conservation of a diabatic saturated two-phase flow can be dealt with

in a simple but realistic manner using a separated flow model approach which assumes that

both liquid and vapor phases travel at the same velocity. From classical thermodynamics,

the vapor quality (x), when represented in terms of mass flow rate ( ) along with two-

phase enthalpy (htp) and entropy (stp) generated by the system can be described by

Equations (4.3-4.5). These equations form the basis of the calculation of entropy generated

by the two-phase ullage region as follows-

(4.3)

(4.4)

m&

lmvmvm

x&&

&

+=

lx)h(1vxhtph −+=

20% and 50%

Fill levels

h∞, T∞h∞, T∞

Water Diabatic saturated two phase liquid

Air

Interphase

Qflux

55

(4.5)

4.2.2.2 Mathematical formulation for entropy calculation of diabatic saturated two phase liquid

The estimation of entropy generation by the diabatic saturated flow is achieved by applying

the second law of thermodynamics to a control volume of length dz as shown in Figure 4.3.

Following relation shows the entropy generated by the system-

(4.6)

Sd ′& is the entropy generation rate per unit length, .

Qδ corresponds to the heat rate

applied to the control volume and Tw is the wall temperature which is assumed to be the

temperature of the system boundary. Steam is a pure substance undergoing an infinitesimal

reversible process of pressure change at constant volume (vtp). Invoking TdS equation [20]

for steam –

(4.7)

Since the entropy of steam (two-phase) is an extensive property, its rate of change can be

expressed as the sum of the rates of change of entropy of the vapor and liquid phases that

constitute the steam-

(4.8)

Figure 4. 3 Schematic of diabatic saturated two-phase liquid in stored container

lx)s(1vxstps −+=

dptpvtpdssatTtpdh +=

)lslmvsvmd(tpdstotalm &&& +=

Liquid Vapor

Heat flux

dz

wT

.δQ)lslmvsvmd(dzSd −+=′ &&&

56

Substituting the values of equations (4.7) and (4.8) in equation (4.6) we obtain-

(4.9)

Assuming the air water system to be at constant ullage pressure at a particular temperature

with negligible kinetic energy and gravity terms, the first law of thermodynamics, when

applied to liquid and ullage regions, gives-

(4.10)

Substituting equation (4.10) in (4.9) we have

(4.11)

This expression can be used to estimate the overall entropy generated by a system subjected

to any thermodynamic change. As our study deals with the estimation of diabatic saturated

flow entropy subject to phase change, the wall temperature of the container can be

expressed in terms of liquid saturation temperature (∆T=Tw-Tsat). This when incorporated

in equation 4.9 gives the following relation-

(4.12)

The ratio can be neglected as the degree of superheat of the walls will be much

less compared to the saturation temperature of the liquid (373 K) for water at one

atmosphere pressure.

The heat input along with the convective heat losses taking place, can be used to estimate

the overall increase in the heat content of the system. Assuming a constant perimeter P of

the tank with respect to z (as shown in figure 4.3), the following relation holds for the heat

content of the system:

wT

.Qδdp

satTtpvtotalm

tpdhsatTtotalm

dzSd −−=′&&

&

lx)dh(1vxdhdxlvh)tpd(htotalmQδ

−++==&

&

dpsatT

tpvtotalm

wT

.Qδ

satT

.QδdzSd

&& −−=′

⎟⎠⎞

⎜⎝⎛−+

⎥⎥⎦

⎤

⎢⎢⎣

⎡

+=′

dzdp

satTtpvtotalm

)sat∆T/T(1sat2T∆T

dzQδSd

&&&

satT∆T

57

(4.13)

Substituting equation 4.13 in equation 4.12 neglecting the ratio of ∆T/Tsat, the entropy of

the diabatic saturated flow can be expressed as follows-

(4.14)

Here h is the wall heat transfer coefficient calculated using the vertical wall Nusselt number

(Nu = hl/k) correlation suggested by Rohsenow et al [11], namely

(4.15)

where Gr is the Grashof number which quantifies the natural convection heat loss from the

system, and Pr is the Prandtl number of the fluid.

Since the pressure drop contribution of the generated entropy is negligible in our case with

constant ullage pressure of stationary tank, the total entropy generated by the system is the

sum of the entropy generated by the diabatic saturated flow and the entropy generated by

the two-phase ullage. This is obtained by combining equations (4.5) and (4.14)

(4.16)

(4.17)

2sathT

P2qlx)s(1vxss +−+=

0.250.59(GrPr)Nu =

Total entropy generation of the system

Heat transfer contribution

2sathT

P2qullagestotals +=

dzPQδh∆q&

== T

Heat transfer contribution

Pressure drop contribution

⎟⎠⎞

⎜⎝⎛−+=′

dzdp

satTtpvtm

2sathT

P2q S d

&&

58

To make the study independent of the parameters, the above relation has been non-

dimensionalised as follows-

(4.18)

where φ is the ratio of total entropy to ullage entropy.

Substituting Equation 4.15 in Equation 4.18 the non dimensional entropy can be expressed

in terms of other non dimensional parameters as -

(4.19)

4.3 Experimental setup

Figure 4. 4 Schematic set-up of the test rig

(1. Thermocouples, 2. Conductivity probe at various test fill levels and top, 3. Humidity sensing probe, 4. Pressure transducer for measuring ullage pressure, 5.Cartridge heaters)

The experiments were carried out in a thermodynamically controlled system, which is

schematically shown in Figure 4.4. The system consisted of a rectangular tank partially

filled with water at various temperatures subjected to natural convection in and around the

system. The dimensions of Plexiglass tank are 260 mm × 180 mm × 50 mm. The bottom

plate was an aluminum block embedded with five 250 W cylindrical cartridge heaters each

of 0.9 cm diameter. A neoprene rubber insulation layer was located below the heater to

ensure that all heat was transferred upwards to the liquid. The temperature was controlled

by a variable thermostat with an operating temperature of -10 ºC to 55 ºC and operating

ullages2satT

P2h∆1 T+=ϕ

ullages2satl T

P2∆Tfk0.250.59(GrPr)1+=ϕ

1

Temperature controller

2

3 4

Data acquisition

system

5

Plexiglass tank

59

humidity of 25% to 85%. The accuracy of this controller was ± 0.5% of the indicated

value. The accuracy with which the temperature controller maintained the preset

temperature was ± 0.2%. The instrumentation schematic used for the present experiment

is also shown in Figure 4. One side of the tank was instrumented with nine equally spaced

K-type thermocouples having a measurement accuracy of ± 0.2%.

Figure 4. 5 Layout of the test rig (1. Constant temperature bath (container), 2. K-Type thermocouples series, 3. Conductivity

Probe, 4. Pressure Transducer, 5. Humidity meter, 6. Data Acquisition System, 7. Temperature controller, 8. Heater assembly, 9. Humidity meter Astable Multivibrator)

The ullage pressure was measured using a 2 mm diameter Kulite XCL8M 100 series

pressure transducer (Figure 4.4) with a measurement accuracy of ± 1%. The pressure

transducer was powered with 5 volts DC, with a gain of 100 for signal amplification. The

relative humidity of the ullage was measured using an auto-timer data logging humidity

meter. The range of this humidity meter was 10% to 95%, and its resolution was 0.1% of

the full scale reading with a response time less than 60 seconds. The sampling rate of the

humidity meter was set at 2 Hz. The layout of the test rig along with various measuring

devices is shown in Figure 4.5. The optimization of the stored air-water was carried out by

estimating mass transfer and entropy generated by the system for a range of temperatures

and two filling levels as shown in Table 4.1.

60

Table 4. 1 Range of Parameters

4.4 Validation

The validation of the present experimental setup was performed for different regimes of

boiling through which the system passed. The set-up was validated for various boiling

regimes through an asymptotic mode of validation [21]. When a liquid is heated in a

controlled environment, the first phase is free convection boiling within the liquid followed

by nucleation, resulting in vapor bubbles that form at certain favorable spots. To validate

the present system, water was heated over its saturation temperature in steps. The heat flux

obtained from the present system was compared with the classical correlation of Rohsenow

and Jakob for nucleate boiling [22] as given in the following equations-

(4.20)

(4.21)

The surface fluid constant, Csf between the polished aluminum surface and water used in

Equation 4.20 was obtained from Pioro [22]. The thermophysical properties of the fluid

were obtained from Liley [23]. The heat transfer coefficients obtained from the present

experimental setup are tabulated in Chapter 3. These heat transfer coefficients were found

to be comparable with those obtained using correlations available in the literature within the

error limit of two-phase experiments.

Parameters Range Fluid Water-Air system

Temperature 19-60 º C Fill level of total height 10%-50%

Tank condition Stationary

( ) 3

1.7

0.5

fPrfghsfC

∆tpfc

σgρfρg

fghfµAQ

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡⎥⎦

⎤⎢⎣

⎡ −=

( ) degW/m∆T5.58AQ1.54h 23

0.75=⎟

⎠⎞

⎜⎝⎛=

61

0

0.0001

0.0002

0.0003

0.0004

0.0005

0.0006

0.0007

0.0008

0.0009

0 5 10 15 20 25 30 35 40 45 50 55Filling level %

Mas

s Tra

nsfe

r kg

/hr

Liquid bulk temperature 19 CLiquid bulk temperature 25 CLiquid bulk temperature 30 CLiquid bulk temperature 40 CLiquid bulk temperature 45 C

0

0.0002

0.0004

0.0006

0.0008

0.001

0.0012

0.0014

0.0016

0 5 10 15 20 25 30 35 40 45 50 55

Filling level %

Mas

s Tra

nsfe

r kg

/hr

Liquid bulk temperature 19 C

Liquid bulk temperature 25 CLiquid bulk temperature 30 C

Liquid bulk temperature 40 CLiquid bulk temperature 45 C

4.5 Results and Discussion

Figure 4. 6 Mass transfer variation with filling levels for an open container

Figure 4. 7 Mass transfer variation with filling levels for a closed container

62

Using mass transfer and entropy generated by the system as factors influencing

thermodynamic optimization, the system has been investigated for a wide range of

temperature and filling levels. Figures 4.6 and 4.7 show the mass transfer variation across

the interface for various fill levels, at different temperatures, with open and closed tank top

respectively. It can be seen in Figure 4.6 that for open containers at 19 °C, 25 °C and 30 °C

the variation of interfacial mass transfer with filling levels are meager as compared to 40 °C

and 45 °C. This meager interfacial mass transfer is because of minimum temperature

difference between the liquid and the surroundings. For 40 °C and 45 °C the mass transfer

increases very abruptly after 30% filling levels. Similarly in Figure 4.7 it can be seen that for

a closed container at 19 °C, 25 °C and 30 °C the variation of interfacial mass transfer with

filling levels are also meager as compared to 40 °C and 45 °C. However after 30% fill level

the interfacial mass transfer reaches an asymptotic limit. In terms of the two-phase physics,

this increase of mass transfer for closed containers at higher temperature can be explained

by Stefan phenomenon taking place at the core of the ullage region, with convection playing

a vital role near the walls. The central zone (as explained in chapter 2) is sandwiched

between the stratified lower zone where heat and mass transfers are predominantly by

conduction and diffusion respectively.

Figure 4.8 shows the variation of total entropy generated per unit volume with temperature,

for open and closed containers, at 20% and 50% fill levels. The total entropy generated per

volume constitutes the entropy generated by the saturated diabatic liquid and the two-phase

ullage (Equation 4.17). It can be seen that the entropy generated per unit volume increases

with increase of temperature. For 20 °C, 25°C and 30 °C the difference in entropy

generated by 20% and 50% filling level for both open and closed containers is very less.

However for 40 °C and 45 °C this difference becomes more prominent. This prominence

in the entropy generation occurs because of the entropy contributed by the ullage region.

The increasing trend of entropy follows a polynomial trend in accordance to its dependence

on heat flux q as described by Equation 4.17. The effect of ullage entropy becomes clear

from Figure 4.9 which shows the variation of the entropy generated per unit volume of the

ullage with temperature for closed systems. This predominant increase of entropy

generation in the ullage region can also be explained by the Stefan phenomenon as

described above.

63

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

15 20 25 30 35 40 45 50

Temperature oC

Ent

ropy

ulla

ge p

er u

nit v

olum

e w

/Km3

entropy 20 percent ullage total per unit volume closed

entropy 50 percent ullage total per unit volume closed

0

0.1

0.2

0.3

0.4

0.5

0.6

15 20 25 30 35 40 45 50Temperature oC

Ent

ropy

per

uni

t vol

ume

w/K

m3

20% filling level - closed container

20% filling level - open container

50% filling level - closed container

50% filling level - open container

Figure 4. 8.Variation of total entropy per unit volume with temperature

Figure 4. 9 Variation of entropy per unit volume in ullage with temperature

64

0

15

30

45

60

75

90

105

120

290 292 294 296 298 300 302 304 306 308 310 312 314 316 318 320

Temperature K

Non

dim

ensi

onal

en

trop

y

20 percent fill close20 percent fill open50 percent fill close50 percent fill open

Figure 4. 10 Variation of non-dimensional entropy generated with temperature

4.5.1 Non Dimensional Entropy generation:

The non dimensional entropy generated by liquid and ullage (Equation 4.19) for 20% and

50% filling levels were correlated with temperature of the liquid in Figure 4.10. The

entropy generated is dependent on Grashoff Number (Gr), Prandtl Number (Pr) and

geometrical parameter (l/d). It is seen that the difference in the entropy generated by high

fill level and low fill level became prominent as the temperature increases. It clearly appears

that for fluid temperatures above 312 K the analysis of entropy generation in conjunction

with the interfacial mass transfer analysis is more relevant.

An important point regarding the present study is that, being dealt with a stationary liquid,

the entropy generation due to fluid friction is not having any contribution to the overall

entropy of the system. Also for low temperature fluids, entropy generation due to heat

transfer was not significant. This was also because of minimum temperature difference

between the liquid and the surroundings. But as the temperature of the liquid increased the

entropy generated became prominent and thus made remarkable contribution to the holistic

optimization of the system. The non dimensional entropy generated by the system also

shows a trend depicted by Equation 4.19, symmetric about the temperature axis reaching an

asymptotic trend after 312 K. This convexity about the temperature axis can be explained

65

by the prominent contribution of the ullage entropy to the overall entropy generated by the

system. Since the non dimensional entropy is the ratio of the total entropy to the ullage

entropy, so as the temperature increases the contribution of ullage entropy also increases

with the prominence contribution of the ratio.

4.6 Holistic optimization

On the basis of the interfacial mass transfer and entropy generated by the system, a search

for optimum temperature is performed in which the challenge is to look for mutual

compensations required for globally optimizing the system holistically. The aim is to

maximize mass transfer and minimize the entropy generated by the system. The system

under consideration is a multi-variable system, having Grashoff Number (Gr), Prandtl

Number (Pr) and geometrical parameter (l/d) as the independent variables. Experimental

data was used to correlate the entropy generated by the system with these independent

variables as given by Equation 4.19.

As seen from the previous explanations the interfacial mass transfer, (mw)total and the

entropy generated by the system S have increasing trends with the increase of temperature.

So considering the direct dependence of these two competing criteria (interfacial mass

transfer and entropy generation) on the independent variable (temperature), the

determination of the individual optima was straight forward. Hence, the optimum value of

(mw)total and S, which could now be termed as [(mw)total]max and Smin and could be determined

by inspection alone without having to employ any sophisticated optimization techniques.

The two competing criteria could be simultaneously considered by employing, a composite

penalty function φ as follows [24]:

(4.22)

where, 0≤α≤ 1

In order to obtain a holistic optimum, φ needed to be minimized, with constants a and b

having either same or different values. The case of a = b = 1 represented equal weightage

for mass transfer maximization and entropy generation minimization. We needed to

determine a value of α at which φ was a minimum and this represented the minimum total

penalty. From eqn.23 it can be seen that φ = f (Gr, Sc, l/d). Hence the optimization could

have proceeded in two ways:

b

SminS

α)(1

a

maxtotal)w(mtotal)w(m

α ⎟⎟⎠

⎞⎜⎜⎝

⎛−+⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛=φ

66

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50

Temperature oC

NM

, 1/

NS

NM for 20% fill closed1/NS for 20% fill closedNM for 20% fill open 1/NS for 20% openNM for 50% fill closed1/NS for 50% fill closedNM for 50% fill open1/NS for 50% fill open

1. Determining l, d for a given fluid temperature

2. Determining the optimum temperature when geometric parameters are held fixed

such that φ was always minimum

In this study we focused on (b). For this situation the only parameter that was allowed to

float was fluid temperature. Though the above looks simplistic, in reality if one tried to rate

a storage device, all geometric and thermal parameters except temperature of the stored

liquid are usually fixed. Hence the procedure adopted in this study for the holistic

optimization was analogous to the rating of a heat exchanger. To visualize the shifts in

optimum according to different weightage assigned to mass transfer and entropy generated

by the system the indices a and b were varied. The optimization of the system was aimed at

evolving at a condition that minimized the total penalty.

This was followed by the determination of optimum temperature by varying NM and 1/NS

[25] with temperature for different storage conditions inside the container. By giving equal

weightage to mass transfer and entropy generated by the system, the optima were achieved.

Figure 11 explains these optima for each fill level of open and closed tanks. It was noticed

that the optimizations for closed tanks were obtained for higher values of NM and 1/NS

when compared with its open counterparts. The vertical line passes through the optimized

mass transfer and entropy generated by the system within the range of parameters

considered for the present study. It was concluded that under the conditions studied, a

temperature of 301 K was optimum to store the liquid.

Figure 4. 11 Variation of NQ and 1/NS with temperature for open and closed tanks

67

Table 4. 2 Overall penalties in mass transfer and entropy generation for holistic optimization of tank with optimum temperature

Table 4.2 elucidates the overall sacrifice to be made in mass transfer enhancement for

storing the liquid at optimum temperatures to improvise the second law efficiency of the

stored fluid. These optima helped us to draw an inference that the effect of entropy

generated by the system should not be always neglected at the cost of maximum mass

transfer taking place from the interface. It is observed that for open containers at both the

fill levels the optimization is critical. It is also observed that the optima shifted towards left

as the fill level increased for both open and closed tanks. For open container the optimum

shifted from 306 K to 303 K and for closed container the optimum shifted from 305 K to

302 K. It is seen that stored liquids showed a better second law efficiency for open

container as compared to its closed counterpart. Roof condensation at high fill level is the

main influencing factor that contributes to this type of optimum.

4.7 Conclusions

An experimental study of a partially filled water tank has been carried out to optimize the

composite objective function. This optimization was attempted by considering two

symbiotic physical parameters viz interfacial mass transfer and the entropy generated by the

system. The optimization was performed to determine the optimum temperature for a

given geometrical condition. It was found that the interfacial mass transfer and the entropy

generated by the system was dependent on liquid-vapor interfacial temperature, fractional

concentration of the evaporating liquid present in the gaseous state, and the surface area of

the liquid vapor interphase. Following were the conclusions that make the present study

unique, and contribute new knowledge in the area of heat and mass transfer optimization of

stored liquids-

1. The interfacial mass transfer variation with filling levels for open and closed

containers at 19 °C, 25 °C and 30 °C are meager as compared to 40 °C and 45 °C

S. No Tank condition

Penalty in mass transfer % (approx)

Reduction in entropy

generation % (approx)

Optimum

temperature K

1 20% fill open 25 75 306 2 20% fill closed 30 70 305 3 50% fill open 22 78 303 4 50% fill closed 28 72 302

68

temperatures due to the minimum temperature differences between the liquids and

their surroundings.

2. For open containers at 40 °C and 45 °C the interfacial mass transfer increases very

abruptly beyond a 30% filling level. However for closed containers beyond a 30%

filling level the interfacial mass transfer reaches an asymptotic limit. This was due

to the Stefan phenomenon taking place at the core of the ullage region, with wall

convection playing a vital role.

3. The difference in entropy generated per unit volume at 20 °C, 25 °C and 30 °C for

20% and 50% filling levels for both open and closed containers is significantly less.

However for 40 °C and 45 °C this difference becomes more prominent. This

considerable hike in the entropy generation occurred because of the entropy

contributed by the ullage region.

4. For 20 °C, 25 °C and 30 °C fluids, entropy generation due to heat transfer was not

significant.

5. The non dimensional entropy generated by liquid and ullage was dependant on

Grashoff Number (Gr), Prandtl Number (Pr) and geometrical parameter (l/d).

6. For fluid temperatures above 312 K the analysis of entropy generation in

conjunction with the interfacial mass transfer analysis was more significant.

7. It was concluded that under the studied conditions there existed a set of optimum

temperatures that will keep the entropy generated by the system to its minimum

possible level by imposing a penalty on the interfacial mass transfer.

8. It was also observed that for both open and closed containers a 50% filling level

showed a better second law optimization compared to a 20% filling level.

9. On a qualitative basis when a holistic optimization is performed the reduction in

entropy generation for open containers is significant and must be accounted for.

4.8 Nomenclature

A base area of the tank, mPP2

Ac cross section area, m2

As surface area, m2

c concentration difference, kg-mol/m3

C molar concentration, kg-mol/m3

69

Cp specific heat of water, J/kg/K

Csf surface fluid constant

D diffusion coefficient m2/s

d fill level

Gr Grashof number, 2µ

2ρ3Tlg ∆β

g acceleration due to gravity, m/s2

h convective heat transfer coefficient, W/m2K

hm convective mass transfer coefficient, m/s

hfg latent heat of vaporization J/kg

k thermal conductivity of material, W m-1 K-1

l characteristic length of tank, m

M molecular weight of species, kg

m mass flow rate, kg/s

N diffusion rate, kg-mol/s-m2

NM ratio of actual mass transfer rate to maximum mass transfer rate

NS ratio of actual entropy generation rate to minimum total entropy generation

rate

P Perimeter, m

Pr Prandtl number of fluid, 'α

ν

p partial pressure, N/m2

R universal gas constant, J K−1mol−1

R’ characteristic gas constant of air, J kg-1K−1mol−1

T temperature, K

∆T excess temperature over saturation, K

V bulk mass velocity upward, m/s

70

x co-ordinate perpendicular to free surface of the liquid

Greek Symbols

α’ thermal diffusivity, m2/s

α weighing factor eqn.(22)

φ objective function

φ non dimensional entropy generated by the system, stotal/sullage

ρ mass density, kg/m3

σ surface tension, N/m

∞ ambient

Subscripts

f fluid

g gas

v vapor

l liquid

min refers to minimum

max refers to maximum

s saturation

t total

v vapor

w water

4.9 References

[1] Bejan, A., 1979. “A study of Entropy Generation in Fundamental convective heat

transfers”. ASME J of Heat Transfer 101, 718-725.

[2] Bejan A., 1980. “Second Law Analysis in Heat Transfer”. Energy, 5721-732.

71

[3] Sarangi, S., and Chowdhury, K., 1982. “On the generation of entropy in a counter

flow heat exchanger”. Cryogenics, 22, 63-65.

[4] Nag, P. K., and Mukherjee, P., 1987. “Thermodynamic optimization of convective

heat transfer through a duct with constant wall temperature”. Int J of Heat and

Mass Transfer, 30 401-405

[5] Bejan, A., 2001. “A thermodynamic optimization of geometry in engineering flow

systems”. Exergy Int. Journal 4 269-277.

[6] Zimparov, V. D., 2001. “Extended performance evaluation criteria for enhanced

heat transfer surfaces: heat transfer through ducts with constant heat flux”. Int J

Heat and Mass Transfer 44, 169-180.

[7] Abbassi, H., 2007. “Entropy generation analysis in a uniformly heated micro

channel heat sink”. Energy 32 1932-1947.

[8] Collado F. J., 2005. “The law of stable equilibrium and the entropy-based boiling

curve for flow boiling”. Energy 30, 807-819.

[9] Vargas, J. V. C., and Bejan, A., 2000. “Thermodynamic optimization of the match

between two streams with phase change”. Energy 25 15-33.

[10] Gradon, L., and Selecki, A., 1977. “Evaporation of a liquid drop immersed in

another immiscible liquid”. The Case of σc < σd, International Journal of Heat

and Mass Transfer 20, 459-466.

[11] Raina, G. K., and Wanchoo, R. K., 1984. “Direct Contact Heat Transfer with Phase

Change: Theoretical Expression for Instantaneous Velocity of a Two-Phase

Bubble”. International Communication in Heat and Mass Transfer 11, 227-237.

[12] Raina, G. K., and Grover, P. D., 1985. “Direct Contact Heat Transfer with Phase

Change: Theoretical Model Incorporating Sloshing Effects”. AIChE Journal, 31

507-510.

[13] Raina, G. K. and Grover, P. D., 1988. “Direct Contact Heat Transfer with Change

of Phase: Experimental Technique”. AIChE Journal 34 1376-1380.

[14] Fang, G., Ward, C. A., 1999. “Temperature measured close to the interface of an

evaporating liquid” Phys. Rev. 59 417–428.

[15] Fedorov, V. I., Luk’yanova, E. A., 2000. “Filling and storage of cryogenic

propellant components cooled below boiling point in rocket tanks at atmospheric

pressure”. Chem. Pet. Eng. 36, 584–587.

[16] Kozyrev, A. V., Sitnikov, A. G., 2001. “Evaporation of a spherical droplet in a

moderate pressure gas”. Phys. Usp. 44, 725–733.

[17] Scurlock, R., 2001 “Low loss Dewars and tanks: liquid evaporation mechanisms and

instabilities”, Coldfacts 7–18.

72

[18] Krahl, R., Adamo, M., 2004. “A model for two phase flow with evaporation”,

Weierstrass-Institut for Angewandte Analysis and Stochastic”, Berlin 899.

[19] Bird, R. B., Stewart, W. E., Lightfoot, E. N., 2007. “Transport Phenomena”, 2nd

Edition, John Wiley and Sons, Inc.

[20] Nag, P. K., 1996. “Engineering Thermodynamics”, Tata McGraw-Hill Publishing

Company Limited

[21] Balaji, C., Hölling, M., Herwig, H., 2007 “Entropy generation minimization in

turbulent mixed convection flows”. International Communications in Heat and

Mass Transfer 34, p. 544–552

[22] Pioro, L., 1999. “Experimental evaluation of constants for the Rohsenow pool

boiling correlation”. International Journal of Heat and Mass Transfer 31, p. 2003-

2013

[23] Liley, P. E., 1984. “Steam Tables in SI Units”, private communication. School of

Mechanical Engineering, Purdue University, West Lafayette, IN,

[24] Burmister, L. C., 1998. “Elements of Thermal- Fluid system design”, Chapter 4

Prentice Hall, New Jersey.

[25] Sasikumar, M. and Balaji, C. 2002. “A holistic Optimization of Convecting-

Radiating Fin Systems”. ASME Journal of Heat Transfer,vol.124, pp. 1110-1116

73

Chapter 5 Effect of mild sway on the interfacial mass transfer rate of stored liquids

Abstract

An exploratory study of two-phase physics was undertaken in a slow moving tank

containing liquid. This study is under the regime of conjugate heat and mass transfer

phenomena. An experiment was designed and performed to estimate the interfacial mass

transfer characteristics of a slowly moving tank. The tank was swayed at varying

frequencies and constant amplitude. The experiments were conducted for a range of liquid

temperatures and filling levels. The experimental set-up consisted of a tank partially filled

with water at different temperatures, being swayed using a 6 degrees of freedom motion

actuator. The experiments were conducted for a frequency range of 0.7-1.6 Hz with

constant amplitude of 0.025 m. The evaporation of liquid from the interface and the

gaseous condensation was quantified by calculating the instantaneous interfacial mass

transfer rate of the slow moving tank. The dependence of interfacial mass transfer rate on

the liquid-vapor interfacial temperature, the fractional concentration of the evaporating

liquid, the surface area of the liquid vapor interface and the filling level of the liquid was

established. As sway frequency, filling levels and liquid temperature increased, the interfacial

mass transfer rate increased. The interfacial mass transfer rate estimated for the swaying

tank compared with the interfacial mass transfer rate of stationary tank shows that vibration

increases mass transfer.

Keywords: concentration gradients, evaporation, condensation, conjugate heat and mass transfer, filling level,

motion actuator

74

5.1 Introduction

In recent years the study of two-phase flow has become increasingly important in a wide

variety of engineering applications for their optimum design and safe operation. Two-

phase flow often involves mass transfer in conjunction with heat transfer. It is, in fact,

almost impossible to think of any dynamic process in which these two phenomena are not

simultaneously involved. The analysis of these phenomena can be usually separated into

two parts – bulk transfer and interface transfer. Interface transfer corresponds to the

mechanisms of heat and mass transfer across phase boundaries. Until recently, the

interfacial heat and mass transfer have not received much attention. However, as interfacial

physics is now involved in many applications from aeronautics to the transportation of

liquid fuels, interfacial heat and mass transfer studies have become critical.

Interfacial heat and mass transfer studies become more complex when associated with

vibration. Vibrations that induce irregular movements of interfaces make the study of

interface transfer more challenging because of sloshing. Sloshing is the free surface

movement of a liquid inside a container. In liquid sloshing the types of motion to which

the free surface is subjected mainly depends on the type of disturbance and the container

shape. These motions include simple planar, non planar, rotational, irregular beating,

symmetric, asymmetric, quasi-periodic and chaotic movements [1]. Although the study of

liquid sloshing dates back to 19th century when Sir Horace Lamb [2] attempted to explain

the fluid dynamics involved, it has only recently been applied in various applications.

For Liquefied Natural Gas (LNG) containers, sloshing is a critical phenomenon for

transportation when tanks are partially filled. These partially filled sloshing tanks generally

result from the loading and unloading of LNG during multiport stops [3]. Marine

transports with different critical motions of ship make LNG sloshing more vulnerable

because of the sudden property change that may lead to inflammability and spontaneous

ignition of the fuel. The gyroscopic effects of ship generate resonance conditions under

different sea states making LNG transportation more vulnerable. Therefore the various

degrees of freedom with which the ship moves make the study of the sloshing tank

challenging, when the hydrodynamic characteristics of the tank are to be analyzed.

To date, understanding the effect of sloshing has been based solely on fluid mechanics.

Natural Gas (NG) is thermodynamically processed to attain its liquid state by cooling to a

75

temperature of −163 °C. In order to maintain LNG at such a low temperature, proper

thermally insulated storage is required. To maintain the thermal equilibrium of the LNG,

the ullage pressure above the liquid containment and the filling levels of the LNG are

critical. Therefore in order to understand the underlying principles and the physical

mechanisms of LNG sloshing, both fluid mechanics and thermodynamics must be studied.

As noted in Chapter 2, the multiphase characteristics of LNG may be well understood by

examining any miscible fluid combination like water-vapor and water system at its

equilibrium. Similar to all other non-reactive liquid-gas combinations, the liquid phase of

natural gas always maintains a dynamic equilibrium with its gaseous phase under all storage

conditions [4]. The two-phase interaction of the liquid with ullage vapor in partially filled

containers mainly depends on the interfacial physics of evaporation and condensation of

the fluids in the container irrespective of the type of fluid. So it can be concluded that the

thermodynamics of the phase change phenomenon makes a significant contribution to the

proper conditioning, transportation and storage of LNG. Hence the study of

thermodynamics becomes critical when studying the sloshing physics.

The mechanisms of the phase change phenomenon that occur when a liquid is immersed in

another liquid were first studied by Gradon and Selecki [5]. Their study identified the

relationship between the terminal radius (radius of the drop before explosion) and the

beginning radius (initial drop radius) for different temperatures of the continuous phase. In

studies of direct contact heat transfer incorporating phase change Raina and Wanchoo [6]

came up with expressions for the instantaneous velocity of a two-phase bubble. Later

Raina and Grover [7] introduced the sloshing effect into the direct contact heat transfer

modeling of a multiphase system. This theoretical model was later validated experimentally

by Raina and Grover [8]. Although this was a leading step in terms of two-phase heat

transfer studies involving sloshing and direct contact between molecules, the study was

limited to two immiscible fluids. The multiphase study of sloshing physics in the vicinity of

miscible fluid combination viz. water-vapor-water system was first addressed by Dent [9].

Dent analyzed the effect of vibration on a horizontal condenser tube vibrating in the plane

of a gravitational field. In this study it was shown that, within the range of parameters

studied there is an increase in condensation of up to a maximum of 15 percent for a

vibrating system in contrast to a non-vibrating system. Rose [10] then studied the heat

transfer mechanisms involved in a drop undergoing condensation. He later concluded that

76

the polyatomicity of water plays an important role in determining the condensation

coefficient [11].

Similar studies of forced convection boiling were undertaken by Tolubinskiy et al. [12].

These experiments were conducted in a heated conduit maintained at a pressure of 0.1 to

0.5 MPa and fluid velocities of 0.5 to 2.0 m/s. The liquid was subcooled in range of 5 to 80

K. The spectrum analyzer and oscillograph showed boiling noise carries information on

the boiling modes both in the noise spectra and in the total intensity. In a similar study

Antonenko et al. [13] identified various boiling modes and showed that heat transfer is

enhanced when a vibration is induced into the system. A detailed study of condensation

heat transfer performance involving various gyroscopic motions was undetaken by Katsuo

et al. [14]. In their study of the condensation of R11 vapor on a vertical smooth tube in

three modes of sinusoidal vibration the effects of motion on the condensation heat transfer

performance were evaluated. A similar type of analytical study was undertaken by

Kravchenko and Fedotkin [15] who analyzed the heat transfer characteristics of a rapidly

moving vapor subjected to film condensation was analyzed. The study was based on the

development of a mathematical model of a rapidly moving vapor in a tube with a significant

temperature and velocity gradient.

With regard to interfacial heat and mass transfer studies, Fang and Ward [16], Fedorov and

Luk’yanova [17], Kozyrev and Sitnikov [18], Scurlock [19], Krahl and Adamo [20] studied

interfacial multiphase heat transfer of cryogenic fluids. These studies addressed methods to

predict the interfacial temperatures and vapor pressures and identify the factors affecting

their measurement. Additional studies were undertaken to quantify the stratified region at

the interface which mainly governs the heat and mass transfer phenomenon. These studies

concluded that although the thickness of the non-condensable gas layer is of the order of a

few nanometers, the presence of this layer makes the temperature of the interfacial liquid

undergoing evaporation significantly higher than the bulk liquid.

Overall, there is little published work addressing the mechanism involved in the change of

phase of a liquid and a gas during storage. In particular, very few studies have addressed the

interfacial heat and mass transfer mechanisms which form an integral part of partially filled

sloshing containers. The study described in this Chapter aims to fill this gap by estimating

the overall mass transfer from a slowly moving water vapor - water interface, taking into

consideration the dynamic equilibrium of the water vapor - water system. The interfacial

77

mass transfer of water vapor – water system is explored keeping in view its analogous

behavior with non-reactive NG –LNG combination. Experiments were conducted using a

1:150 scaled down model of an actual LNG tank being swayed using a 6 degrees of freedom

motion actuator. The experiments make use of a range of measuring instruments to

completely characterize the interface mass transfer physics.

5.2 Theoretical Background

In the present study, the analytical prediction of mass transfer was based on the principle of

conservation of mass, momentum and energy. A multi-component system was constructed

with variations in concentration and temperature, continuously transferring mass across its

interface. The ullage region above the liquid was composed of water vapor and air. In this

two phase system, the concentration of water molecules was higher just above the liquid

surface compared to that in the main portion of the air, leading to diffusion of water

through air. The concentrations of liquid water molecules and water vapor mainly govern

the mass transfer across the interface. The mass transfer of water from the liquid phase to

the vapor phase is due to upward mass diffusion of water into the gaseous phase.

Following are the assumptions that govern the diffusion of water vapor through air:

At a particular instant of time-

1. The system is under steady state conditions (quasi-steady state).

2. The temperature of the liquid and water vapor varies along y-direction with

negligible change along x and z directions.

3. The vapor concentration varies along y-direction with negligible change along x and

z directions.

4. Both fluids (air and water vapor) behave as perfect gases.

5. Air that flows across the free surface has negligible solubility in water.

Figure 5.1 shows the mass diffusion of water vapor taking place through ullage between

planes y1 and y2 when the tank moves with an angular frequency of ω rad/sec. The tank

movement induces standing waves thereby generating slow moving interface.

78

Figure 5. 1 Diffusion of water vapor through air between two planes y1 and y2 From Thiagarajan et al (21) the maximum wave elevation is given by-

(5.1)

The maximum wave elevation ηmax is a function of x and t. So ηmax at x=0 is constant. For

first mode sway oscillation, n = 1. Therefore the wave elevation η can be written in terms

of ηmax as-

(5.2)

From chapter 2, 1-Dimensional interfacial mass flux φ in the ullage region is given by –

(5.3)

∑∞

=⎟⎠⎞

⎜⎝⎛

−⎟⎠⎞

⎜⎝⎛−

==1n

2n

2

2

xmax bcos

)ω(ωω

btanh

nπ)cos12(

aRAOt)(x,η xnhnn πππ

⎥⎦⎤

⎢⎣⎡=

bxcost cosmaxηt)η(x, πω

∫−

=⎥⎥⎦

⎤

⎢⎢⎣

⎡∫ ∫

+−

w2p

w1p wptpwdp

tpwMGTD2y

1y

η1y

1ydydyφ

O z

b x

y

w

Water vapor and air

Water

Plane 1

Plane 2

ηmax

Acosωt

79

(5.4)

Equation 5.4 when written in the form of interfacial mass transfer from the free surface

gives the following relation-

(5.5)

Where dAc is an elemental cross sectional area of the tank from which interfacial mass

transfer is taking place.

Total interfacial mass transfer is given by-

(5.6)

Substituting equation 5.2 in 5.6 we get-

(5.7)

Let = I1, y2-y1 = A and ηmaxcosωt = B Therefore

(5.8)

(5.9)

[ ]⎥⎥⎦

⎤

⎢⎢⎣

⎡

−

−=−−

w1ptpw2ptp

logtpwMGTDη1y2yφ

⎥⎥⎦

⎤

⎢⎢⎣

⎡

−

−

−−=

w1ptpw2ptp

logtpwMη]1y2GT[y

cDdAwdm

∫⎥⎥⎦

⎤

⎢⎢⎣

⎡

−−⎥⎥⎦

⎤

⎢⎢⎣

⎡

−

−=

b

0 η1y2ywdx

w1ptpw2ptp

logtpwMGTD

wm

∫⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−⎥⎥⎦

⎤

⎢⎢⎣

⎡

−

−=

b

0bπxcoscosωmaxη1y2y

dx

w1ptpw2ptp

logtpwMGTDw

wmt

1Iw1ptpw2ptp

logtpwMGTD

wm⎥⎥⎦

⎤

⎢⎢⎣

⎡

−

−=

∫⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−

b

0bπxcoscosωmaxη1y2y

dx

t

∫−

=b

0bπxBcosA

dx1I

80

(5.10)

where α = B/A

Let ubπx

= , so duπbdx =

When x = 0, u = 0; x = b, u = π

(5.11)

Let 2IπAb

1I = , where (5.12)

, where 2utant = (5.13)

(5.14)

(5.15)

(5.16)

Substituting value of I2 in equation 5.12 we get-

2B2A

b1I

−= (5.17)

(5.18)

(5.19)

∫−

=b

0bπxαcos1

dxA1

1I

∫−

=π

0 αcosu1du

πAb

1I

∫−

=π

0 αcosu1du

2I

∫++

=π

0 2αtα-2t1

2dt2I

π

0α1α1t1tan

2α1

22I ⎟⎟

⎠

⎞⎜⎜⎝

⎛

−+−

−=

∞−

−= 1tan

2α1

22I

2π

2α1

22I

−=

( ) ωt2cos2maxη2

1y2y

1

w1ptpw2ptp

logtpwMGT

Dbwwm

−−⎥⎥⎦

⎤

⎢⎢⎣

⎡

−

−=

( )21y2y

ωt2cos2maxη

1

1

w1ptpw2ptp

logtpwM)1y2GT(y

Dbwwm

−−

⎥⎥⎦

⎤

⎢⎢⎣

⎡

−

−

−=

81

(5.20)

(5.21)

Equation 5.21 gives the overall mass transfer of water vapor from the standing wave

interface. The mass transfer at the interphase is determined using this expression. This

relation when compared with interfacial mass transfer of stationary tank, states that the

mass transfer through swaying tanks is always greater than the mass transfer from stationary

tanks. The two planes (y1 and y2) through which the phenomenon of mass transfer takes

place will have different pressures which can be calculated thermodynamically. The partial

pressure of water vapor near the surface is assumed to be saturated pressure corresponding

to the wet bulb temperature. The wet bulb temperature is the same as the dry bulb

temperature for saturated water at the interface. The partial pressure of water vapor

through the region of mass transfer is determined using the relation between partial

pressure and relative humidity as stated by the psychrometry [22].

( )21

21y2y

ωt2cos2maxη

1w1ptpw2ptp

logtpwM)1y2GT(y

Dbwwm

−

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−−

⎥⎥⎦

⎤

⎢⎢⎣

⎡

−

−

−=

( ) ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛−−−−−+

−+

⎥⎥⎦

⎤

⎢⎢⎣

⎡

−

−

−= 2

1y2y

ωt2cos2maxη

211

w1ptpw2ptp

logtpwM)1y2GT(y

Dbwwm

82

5.3 Experimental setup

Figure 5. 2 Schematic set-up of instrumented tank (1. Thermocouples, 2. Conductivity probe at various test filling levels and top,

3. Humidity sensing probe, 4. Pressure transducer for measuring ullage pressure, 5.Cartridge heaters, 6. Hexapod)

The experiments were carried out in a thermodynamically controlled system. A schematic

of this system is shown in Figure 5.2. The set-up consisted of an instrumented tank being

swayed using a six degrees of freedom motion actuator (Hexapod) at a particular frequency

and amplitude. The tank was made of plexiglass with dimensions of 260 mm × 180 mm ×

50 mm. The bottom plate was an aluminum block embedded with five 250 W cylindrical

cartridge heaters each of 0.9 cm diameter. A neoprene rubber insulation layer was

positioned below the heater to ensure that all heat was transferred upwards to the liquid.

The temperature was controlled by a variable thermostat with an operating temperature of -

10 ºC to 55 ºC and operating humidity of 25% to 85%. The accuracy of this controller was

± 0.5% of the indicated value. The accuracy with which the temperature controller

maintained the preset temperature was ± 0.2%. One side of the tank was fitted with nine

equally spaced K-Type thermocouples. As the dynamics mainly occur at the interface, the

three mean water level thermocouples were chosen to have a high response time of 15

seconds. The other thermocouples had a measurement accuracy of ± 0.2% and a response

time of 30 seconds.

1

Temperature controller

2

3 4

Data acquisition

system

5

Plexiglass tank

6

83

Figure 5. 3 Schematic of the experimental setup with position of moisture sensing probe

The ullage pressure was measured using a 2 mm diameter Kulite XCL8M 100 series

pressure transducer (Figure 5.2) with a measurement accuracy of ± 1%. The pressure

transducer was powered by 5 volts DC, and the signal was amplified 100x. In order to

record the relative humidity variation along the tank height from the free surface, a set of

moisture sensing probes were used in combination with a relative humidity meter.

Moisture sensing probes work on the principle of DC capacitance using Astable

Multivibrator circuitry [23]. The overall mass transfer characteristics of the system were

determined using the humidity recorded by these probes which were placed at 20%, 50%,

80% and 100% filling level (roof top) as shown in Figure 5.3. As these probes sense the

variable concentration of moist air in a conduit involving heat and mass transfer, they

indicate the overall concentration profile of the system with a decaying value of moisture

from the free surface to the top of the tank. The moisture sensing probes were cross

calibrated using an auto-timer data logging relative humidity meter. The range of the

humidity meter was 10% to 95%, its resolution was 0.1% of the full scale reading with a

response time less than 60 seconds and its sampling rate was set at 2 Hz. Further details of

the meter are provided in Chapter 2.

C∞, h mass transfer

Mass diffusion

Cb

Moisture sensing probes with uniform cross section at 20%, 50%, 80% and 100% filling levels

84

Table 5. 1 Natural sloshing periods of tank for different filling levels

The hexapod motion actuator used was the MOOG 6DOF 2000e system, capable of

oscillating a 1 tonne payload at six degrees of freedom, with a maximum velocity of 0.5 m/s

and acceleration of 0.6g. This motion actuator fulfilled the experimental tank sloshing

requirements by generating a suitably slow moving interface. The base of this model was

able to impart programmed motions into the tank, and allowed the response quantity

(interfacial mass transfer) to be measured. The natural frequency of sloshing in the tank is

given by-

2

tanhnT

gn n ha a

ππ π

=⎛ ⎞⎜ ⎟⎝ ⎠

(5.22)

where h is the filling level. For the 1:150 scaled tank with dimensions of 260 mm × 180

mm × 50 mm, the natural periods of the first mode (n = 1) were estimated as shown in

Table 5.1. The sloshing frequencies used to generate standing waves were selected

according to Table 5.1. These slow moving standing waves at different temperatures

formed the basis of the estimation of interfacial mass transfer. The amplitude of motion

was kept constant at 0.025 m. The frequencies obtained from Table 5.1 correspond to

single degree of freedom sway motion only. To obtain more accurate motion parameters,

precise calibration of the hexapod was performed by tracking the Light Emitting Diodes

located on the hexapod. It was observed that the hexapod at particular sway amplitudes

had a limiting frequency beyond which it was unable to provide the desired uniform output.

On the basis of the operating velocity and acceleration limits the hexapod was tested for

varying levels of payload. These operating conditions formed the basis of the range of

parameters to be studied. The operating conditions of the hexapod obtained by calibration

always had a lag in output with respect to input. Table 5.2 gives the matrix of input and

output operating conditions generated by the hexapod for the present experiments.

Frequency, Hz

Mode No Input Amplitude, m

Output Amplitude, m

0.7 1 0.027 0.025 0.8 1 0.028 0.025 1.1 1 0.030 0.025 1.6 1 0.030 0.025 1.7 1 0.030 0.025

85

Table 5. 2 Hexapod Operating Conditions (Input/Output Matrix)

Figure 5. 4 Layout of the experimental set up (1.Constant temperature bath (container), 2. K-Type thermocouples series, 3. Conductivity Probe, 4. Pressure Transducer, 5. Temperature controller, 6. Heater assembly, 7. Humidity

meter Astable Multivibrator, 8. Ultrasonic probe, 9. Hexapod)

Natural sloshing periods Filling levels (%) Depth of filling (m)

T1-sec f1-Hz T2-sec f2-Hz 10 0.018 1.25 0.8 0.64 1.57

20 0.036 0.90 1.1 0.49 2.05

40 0.072 0.68 1.45 0.42 2.37

50 0.09 0.65 1.55 0.41 2.42

5

3

4

2 1

7

6

9

8

86

The lay out of the experimental setup along with various measuring devices is shown in

figure 5.4. To measure the instantaneous position of the swaying tank an ultrasonic probe

(model SU1-B1-0A from Micro Detectors) was positioned on a tripod facing one side of

the tank. This self powered displacement sensor has a power consumption of 35 mA at 30

v DC peak voltage. The probe has a linearity error of 0.3% and measurement repeatability

of ±2 mm. A Custom made amplifier was used to amplify and filter the pressure data

obtained from the Kulite pressure transducer and an anti-aliasing filter was used to clean up

the signal and remove noise. The data were acquired at a sampling rate of 49 Hz through

the Agilent data acquisition system U2331A 64 multifunction DAQ 12 bits and 3MS/s and

stored on an Intel dual core 8 M personal computer. The thermal inertia of the fluids along

with the response time of thermocouples makes this sampling rate adequate to capture all

the intricate details of interfacial mass transfer. A JVC SD-Video camcorder with output in

MPEG format non-interlaced was set up to capture the global phenomenon of the interface

movement.

At the commencement of the experiments, the ullage was kept at atmospheric pressure. As

the temperature of the stored fluid was increased from 19 ºC to 50 ºC in step of 10 ˚C, the

ullage pressure also increased. At each 10 ˚C step in temperature, experiments were carried

out for a range of filling levels as shown in Table 5.3. This was followed by the

characterization of heat and mass transfer across the interface.

Table 5. 3 Range of Parameters

5.4 Uncertainty Analysis

The interfacial mass transfer rate is a function of interfacial temperature, relative humidity,

ullage pressure and surface area. These variables in turn are dependent on other

thermodynamic parameters like saturation pressure and saturation temperature. Hence the

accuracy of measurements of these parameters has an impact on the overall uncertainty

Parameters Range

Fluid Water-Air system Temperature 19-50 º C

Filling level of total height 10%-50%

Sway Amplitude 0.025 m Sway Frequency 0.6 Hz to 1.6 Hz

87

with which interfacial mass transfer is estimated. The uncertainty in the interfacial mass

transfer was computed by perturbing the values of the independent variables between their

possible extremes as dictated by the respective uncertainties in their measurement [24]. The

uncertainty in estimation of the mass transfer as obtained for the extremities was within ±

4.5%. The uncertainties in measurements and calculated parameters are shown in Table

5.4.

Table 5. 4 Uncertainty in the physical quantities of present experimental study

Parameters Range

Area ± 4% Temperature ± 0.2 %

Pressure ± 1% Relative Humidity ± 3%

Interfacial Mass Transfer rate ± 4.5%

5.5 Validation

The present experimental system was validated for various boiling regimes using an

asymptotic mode of validation [25]. When a liquid is heated in a controlled environment,

the first phase is free convection boiling within the liquid followed by nucleation, resulting

in vapor bubbles that form at certain favorable spots. To validate the present system, water

was heated over its saturation temperature in steps of 10 ˚C. The heat flux obtained from

the system was then compared with the classical correlation of Rohsenow and Jakob for

nucleate boiling [26] as given in the following equations-

(5.23)

(5.24) The surface fluid constant to be used between the polished aluminum surface and water in

the correlation by Rohsenow was obtained from Pioro [27]. The thermophysical properties

of the fluid were obtained from Liley [28]. The heat transfer coefficients obtained from the

present experimental setup are tabulated in Chapter 3. These heat transfer coefficients were

found to be comparable with those obtained using correlations available in the literature

within the error limit of two-phase experiments.

( ) 3

1.7

0.5

fPrfghsfC

∆tpfc

σgρfρg

fghfµAQ

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡⎥⎦

⎤⎢⎣

⎡ −=

( ) degW/m∆T5.58AQ1.54h 23

0.75=⎟

⎠⎞

⎜⎝⎛=

88

5.6 Results and Discussion

Figure 5.5 shows the standing wave generated at 0.025 m amplitude, 0.7 Hz sway frequency

at 20% and 50% filling levels of the liquid at 30 °C. These slow moving standing waves

formed the basis of the interfacial mass transfer calculations.

Figure 5. 5 Standing wave generation by swaying liquid at 30 ° C with Amplitude 0.025m,

Frequency 0.7 Hz

5.6.1 Influence of sway frequency on mass transfer

The mass transfer at the interphase is determined using the expression described in the

theoretical background through equation 5.21. Figure 5.6 shows the variation of interfacial

mass transfer rate with time for different sway frequencies of tank at a temperature of 30°C

and 10% filling level. The bold line depicts the moving average values of the interfacial

mass transfer rate to ensure that variations in the mean are aligned with the variations in the

data rather than being shifted in time. It can be seen that as the sway frequency increases

from 0.6 Hz to 1.1 Hz the interfacial mass transfer rate also increases. The increase in

average interfacial mass transfer (Root Mean Square value) from sway frequency of 0.6 Hz

to 1.1 Hz is nearly 2.25 times. For the sway frequency of 0.6 Hz and 0.7 Hz the mass

transfer varied within 19.7% and 23.9 % range between maximum and minimum values

respectively, where as for 1.1 Hz this range got reduced to 12.1%. For sway frequency of

0.6 Hz the frequency of interfacial mass transfer rate change was 0.6 cycle/ sec but for 0.7

Hz the frequency with which the interfacial mass transfer rate was varying changed to 0.7

cycle/ sec. Unlike to these variations for 1.1 Hz sway frequency the interfacial mass transfer

rate frequency varied with 1.17 cycles/ sec between its maximum and minimum values.

In terms of the two-phase physics, this increase of mass transfer with increasing sway

frequency can be correlated with breaking of the thermal boundary layer and saturation

pressure variation. As sway frequency increases the breaking of thermal boundary layer

20% filling level 50% filling level

89

becomes faster thereby allowing more liquid to vaporize to its vapor form. Besides these

with the increase in sway frequency the ullage pressure also varies at a faster rate. These

two contributing factors aggregate to increase the interfacial mass transfer rate.

Figure 5. 6 Interfacial mass transfer variation at 10% filling level and 30° C for different sway frequencies

1.15E-04

1.25E-04

1.35E-04

1.45E-04

1.55E-04

1.65E-04

70 71 72 73 74 75 76 77 78 79 80time,s

mas

s tr

ansf

er r

ate,

kg

/h

r

0.7 Hz sway frequency

RMS

2.50E-04

2.55E-04

2.60E-04

2.65E-04

2.70E-04

2.75E-04

2.80E-04

2.85E-04

2.90E-04

2.95E-04

3.00E-04

70 71 72 73 74 75 76 77 78 79 80

time, s

mas

s tr

ansf

er r

ate,

kg

/h

r

1.1 Hz sway frequency

RMS

1.10E-04

1.15E-04

1.20E-04

1.25E-04

1.30E-04

1.35E-04

1.40E-04

1.45E-04

70 71 72 73 74 75 76 77 78 79 80

time, s

mas

s tr

ansf

er r

ate,

kg

/h

r0.6 Hz sway frequencyRMS

90

1.38E-01

1.40E-01

1.42E-01

1.44E-01

1.46E-01

1.48E-01

1.50E-01

70 71 72 73 74 75 76 77 78 79 80

time,s

ulla

ge

hei

gh

t va

riat

ion

, m

Table 5. 5 Mean Interfacial mass transfer comparison between stationary and vibrating tanks

The interfacial mass transfer rate determined for swaying tank at different swaying

frequencies has also been compared with stationary tank as described in Chapter 2. Table

5.5 shows the effect of vibration on the interfacial mass transfer rate for 10% filling level

liquid at 30 °C. For the swaying tanks the interfacial mass transfer tabulated above is the

Root Mean Squared (RMS) interfacial mass transfer calculated as follows-

(5.25)

where n is number of observations.

It can be seen that the mass transfer rate has increased by 3.1 times from the situation of

stationary tank with open top to swaying tank with 1.1 Hz swaying frequency.

Figure 5. 7 Ullage height variation for10% filling level tank swaying at 0.7 Hz frequency

Filling Level – 10%, liquid temperature – 30 ° C

Sway Frequency (Hz) Mass transfer rate (kg/ hr)

0 (Stationary, open) 0.000090

0 (Stationary, closed) 0.000110

0.6 0.000125

0.7 0.000140

1.1 0.000280

n

2wnm2

w2m2w1m

rmsm+−−−−−−++

=

91

This concludes that within the range of parameters studied effect of vibration has an impact

on the interfacial mass transfer rate. The increase in mass transfer with increase in

frequency can be attributed to the variation of ullage pressure as explained in the previous

section. As the tank swayed it induced elevation in the water column there by creating a

variable ullage region. This variable ullage region contributed to the change in interfacial

mass transfer of the swaying tanks. Figure 5.7 shows the variation of ullage height for the

tank with 10% filling level and swaying at 0.7 Hz frequency. This variation has been

recorded for all the sway frequencies under considerations.

5.6.2 Influence of tank filling levels on mass transfer

Figure 5.8 shows the variation in interfacial mass transfer rates with time for various filling

levels of liquid when the temperature was kept constant at 30 °C. As the filling level

increases from 10% to 50% filling level the interfacial mass transfer rate also increases.

Table 5.6 shows the RMS interfacial mass transfer rates of 10%, 20% and 50% filling level

containers swaying at a frequency of 0.7 Hz. For 10% filling level the mass transfer varied

within 23.9% range between the maximum and minimum values and for 20% the range got

increased to 25.23%. But for 50% filling level this range got reduced to 6.38 %. The

frequency with which interfacial mass transfer rate changed with time between the

maximum and minimum values is 0.7 cycle/sec for 10%, 20% and 50% filling levels.

Therefore it can be concluded that interfacial mass transfer increases as the quantity of

liquid in the container increases.

Table 5. 6 Effect of filling level on mean interfacial mass transfer

In terms of two-phase physics, this increase in mass transfer with increasing filling level can

be attributed to the Stefan phenomenon. The Stefan phenomenon takes place at the centre

of the ullage region, with convection playing a vital role near the walls. The density of the

air-vapor mixture in the immediate vicinity of the wall surface and further away from it

determines the convection pattern in the system. The density driven convective current in

the present experiment is significant because of the difference in densities of water and air-

Sway Frequency (Hz)- 0.7 Hz, liquid temperature – 30 ° C

Filling Level % Mass transfer rate (kg/ hr)

10 0.000142

20 0.000165

50 0.00031

92

vapor mixture (nearly four times). With increasing filling levels the quantity of water and

air-vapor mixture increases. This creates a prominent bulk movement between the fluids.

The bulk movement of the air-vapor mixture over a distance stops when the density of the

lifted mixture becomes equal to that of the mixture away from the wall. In another

situation the movement of the mixture continues until the density becomes equal. Since for

50% filling level more liquid is present in the container the current movement continues for

more time, resulting in a higher interfacial mass transfer rate.

5.6.3 Influence of liquid temperature on mass transfer

Figure 5.9 shows the variation of interfacial mass transfer rate with time for different

temperatures of liquid at 10% filling level. It can be seen that as the liquid temperature

increases from 20°C to 50°C the interfacial mass transfer rate increases by about 13.52

times. For liquid temperatures of 20°C the mass transfer varied within a 30.9% range

between the maximum and minimum values and for 30°C this range got reduced to

25.23%. For liquid temperature of 40°C this range was reduced to 17.5% and for 50°C this

variation was within 14.32%. The frequency with which the interfacial mass transfer rate

changed with time was 0.7 Hz through out the observation.

In terms of two-phase physics, this increase of mass transfer with increasing temperature

can be attributed to the formation of a thermal boundary layer formation. As the

temperature of the tank walls rise, the thermal boundary layer nearest to the heated wall

heats up faster, compared to the central tank axis. As explained earlier, the heated air-vapor

volume lifts upward a distance that is in the order of the thermal boundary layer thickness.

As the temperature increases, the thermal boundary layer thickness also increases. This

makes the density driven mass of the air-vapor bulk to move more towards the core of the

ullage region. Therefore this density-driven convection current leads to an increase in the

interfacial mass transfer rate of the liquid.

93

1.15E-04

1.20E-04

1.25E-04

1.30E-04

1.35E-04

1.40E-04

1.45E-04

1.50E-04

1.55E-04

1.60E-04

75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90time,s

mas

s tr

ansf

er r

ate,

kg

/h

r

10 % filling levelRMS

1.40E-04

1.50E-04

1.60E-04

1.70E-04

1.80E-04

1.90E-04

2.00E-04

75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90time s

mas

s tr

ansf

er r

ate,

kg

/h

r

20 % filling levelRMS

0.0003

0.000305

0.00031

0.000315

0.00032

0.000325

75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90

time s

mas

s tr

ansf

er r

ate,

kg

/h

r

50 % filling levelRMS

Figure 5. 8 Interfacial mass transfer variation with filling levels at 30° C and 0.7 Hz sway frequency

94

7.50E-05

8.50E-05

9.50E-05

1.05E-04

1.15E-04

1.25E-04

75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90

time,s

mas

s tr

ansf

er r

ate,

kg

/h

r

Temperature = 20 CRMS

1.40E-04

1.50E-04

1.60E-04

1.70E-04

1.80E-04

1.90E-04

75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90

time s

mas

s tr

ansf

er r

ate,

kg

/h

r

Temperature = 30 CRMS

3.15E-04

3.25E-04

3.35E-04

3.45E-04

3.55E-04

3.65E-04

3.75E-04

3.85E-04

3.95E-04

4.05E-04

75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90time s

mas

s tra

nsfe

r rat

e, kg

/hr

Temperature = 40 C

RMS

1.05E-03

1.10E-03

1.15E-03

1.20E-03

1.25E-03

1.30E-03

1.35E-03

1.40E-03

75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90

time,sec

mas

s tra

nsfe

r rat

e, k

g/hr

Temperature = 50 CRMS

Figure 5. 9 Interfacial mass transfer variation with temperature for 10% filling level and 0.7 Hz sway frequency

95

On the basis of this vapor-air mixture movement in the heated tank Malyshev and Zlobin [29]

have described three zones of mass transfer (Figure 5.10) as follows-

1. Zone I immediately above the interface, having negligible buoyancy currents and

characterized by appreciable vapor concentration.

2. Zone II above Zone I mainly governed by convection at the wall with very small vapor

concentration gradient along the tank height.

3. Zone III above Zone II extending to the top wall characterized by minimum vapor

concentration and governed by conductive heat and mass transfer phenomenon.

Figure 5. 10 Model of the heat and mass transfer in the tank with characteristic zones

The height of Zone II as measured in the present experiments varied from 10 cm above the

stratified layer (Zone I) for 10 % filling level and 5.5 cm above this layer for 50 % filling level.

Zone II is sandwiched between the stratified layer (Zone I) influenced by diffusive mass

transfer (density and temperature difference) and Zone III, and has entirely different

thermophysical properties. This zone mass transfer takes place only because of the density

gradient with little condensation effect. The width of this zone depends on the concentration

gradient, the heating rate, and the ratio of the molecular weights of water vapor and air.

5.7 Conclusions

This experimental study of a tank partially filled with water has been carried out with different

combinations of sway frequencies, filling levels and water temperatures. These experiments

were conducted on a 1:150 scaled down model of an actual LNG tank being swayed on

x

I

III

II

96

hexapod. The experiment makes use of a range of measuring instruments to characterize the

physics of interfacial mass transfer. The thermodynamic variables of the fluids were mapped

over a wide range of parameters. Following are the conclusions that make the present study

unique, and contribute new knowledge in the area of heat and mass transfer of partially filled

swaying tanks-

1. The interfacial mass transfer is mainly governed by the concentration gradient across

the liquid and gaseous phases, irrespective of the container being swayed or stationary.

2. The interfacial mass transfer rate increases as the sway frequency increases from 0.6 Hz

to 1.1 Hz. The average interfacial mass transfer increased by 2.25 times when sway

frequency increases from 0.6 Hz to 1.1 Hz. For the sway frequency of 0.6 Hz and 0.7

Hz the mass transfer varied within 19.7% and 23.9 % range between maximum and

minimum values respectively, whereas for 1.1 Hz this range gets reduced to 12.1%.

For sway frequency of 0.6 Hz the interfacial mass transfer rate changed with time with

a frequency of 0.6 Hz where for 0.7 Hz sway frequency the mass transfer changed with

a frequency of 0.7 Hz. As sway frequency changed to 1.1 Hz frequency the interfacial

mass transfer rate changed at a frequency of about 1.17 cycles/ sec between the

maximum and minimum values.

3. With the increase in the filling level the interfacial mass transfer rate also increases. For

liquid at 30 °C with 0.7 Hz sway frequency and filling level of 10% the mass transfer

varied within range of 23.9% between maximum and minimum values where as for

20% filling level the range increases to 25.23%. But for 50% filling level this range gets

reduced to 6.38% between maximum and minimum values. The frequency with which

interfacial mass transfer rate changed with time between the maximum and minimum

values is 0.7 cycle/sec for 10%, 20% and 50% filling levels.

4. As the liquid temperature increases the interfacial mass transfer rate also increases.

For 10% filling level the mass transfer rate increases by 13.52 times from 20°C to 50°C

liquid temperature. For liquid temperature at 20°C the mass transfer varied within

30.9% range between maximum and minimum values where as for 30°C this range got

reduced to 25.23%. For liquid temperature of 40°C this range was 17.5% and for 50°C

this variation was within 14.32% between the maximum and minimum values.

97

5. The central zone of mass transfer prevails for a region with maximum wall convection

and minimum concentration gradient for an approximate height of 10 cm and 5.5 cm

for 10 % and 50 % filling levels respectively.

6. Within the range of parameters studied the sway vibration increases the interfacial mass

transfer rate. For 10% filling level liquid at 30 °C the mass transfer rate has increased

by nearly 3.1 times from the situation of stationary tank with open top to swaying tank

with 1.1 Hz. This increase in interfacial mass transfer of swaying tank can be explained

by the generation of variable ullage region.

5.8 Nomenclature

A base area of the tank, m2

a amplitude, m

b tank width, m

As surface area, m2

Cp specific heat of water, J/kg/K

Csf surface fluid constant

D diffusion coefficient m2/s

g acceleration due to gravity, m/s2

h convective heat transfer coefficient, W/m2K (eq 5.5), filling level, m (eq5.3)

hfg latent heat of vaporization J/kg

M molecular weight of species, kg

m mass flow rate, kg/s

n mode number

p partial pressure, N/m2

G universal gas constant, J K−1mol−1

T temperature, K

∆T excess temperature over saturation, K

98

V bulk mass velocity upward, m/s

w tank depth, m

x co-ordinate perpendicular to free surface of the liquid

Greek Symbols

µ dynamic viscosity of water, kg/ms

ρ mass density of water, kg/m3

σ surface tension, N/m

θ angle of tilt

ω sway frequency, rad/s

ωn natural frequency of oscillation, rad/s

Subscripts

a air

f fluid

g gas

s saturation

t total

v vapor

w water

5.9 Acknowledgement

The study was funded by an Australian Research Council Discovery Grant-DP0881447.

5.10 References

[1] Raouf Ibrahim, A., 2005. Liquid Sloshing Dynamics Theory and Applications.

Cambridge Press University.

99

[2] Sir Horace Lamb, 1932. Hydrodynamics, Cambridge Mathematical Library,

Sixth Edition.

[3] Faltinsen, O. M., Olsen, H. A., Abramson, H. N., and Bass, R. L., 1974.

“Liquid sloshing in LNG carriers. DNV Classification and Registry of

Shipping,” 85.

[4] Bird, R. B., Stewart, W. E., Lightfoot, E. N., 2007. Transport Phenomena, 2nd

Edition, John Wiley and Sons, Inc.

[5] Gradon, L., and Selecki, A., 1977. “Evaporation of a liquid drop immersed in

another immiscible liquid. The Case of σc < σd,” International Journal of

Heat and Mass Transfer, 20

[6] Raina, G., K., and Wanchoo, R., K., 1984. “Direct Contact Heat Transfer with

Phase Change: Theoretical Expression for Instantaneous Velocity of a Two-

Phase Bubble,” International Communication in Heat and Mass Transfer, 11.

[7] Raina, G. K., and Grover, P., D., 1985. “Direct Contact Heat Transfer with

Phase Change: Theoretical Model Incorporating Sloshing Effects,” AIChE

Journal, 31.

[8] Raina, G. K., and Grover, P. D., 1988. “Direct Contact Heat Transfer with

Change of Phase: Experimental Technique,” AIChE Journal 34.

[9] Dent, J. C., 1969-1970. “Effect of Vibration on condensation heat transfer to a

Horizontal Tube. Proceedings Institution of Mechanical Engineers,” 184.

[10] Rose, J., W., 1981. “Dropwise Condensation Theory, International Journal of

Heat and Mass Transfer,” 24.

[11] Rose, J. W., 1987. “On interphase matter transfer, the condensation coefficient

and dropwise condensation,” Proceedings of Royal Society of London, 11.

[12] Tolubinskiy, V. I., Kichigin, A. M., Povsten, S. G., and Moskalenko, A. A.,

1980. “Investigation of Forced-Convection Boiling by the Acoustical Method,”

Heat Transfer- Soviet Research, 12.

[13] Antonenko, V. A., Chistyakov, Yu., G. and Kudritskiy, G. R., 1992.

“Vibration-Aided boiling heat transfer, Heat Transfer” Soviet Research, 24.

100

[14] Nishiyama, K., Murata, A., and Kajikawa, T., 1986. “The Study of

Condensation Heat Transfer Performance Due to Periodic Motion of a

Vertical Tube,” Heat Transfer: Japanese Research, 15.

[15] Kravchenko, V. A., and Fedotkin Yu. Y., 1990. “Film Condensation Heat

Transfer of a Rapidly Moving Vapor,” Heat Transfer- Soviet Research, 22.

[16] Fang, G., Ward, C. A., 1999. “Temperature measured close to the interface of

an evaporating liquid,” Phys. Rev. 59, 417–428.

[17] Fedorov, V. I., Luk’yanova, E. A., 2000. “Filling and storage of cryogenic

propellant components cooled below boiling point in rocket tanks at

atmospheric pressure,” Chem. Pet. Eng. 36, 584–587.

[18] Kozyrev, A. V., Sitnikov, A. G., 2001. “Evaporation of a spherical droplet in a

moderate pressure gas,” Phys. Usp. 44, 725–733.

[19] Scurlock, R., 2001. Low loss Dewars and tanks: liquid evaporation mechanisms

and instabilities, Coldfacts 7–18.

[20] Krahl, R., Adamo, M., 2004. “A model for two phase flow with evaporation,”

Weierstrass-Institut for Angewandte Analysis and Stochastic, Berlin 899.

[21] Thiagarajan, K. P., Rakshit, D., Repalle, N., 2011. “The air-water sloshing

problem: fundamental analysis and parametric studies on excitation and fill

levels,” International Journal of Ocean Engineering, 38, 498-508.

[22] Bird, R. B., Stewart, W. E., 2007 Lightfoot, E.N., Transport Phenomena, 2nd

Edition, John Wiley and Sons, Inc.

[23] Parameswaram, M.., Baltes, H. P., Brett, M. J., Fraser, D. E. and Robinson, A.

M., 1988. “A Capacitive Humidity Sensor based on CMOS Technology with

adsorbing film.” Sensors and Actuators, 15, 325-335.

[24] Coleman, H. W., and Steele, W. G., 1989. “Experimentation and uncertainty

analysis for engineers,” 2nd edition John Willey and Sons.

101

[25] Balaji, C., Hölling, M., Herwig, H., 2007. “Entropy generation minimization in

turbulent mixed convection flows.” International Communications in Heat and

Mass Transfer 34, p. 544–552

[26] Incropera, F. P., Dewitt, D. P., Bergmen, T. L., Lavine, A. S., 2005.

Introduction to Heat Transfer 5th ed, John Willey and Sons.

[27] Pioro, L., 1999. “Experimental evaluation of constants for the Rohsenow pool

boiling correlation.” International Journal of Heat and Mass Transfer 31, p.

2003-2013

[28] Liley, P. E., 1984. Steam Tables in SI Units, private communication. School of

Mechanical Engineering, Purdue University, West Lafayette, IN

[29] Malyshev, V. V., and Zlobin, E. P., 1972. “Evaporation of liquid hydrocarbons

in heated closed containers.” Translated from Inzheneruo-Fizicheskii Zhurnal,

Vol. 23, No. 4, p. 701-708.

102

Chapter 6

Conclusions and suggestions for future work

6.1 Introduction

On the basis of the studies quantifying thermodynamic behavior of stored liquids, broad

conclusions that have been arrived at are presented in this chapter. This study was aimed at

designing and performing controlled experiments to estimate the interfacial mass transfer

characteristics of a liquid-vapor system. The parameters which were studied are the liquid

temperature, fractional concentration of the evaporating liquid, the surface area of the liquid

vapor interface, the fill level of the liquid and the frequency of oscillations. The previous

chapters mainly dealt with the description of the problems along with the findings of the study

in sections. The following sections present comprehensive conclusions of the studies and list

out suggestions for future extensions of the present work.

6.2 Broad conclusions of the Present Study

The major conclusions that are drawn from the present study are as follows -

1. The interfacial mass transfer is mainly governed by the concentration gradient across

the liquid and gaseous phases, irrespective of the container being open or closed.

2. For stationary tanks, the central zone of mass transfer prevails for a region with

maximum wall convection and minimum concentration gradient for an approximate

height of 8.5 cm and 5 cm for 10 % filling level and 50 % filling level respectively.

103

3. The interface mass transfer in a stationary closed container was almost 50 % more than

its open counterpart.

4. For stationary tanks, entropy generation of the ullage is greater than the saturated

liquid.

5. In stationary tanks, ullage entropy is governed by a region of mist with high humidity

(dryness fraction) contributing to the maximum part of the overall entropy.

6. For stationary tanks, the interfacial mass transfer variation with filling levels for open

and closed containers at 19 °C, 25 °C and 30 °C are meager as compared to 40 °C and

45 °C because of minimum temperature difference between the liquid and the

surroundings.

7. For open stationary tanks at 40 °C and 45 °C the interfacial mass transfer increases

very abruptly after 30% filling levels. However for closed containers after 30% fill level

the interfacial mass transfer reaches an asymptotic limit. This was due to the Stefan

phenomenon taking place at the core of the ullage region, with wall convection playing

a vital role.

8. For stationary tanks, the difference in entropy generated per unit volume at 20 °C, 25

°C and 30 °C for 20% and 50% filling level for both open and closed containers is

significantly less. However for 40 °C and 45 °C this difference becomes more

prominent. This considerable hike in the entropy generation occurred because of the

entropy contributed by the ullage region.

9. In stationary tanks for 20 °C, 25 °C and 30 °C fluids, entropy generation due to heat

transfer was not significant.

10. The non dimensional entropy generated by liquid and ullage was dependant on

Grashoff Number (Gr), Prandtl Number (Pr) and geometrical parameter (l/d).

104

11. In stationary tanks, the “holistic” optimization was more relevant for fluid temperature

above 312 K.

12. It was concluded that under the studied conditions there was a set of optimum

temperatures that will keep the entropy generated by the system to its minimum

possible level by imposing a penalty on the interfacial mass transfer.

13. It was also observed that for both open and closed stationary tanks 50% fill level

showed a better second law optimization as compared to 20% filling level.

14. For stationary containers when a holistic optimization is performed the reduction in

entropy generation for open containers is significant and must be accounted for.

15. The interfacial mass transfer is mainly governed by the concentration gradient across

the liquid and gaseous phases, irrespective of the container being swayed or stationary.

16. The interfacial mass transfer rate increases as the sway frequency increases from 0.6 Hz

to 1.1 Hz. The average interfacial mass transfer increased by 2.25 times when sway

frequency increases from 0.6 Hz to 1.1 Hz. For the sway frequency of 0.6 Hz and 0.7

Hz the mass transfer varied within 19.7% and 23.9 % range between maximum and

minimum values respectively, whereas for 1.1 Hz this range gets reduced to 12.1%.

For sway frequency of 0.6 Hz the interfacial mass transfer rate changed with time with

a frequency of 0.6 Hz where for 0.7 Hz sway frequency the mass transfer changed with

a frequency of 0.7 Hz. As sway frequency changed to 1.1 Hz frequency the interfacial

mass transfer rate changed at a frequency of about 1.17 cycles/ sec between the

maximum and minimum values.

17. With the increase in the filling level the interfacial mass transfer rate also increases. For

liquid at 30 °C with 0.7 Hz sway frequency and filling level of 10% the mass transfer

varied within range of 23.9% between maximum and minimum values where as for

20% filling level the range increases to 25.23%. But for 50% filling level this range gets

reduced to 6.38% between maximum and minimum values. The frequency with which

105

interfacial mass transfer rate changed with time between the maximum and minimum

values is 0.7 cycle/sec for 10%, 20% and 50% filling levels.

18. As the liquid temperature increases the interfacial mass transfer rate also increases.

For 10% filling level the mass transfer rate increases by 13.52 times from 20°C to 50°C

liquid temperature. For liquid temperature at 20°C the mass transfer varied within

30.9% range between maximum and minimum values where as for 30°C this range got

reduced to 25.23%. For liquid temperature of 40°C this range was 17.5% and for 50°C

this variation was within 14.32% between the maximum and minimum values.

19. The central zone of mass transfer prevails for a region with maximum wall convection

and minimum concentration gradient for an approximate height of 10 cm and 5.5 cm

for 10 % and 50 % filling levels respectively.

20. Within the range of parameters studied the sway vibration increases the interfacial mass

transfer rate. For 10% filling level liquid at 30 °C the mass transfer rate has increased

by nearly 3.1 times from the situation of stationary tank with open top to swaying tank

with 1.1 Hz. This increase in interfacial mass transfer of swaying tank can be explained

by the generation of variable ullage region.

6.3 Suggestions for future work

Estimations of interfacial mass transfer from liquids stored in containers under various

conditions have been carried out in this study. For stationary tanks the study also considered

the temperature estimation of the system by optimizing interfacial mass transfer and entropy

generated by the system. In the context of extending the work for future studies the following

points can be considered:

1. The study considered thermodynamic mass transfer analysis of stored liquids using air

and water as the fluids. This study can further be refined by making utilization of actual

LNG.

2. A CFD model estimating interfacial mass transfer from LNG can be used as a

precursor of the experiments.

3. High precision RTDs in combination with relative humidity meter will increase the

accuracy of measurements.

106

4. The thermodynamic optimization of interfacial mass transfer maximization and

entropy generation minimization can be made multidimensional if it is done using

Artificial Neural Networks or Genetic Algorithms.

a-i

Appendix-1

a-ii

Proceedings of the 20th National and 9th International ISHMT-ASME Heat and Mass Transfer Conference

January 4-6, 2010

10HMTC380

An experimental study on the interface mass transfer governing thermodynamics of stored liquids

Dibakar Rakshit †a, Ramesh Narayanaswamy b, Tam Truong a, Krish P Thiagarajan a a The University of Western Australia, Crawley, WA, Australia, b Curtin University of Technology, WA, Australia

ABSTRACT

Heat transfer processes that occur in mixtures of more than one substance always deals with mass transfer phenomenon. The thermodynamics of two phase physics involving different liquid-gas combinations can be studied under the regime of these conjugate heat and mass transfer phenomena. Evaporation from liquid surface depends on: (i) liquid-vapor interfacial temperature, (ii) fractional concentration of the evaporating liquid present in gaseous state, and (iii) the surface area of the liquid vapor interphase.

With increasing demand for Liquefied Natural Gas (LNG), transportation of LNG through ships is in high demand with even low filling levels. This type of transportation is prone to sloshing due to ship motion in a seaway. The ullage pressure above the liquid containment plays a critical role in making a proper selection of the various fill levels and temperatures with which LNG can be transported from one place to another. Multiphase characteristics of LNG may be understood by examining a miscible fluid combination viz. water-vapor water system. In particular, the roles played by vibration in the heat transfer mechanisms such as boil-off during heating and condensation when in direct contact with the atmosphere, needs to be understood. In the present study an attempt has been made to understand the underlying principle of an evaporating and condensing system involving two-phase heat and mass transfer by studying its effect on a stationary liquid tank. An experiment with an air-water system has been designed and conducted to explain the characteristics of heat and mass transfer at various fill levels and temperatures. The sophistication lies in the instrumentation of the experiment where conductivity probes accompanied by a humidity meter is used in combination with thermocouples and ullage pressure measuring device to map the entire physics of interface mass transfer. This is

followed by an estimation of overall mass transfer from the interface in analogy with the multiphase heat transfer taking place across the interphase. NOMENCLATURE A base area of the tank, m2 C molar concentration kg-mol/m3 Cp specific heat of water, J/kg/K Csf surface fluid constant D diffusion coefficient m2/s

Gr Grashof number, 2µ

2ρ3Tlg ∆β

g acceleration due to gravity, m/s2 h convective heat transfer

coefficient, W/m2K hfg latent heat of vaporization J/kg l characteristic length of tank, m M molecular weight of species kg m mass flow rate kg/sec N diffusion rate kg-mol/s-m2

Pr Prandtl number of fluid α

ν

p partial pressure N/m2 R universal gas constant JK−1mol−1 R’ characteristic gas constant of air

Jkg-1K−1mol−1 T temperature, K ∆T excess temperature over

saturation v bulk mass velocity upward, m/s x co-ordinate perpendicular to free

surface of the liquid Greek Symbols α thermal diffusivity, m2/s β isobaric cubic expansivity of the

fluid,

pT

ρ

ρ

1⎟⎠⎞

⎜⎝⎛∂

∂− , K-1

a-iii

µ dynamic viscosity of water, kg/ms

ν kinematic viscosity, m2/s ρ mass density of water, kg/m3

σ surface tension, N/m Subscripts a air w water t total d dry v vapor s saturation INTRODUCTION Liquefied Natural Gas (LNG) is a significant energy resource, which requires careful transportation and storage. With the increasing demand of LNG sometimes the LNG cargos are transported with partially filled tanks. These partially filled tanks when transported from one place to another cannot maintain its thermodynamic stability, and thus in its long run the LNG properties gets modified. The partially filled tanks generally result from several loading and unloading of the LNG when multiport stops are made [1]. As real time experiments require huge resources in terms of instrumentation and time, experimentation using scale models explain the physics of the problem very precisely. So it has become a standard practice in marine technology to carry out experiments on scaled models of the actual system. Major companies like Det Norske Veritas [2] and the American Bureau of Shipping [3] exploit every possible way of finding the solutions to the real time marine science problems through scaled experimentations. Bass et al. [4] in their study of LNG sloshing made an important milestone in the research of sloshing. They carried out several experiments to capture the scaling effects of the system. They prescribed various fluids which can closely mimic the phenomenon in real life situation. The design of their experiments was based on water-air fluid system. The various degrees of freedom with which the ship moves make the study of the LNG tank challenging when its thermodynamic characteristics are to be analysed. So to understand the underlying principle and the physical mechanisms of the fluid mechanics and thermodynamics involved in a LNG tank, appropriate modelling and experiments need to be performed. As the thermodynamics of the phase change phenomenon has a significant contribution in the proper conditioning, transportation and storage of LNG, its study becomes critical in terms of understanding the underlying physics. As far as the heat transfer and thermodynamic studies of LNG are concerned, it has mainly been done in the designing of the LNG vessel which is to be maintained at sub-zero temperatures. Kobayashi and Tsutsumi [5] studied the heat gain characteristics of the supporting skirt of MOSS type

of LNG tank. In their study they inferred that the reduction of the evaporation rate of the liquefied cargo by insulation enhancement is not a practical solution because of the type of material used for insulation. Chen et al. [6] studied thermodynamics of the LNG stations designed with prevention of venting of natural gas from LNG tanks. Bates and Morrison [7] studied the phase change phenomenon of the stored LNG that affects the composition in the long run when mixed with fresh LNG. Boukeffa et al. [8] in their study of heat transfer across a cryostat experimentally and numerically attempted to predict the velocity and temperature profile across the cryostat neck which is the most critical region of heat leakage in a cryostat. Another study by Boumaza et al. [9] to see the heat transfer effects on cryogenic tank concluded that the pattern of cryogenic heat transfer showed some discrepancies with the theoretical calculation of heat transfer across the system. The mechanism of evaporation of a liquid immersed in another liquid was first studied by Gradon and Selecki [10]. Raina and Wanchoo [11] in their studies with direct contact heat transfer incorporating phase change came up with expressions for instantaneous velocity of a two-phase bubble. Raina and Grover [12] introduced sloshing effect into the direct contact heat transfer modelling of a multiphase system. This theoretical model was later validated through experiments by Raina and Grover [13]. An inception of the study of two phase miscible fluid combination in vibration was made by Dent [14] explaining the effect of vibration on a horizontal condenser tube vibrating in the plane of gravitational field. Similar studies in the regime of the forced convection boiling were done by Tolubinskiy et al. [15]. Running on the same lines later Antonenko et al. [16] identified the various boiling phases which shows an enhancement in heat transfer when a vibration is induced in the system. A detailed study of condensation heat transfer performance involving various gyroscopic motions was done by Katsuo et al. [17]. A similar type of analytical study was made by Kravchenko and Fedotkin [18] in which heat transfer characteristics of a rapidly moving vapor subjected to film condensation was analysed. The literature review for this novel approach of incorporating the thermodynamics of the problem shows that published works involving the phase change mechanism of LNG containers in marine technology are very scarce. The two- phase interaction of the liquid with ullage vapors in LNG containers mainly depends on the interfacial physics of the evaporating and the condensing fluids that take part in the process. Therefore to understand its mechanism a heat and mass transfer mapping of the interfacial system is required. As far as heat transfer studies of multiphase interfacial conditions are concerned there are some studies been made in

a-iv

this field by Fang and Ward [19], Fedorov and Luk’yanova [20], Kozyrev and Sitnikov [21], Scurlock [22], Krahl and Adamo [23]. These studies are based on the methods of predicting the interfacial temperature, vapor pressure with various factors affecting their measurement. Besides these some studies were made to quantify the stratified region at the interphase which mainly governs the heat transfer and mass transfer phenomenon. Although the thickness of this layer is of the order of a few nanometers but presence of this non condensable gas layer makes the temperature of the interfacial liquid undergoing the process of evaporation much higher than the bulk liquid. In most of these studies the regime of evaporation and condensation of the interphase is investigated without much consideration to the effects of temperature variation and fill levels at which the liquid is stored or transported. Present investigation will make an attempt to fill the gap in this direction where a consolidated experimental study has been made to understand the interphase physics of evaporation and condensation of stored liquids keeping in view the fill level effects. These experiments are conducted in a 1:5 scaled down tank of an existing hydrodynamic experimental model of an actual LNG tank. The tank is rectangular in geometry, and the two phases that are investigated are air and water. THEORETICAL BACKGROUND The analytical prediction of mass transfer is based on the principle of liquid-gas interphase tracking. This theory works on the principle of conservation of mass, momentum and energy where a system contains two or more components having variation in concentrations. The transportation of one constituent from a region of higher concentration to that of a region of lower concentration mainly governs the mass transfer. In this system of two phases the concentration of water molecules are higher just above the liquid surface compared to that in the main portion of air stream leading to the process of diffusion. Thus the concentration of water molecules in combination with its vaporized state governs the mass transfer across the interphase. The assumptions that govern the diffusion of water vapor through air are as follows-

1. System is under steady state and isothermal conditions.

2. Total pressure within the system is constant.

3. Both fluids (air and water vapors) behave as perfect gases.

4. Air which flow across the surface has negligible solubility in water.

5. Slight movement of air over the ullage does not bring any change in concentration profile.

Under steady state condition, the upward water evaporation into air is balanced by downward diffusion of air so that concentration at any location from the water surface remains constant as suggested by John Lienhard [24]. From Fick’s law of diffusion for two gaseous species ‘b’ and ‘c’ (air and water vapor) undergoing diffusion the diffusion rates are-

dx

dCDN b

bcb −= (1)

dxcdC

cbDcN −= (2)

Invoking the Universal Gas law

⎟⎟⎠

⎞⎜⎜⎝

⎛−==

RTbM

bpdx

dbcD

Abm

bN (3)

Similarly for species ‘c’ the diffusion rate will be

⎟⎠⎞

⎜⎝⎛

−==RT

Mp

dx

dD

A

mc

N cccb

c (4)

It is worth mentioning here that the water evaporates and at the same time diffuses upward through the air. Downward mass diffusion of air is given by-

dxadp

RTaAM

Dam −= (5)

Since there can be no net mass movement of air downward at the surface of water, the bulk mass movement upward with a velocity is large enough to compensate for the mass diffusion of air downward. Bulk mass transfer of air is equal to

AvRT

aMapAvaρ −=− (6)

The mass transfer of water is due to (i) upward mass diffusion of water and (ii) water vapors carried upward along with upward bulk movement of moving air. Thus the total mass transfer of water vapor is

Adx

adp

ap

D

RTwMwp

dxwdp

wMRT

ADtotal)w(m +−=

(7) From Dalton’s law of partial pressures,

dxwdp

dxadp

dxtdp

+= (8)

Since total ullage pressure of the system remains constant at a particular temperature dpt/dx=0 and therefore

a-v

dxwdp

dxadp

−= (9)

Hence equation (7) can be rewritten as

( )⎭⎬⎫

⎩⎨⎧

+−=

wptptp

dxwdp

wMRT

DAtotalwm (10)

This depicts a binary liquid-gaseous system undergoing process of diffusion and behaving as ideal gaseous components of a practically stagnant system. The expression when integrated between two planes x1 and x2 from where the mass transfer is taking place can be rewritten as

⎟⎟⎠

⎞⎜⎜⎝

⎛

−

−

−=

w1ptpw2ptp

elog)1x2(x

tpwM

RT

DAtotal)w(m

(11) Thus the mass transfer of the interphase can be calculated using the above derived expression without any loss of generality. The total pressure of the ullage through which the mass transfer is taking place can be calculated using the ideal gas equation,

T'ρRp = (12) The density of the fluid system is calculated by the equivalence effect of the present water vapor using following equation-

TvRvp

TdRdp

humidairρ += (13)

The two planes through which the phenomenon of mass transfer takes place will have different pressures which can be calculated thermodynamically. The partial pressure of the water vapor near the surface is assumed to be saturated pressure corresponding to the wet bulb temperature (same as the dry bulb temperature for saturated water) of the interface which is obtained by the interfacial K-type thermocouple. The partial pressure of water vapor through the region of mass transfer is determined by using the relation between partial pressure and relative humidity. Malyshev and Zlobin [25] had described that the theoretical background outlined above provided meaningful results to their experimental investigations as well. Their analytical explanation was validated by experimental data for a closed container. It showed that in a closed container filled with liquid the convective mass transfer region governing the Stefan’s phenomenon are sandwiched between two zones having entirely different physics. These two zones are the one consisting of the stratified layer just above the interphase and the other near the top wall with conductive heat and mass transfer governing the physics. The central

zone where the convection at walls is significant is characterized by very small vapor gradient along the container height. This central zone mainly signifies the mass transfer taking place because of the density gradient with minimal condensation. The width of this zone depends on the concentration gradient, on the heating rate and on the ratio of the molecular weights of vapor and air. In the present investigation also a comparison has been made in the mass diffusion results obtained by keeping the system open in contrast to its closed counterpart to evaluate any difference between them. EXPERIMENTAL SET UP The experiments are carried out in a thermodynamically controlled system as shown in Figure 1. The system consists of a plexiglass tank of dimensions 260 mm × 180 mm × 50 mm with heating arrangements at the bottom with the removable top. The bottom plate is an aluminum block embedded with 5 cylindrical cartridge heaters of 0.9 cm diameter and wattage of 250 W each to heat the plate. To avoid any direct contact with the aluminum plate proper thermal insulation has been provided all along the heated plate. The system is heated under a controlled environment with a temperature controller associated with it. The temperature controller comprises of a variable thermostatic arrangement that allows the desired temperature of the fluid to be reached. The temperature of the plate can be set manually to the desired value with the help of a digital switch provided on the temperature controller. Once the required temperature of the plate is attained the thermostatic arrangement cuts off the power unless the plate reaches a temperature less than the demarcated value. The accuracy with which the temperature controller maintains the designated temperature is ± 0.2%. To measure the temperature of the fluid, interphase and the ullage contained in the tank thermocouples are used. One side of the tank is instrumented with 9 equally spaced K-Type thermocouples also having a measurement accuracy of ± 0.2%. The top plexiglass consists of two slots having arrangements to measure the ullage pressure and relative humidity of the ullage. The ullage pressure is measured using a 2mm diameter Kulite XCL8M 100 series pressure transducer that can measure pressure with a measurement accuracy of ±1%. The pressure transducer used is being powered with 5 volts DC with a gain of 100 to avoid any noise.

a-vi

Table 1 Range of Parameters

Figure 1. Schematic setup of thermodynamically controlled test rig, 1. Thermocouples, 2. Conductivity probe at various test fill levels and top, 3. Humidity sensing probe, 4. Pressure transducer for measuring ullage pressure, 5.cartridge heaters to give requisite temperature rise of the fluid Relative humidity of the ullage region is measured using RS auto-timer data logging humidity meter which can measure relative humidity with ± 3% accuracy. Besides these, a series of conductivity probes which works on the principle of a variable voltage capacitor resistor system has been used. These probes show a voltage change on the basis of the mist being sensed by them. At the commencement of the experiments the ullage pressure is kept at atmospheric pressure. However as the temperature of the stored fluid is increased the ullage pressure also changes. The temperature of the liquid is raised from 19ºC to 60ºC in steps of 5˚C. For each set of experiments steady state condition of the thermostatic arrangement is attained over a period of time before starting the actual experiment. At each temperature the experiments are carried out for two different filling levels as shown in Table 1. These filling levels are decided on the basis of the preliminary CFD analysis made in the context of hydrodynamic studies of stored liquids by Rakshit et al [26]. This is followed by the characterization of the heat and mass transfer studies across the interphase.

RESULTS AND DISCUSSION Validation For the problem under consideration, results in the open literature are scarce. The validation of the present experimental setup is done for different regimes of boiling through which the system has passed through. Since the present experiment is designed in such a manner that the heat and mass transfer phenomenon of the two participating phases can be studied through an elevated temperature hence the setup has been validated for various boiling regimes through an asymptotic mode of validation [27]. Since in a boiling experiment it is very difficult to predict the fluid behavior hence most of the validations are based on the basis of the given empirical relations obtained by an experiment. When a liquid is heated under controlled environment the first phase is the free convection boiling where heat transfer is solely due to free convection and the film coefficient follows the correlation of Nusselt number in terms of Grashof number (Gr) and Prandtl number (Pr). However the predominance in boiling starts with temperature excess when the liquid is heated for a long time with nucleation playing an important role. When the liquid is overheated in relation to its saturation temperature, vapor bubbles are formed at certain favorable spots called the nucleation or active sites which may be the wall surface irregularities, air bubbles and the particles of dust. To validate the present system, water is heated over its saturation temperature in steps. The heat flux obtained from the present system is compared with the classical correlation of Rohsenow and Jakob for the nucleate boiling [28] as given in the following equations-

( )3

1.7fPrfghsfC

∆tpfc0.5

σgρfρg

fghfµA

Q⎥⎥

⎦

⎤

⎢⎢

⎣

⎡⎥⎦

⎤⎢⎣

⎡ −=

(14)

( ) deg2W/m3∆T5.580.75

A

Q1.54h == ⎟

⎠⎞

⎜⎝⎛

(15) The surface fluid constant as required between polished aluminum surface and water in the correlation by Rohsenow is obtained from the experimental investigation done by Pioro [29]. All thermophysical properties of the fluids are calculated from steam tables of Liley [30]. The heat transfer coefficients as obtained from the present experimental setup are tabulated in Table 2. The values of the heat transfer coefficients are comparable with those obtained through the correlations present in the literature within the error limits of a boiling experiment.

S.No Parameters Range

1 Fluid Water-Air system

2 Temperature 19-60 º C

3 Fill level of total height 20%, 50%

4 Tank condition Stationary

1

Temperature

2

34

5

a-vii

19.5

22

24.5

27

29.5

32

34.5

37

39.5

42

44.5

47

49.5

52

54.5

57

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5

Time (Hours)

Tem

pera

ture

(C)

57 C Bulk Temp

49 C Bulk Temp

48 C Bulk Temp

15161718192021222324252627282930313233343536373839404142

0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5 2.75 3 3.25 3.5 3.75 4 4.25 4.5 4.75 5

Time (Hours)

Tem

pera

ture

(C)

BulkInterphaseUllage 1Ullage 2

20

22.5

25

27.5

30

32.5

35

37.5

40

42.5

45

47.5

50

52.5

55

57.5

60

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5

Time (Hours)T

empe

ratu

re (C

)

57 C Bulk Temp

49 C Bulk Temp

48 C Bulk Temp

18.5

21

23.5

26

28.5

31

33.5

36

38.5

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5

Time (Hours)

Tem

pera

ture

(C)

57 C Bulk Temp

49 C Bulk Temp

48 C Bulk Temp

The test rig along with various measuring devices is described in Figure 2. Figure 3 shows the transient temperature history for the bulk, interphase and ullage region of the tank at 20% filling level. The liquid after reaching a constant temperature level shows some undulations with a periodicity of about 7 cycles per hour and was found to be repeatable for different sets of experiments. Similarly the ullage region also reports some irregular time dependant variation with periodic spikes. This is the result of the convection cells which form at the interphase because of the density gradient between the liquid and the gaseous phase as suggested by [22]. The blue line depicts the time the system at constant temperature is allowed to come to its equilibrium with the environment. The temperature decay to the environment for 20 % filling level of the stored liquid for its bulk, interphase and ullage region are measured over a period of time. These are reported in Figures 4 - 6 for different temperatures of the stored fluid.

Table 2 Comparison of the heat transfer

coefficients

Figure 2. Test rig 1. Constant temperature bath (container), 2. k type thermocouples series, 3. Conductivity Probe, 4. Pressure Transducer, 5. Humidity meter, 6. Data Acquisition System, 7. Temperature controller, 8. Heater assembly, 9. Humidity meter Astable Multivibrator

Figure 3. Transient temperature history for 20% filling level

Figure 4. Temperature decay in the bulk over a period of 8.5 hours

Figure5.Temperature decay in the interphase over a period of 8.5 hours

Figure 6. Temperature decay in the ullage over a period of 8.5 hours

S.No Temperature differential

hcalculated (W/m2K)

hliterature (W/m2K)

1 10 9615.38 13236.22 2 11 8741.25 7426.98 3 12 8012.82 9642.24 4 14 6868.13 8409.02

a-viii

0

0.000125

0.00025

0.000375

0.0005

0.000625

0.00075

0.000875

0.001

0.001125

0.00125

0.001375

290 292.5 295 297.5 300 302.5 305 307.5 310 312.5 315 317.5 320

Temperature K

Mas

s tra

nsfe

r K

g/hr

20 percent fill level

20 percent fill level closed container

50 percent fill level

50 percent fill level closed container

0

0.000125

0.00025

0.000375

0.0005

0.000625

0.00075

0.000875

0.001

0.001125

0.00125

0.001375

1000 1500 2000 2500 3000 3500

Water vapor saturation pressure (Pa)

Mas

s tra

nsfe

r (K

g/hr

)

20 percent fill level container open20 percent fill level closed container50 percent fill level container open50 percent fill level container closed

The temperature drop is almost uniform with some irregularities seen from time to time. It is observed here that even after a period of 8.5 hours there is a temperature drop of 2 degrees between bulk and the ullage for each temperature level. This drop is consistent for all the temperature levels and even over longer period of time as observed by the results. Thus it can be inferred that even if the bulk and the ullage is allowed to be in equilibrium over a long period of time the temperature difference exists and this leads to heat and mass transfer from and to the bulk liquid all through out the storage time. It is also seen that the fluid with highest degree of heating cools more as compared to the lower heated fluids. Influence of tank fill levels on mass transfer The mass transfer at the interphase is determined using the expression described in the theoretical background through equation 11. The mass diffusivity of air through water as required in the relation is calculated through the empirical correlation suggested by Martínez [31]. The datum coefficients are obtained from the experimental results mentioned by Rohsenow and Hartnett [32]. Figure 7 explains the mass transfer phenomenon taking place across the interphase at various temperatures for two filling levels with and without the top cover. It can be seen that at lower temperature the mass transfer across the interphase is same for an open and a closed container in contrast to its high temperature counter parts where the mass transfer increases by about 50% for the 20% filling level and by about 20% for 50% filling level. This is because of the increase in the saturation pressure of the water vapor in the ullage region for a closed container in comparison to an open container. This becomes clear from Figure 8 which gives the variation of the interphase mass transfer with the saturation pressure of water vapor in the ullage.

Figure 7. Interphase mass transfer variation with temperature

Figure 8. Mass transfer variation with vapor pressure of ullage In terms of the two-phase physics, this increase of mass transfer in the closed container is due to the predominant Stefan phenomenon taking place at the core of the ullage region with convection playing a vital role near the walls. The height of this zone is about 7 cm and 5 cm above the stratified layer for 20% filling level and 50% filling level respectively. This zone is coupled in between the stratified nano layer zone and the top wall zone where heat and mass transfer is predominantly by conduction. This proposition is in good agreement with Malyshev and Zlobin [25]. CONCLUSIONS An experimental study of a closed container partially filled with water has been carried out with different combinations of the filling levels and temperatures. The thermodynamic variables of the filling fluids are mapped over a wide range of parameters. Following are the conclusions that make the present study unique, and seek to contribute further insight and new knowledge in the area of heat and mass transfer of stored liquid –

1. The interphase mass transfer is mainly governed by the concentration gradient across the liquid and gaseous phases irrespective of the container being open or closed.

2. The central zone of mass transfer prevails for a region with maximum wall convection and minimum concentration gradient for an approximate height of 7 cm and 5 cm for 20% filling level and 50% filling level respectively.

3. The interphase mass transfer in a closed container is almost 40% more than its open counterpart.

4. A minimal temperature drop of 2 degrees is observed between bulk and the ullage for each temperature level leading to a continuous heat and

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mass transfer from and to the bulk liquid all through out the storage time. ACKNOWLEDGEMENTS The study was funded by the Australian Research Council. REFERENCES 1. Faltinsen, O., M., Olsen, H., A.,

Abramson, H., N., and Bass, R., L., 1974. “Liquid sloshing in LNG carriers”. DNV Classification and Registry of Shipping, 85.

2. Stuart, B., 2004. “New research into LNG tank sloshing”. Classification News, DNV Corporate Publication, 1: p.6-7.

3. American Bureau of Shipping, 2006. “Strength assessment of membrane-type LNG containment systems under sloshing loads”. Incorporated by Act of Legislature of the State of New York. April

4. Bass, R. L., Bowles, E. B., Trudell, R.W., Navickas, J., Peck, J. C., Yoshimura, N., Endo, S., Pots, B. F. M., 1985. “Modelling Criteria for Scaled LNG Sloshing Experiments”. Transaction of the ASME, 107, June, p. 272-280.

5. Kobayashi, Y., and Tsutsumi, T., 2000. “Heat Transfer Characteristic of the Spherical Tank Skirt of LNG and LH2 Having a Hybrid Construction”. Transactions of the West Japan Society of Naval Architects, 99, p. 395-402.

6. Chen, Q.,S., Wegrzyn, J., Prasad, V., 2004. “Analysis of temperature and pressure changes in liquefied natural gas (LNG) cryogenic tanks”. Cryogenics, 44, p. 701–709.

7. Bates, S. and Morrison, D. S., 1997. “Modelling the behavior of stratified liquid natural gas in storage tanks: a study of the rollover phenomenon”. Int. J. Heat Mass Transfer, 40(8), p. 1875-1884.

8. Boukeffa, D., Boumaza, M., Francois, M., X., and Pellerin, S., 2001. “Experimental and Numerical Analysis of heat losses in a liquid Nitrogen Cryostat.” Applied Thermal Engineering, 21, p. 967-975.

9. Khemis, O., Boumaza, M., Ali, A., M., Francois, M., X., 2003. “Experimental analysis of heat transfers in a cryogenic tank without lateral insulation”. Applied Thermal Engineering, 23, p. 2107–2117.

10. Gradon, L., and Selecki, A., 1977. “Evaporation of a liquid drop immersed in another immiscible liquid. The Case of σc < σd”. International Journal of Heat and Mass Transfer, 20, p. 459-466

11. Raina, G., K., and Wanchoo, R., K., Direct Contact, 1984. “Heat Transfer with Phase Change: Theoretical Expression for Instantaneous Velocity of a Two-Phase Bubble”. International Communication in Heat and Mass Transfer, 11, p.227-237.

12. Raina, G., K., and Grover, P., D., 1985. “Direct Contact Heat Transfer with Phase Change: Theoretical Model Incorporating Sloshing Effects”. AIChE Journal, 31, p. 507-510.

13. Raina, G., K., and Grover, P., D., 1988. “Direct Contact Heat Transfer with Change of Phase: Experimental Technique”. AIChE Journal. 34, p. 1376-1380.

14. Dent, J., C., 1969-1970. “Effect of Vibration on condensation heat transfer to a Horizontal Tube”. Proceedings Institution of Mechanical Engineers, 184 Pt 1, No.5: p.99-105.

15. Tolubinskiy, V., I., Kichigin, A., M., Povsten, S., G., and Moskalenko, A., A., 1980. “Investigation of Forced-Convection Boiling by the Acoustical Method”. Heat Transfer- Soviet Research, Vol. 12, No.2, March-April, p 141-146.

16. Antonenko, V., A., Chistyakov, Yu., G. and Kudritskiy, G., R., 1992. “Vibration-Aided boiling heat transfer”. Heat Transfer- Soviet Research, Vol. 24, No.8, p 1147-1151.

17. Nishiyama, K., Murata, A., and Kajikawa, T., 1986. “The Study of Condensation Heat Transfer Performance Due to Periodic Motion of a Vertical Tube”. Heat Transfer: Japanese Research, Volume 15, Issue 6. p. 32-43.

18. Kravchenko, V., A., and Fedotkin Yu., Y., 1990. “Film Condensation Heat Transfer of a Rapidly Moving Vapor”. Heat Transfer- Soviet Research, Vol. 22, No-7, p.963-969.

19. Fang, G. Ward, C., A., 1999. “Temperature measured close to the interface of an evaporating liquid”. Phys. Rev. 59 p.417–428.

20. Fedorov, V.,I., Luk’yanova, E.,A., 2000. “Filling and storage of cryogenic propellant components cooled below boiling point in rocket tanks at atmospheric pressure”. Chem. Pet. Eng. 36, p. 584–587.

21. Kozyrev, A., V., Sitnikov, A., G., 2001. “Evaporation of a spherical droplet in a moderate pressure gas”. Phys. Usp. 44 p.725–733.

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22. Scurlock, R., 2001. “Low loss Dewars and tanks: liquid evaporation mechanisms and instabilities”. Coldfacts p. 7–18.

23. Krahl, R., Adamo, M., 2004. “A model for two phase flow with evaporation”. Weierstrass-Institut for Angewandte Analysis and Stochastic, Berlin, p. 899.

24. John, H., Lienhard, IV/ John, H., Lienhard, V, 2005. “Heat Transfer Text Book”. Cambridge Press.

25. Malyshev, V., V., and Zlobin, E., P., 1972. “Evaporation of liquid hydrocarbons in heated closed containers”. Translated from Inzheneruo-Fizicheskii Zhurnal, Vol. 23, No. 4, p. 701-708, October,

26. Rakshit, D, Repalle, N, Putta, J and Thiagarajan, K P, 2008. “A numerical study on the impact of filling level on liquid sloshing pressures”. Proceedings 27th Int. Conf. Offshore Mech Arctic Eng, Paper 2008-57557, Estoril, Portugal.

27. Balaji, C., Hölling , M., Herwig H., 2007. “Entropy generation minimization in turbulent mixed convection flows”. International Communications in Heat and Mass Transfer 34, p. 544–552

28. Incropera, F., P., Dewitt, D, P., Bergmen, T., L., Lavine, A., S., 2005. “Introduction to Heat Transfer”. 5th ed, John Willey and Sons.

29. Pioro, L., 1999. “Experimental evaluation of constants for the Rohsenow pool boiling correlation”. International Journal of Heat and Mass Transfer 31, p. 2003-2013

30. Liley, P., E., 1984. “Steam Tables in SI Units, private communication”. School of Mechanical Engineering, Purdue University, West Lafayette, IN, May,

31. Unpublished, Martínez, Herranz, Isidoro, Mass diffusivity data by Polytechnic University of Madrid See also URL http://webserver.dmt.upm.es/~isidoro/

32. Rohsenow, W., M., Hartnett, J., P., Cho, Y., I., 1988. “Hand book of Heat Transfer”. 3rd ed, McGraw-Hill New York.

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Appendix-2

a-xii

Proceedings of the 27th International Conference on Offshore Mechanics and Arctic Engineering OMAE2008

June 15-20, Estoril, Portugal

OMAE2008-57557

A numerical study on the impact of filling level on liquid sloshing pressures

Dibakar Rakshit, Nitin Repalle, Jagadeesh Putta and Krish P Thiagarajan† School of Mechanical Engineering

The University of Western Australia, Crawley, WA, Australia

ABSTRACT Sloshing is a critical phenomenon in Liquefied Natural Gas (LNG) containments in terms of impact loads on the walls of the tank. The peak impact pressures and its location in the tank during sloshing depend on tank fill levels, period and amplitude of oscillation of the tank. In the present study, numerical investigations were carried out on the sloshing motions in a two-dimensional tank at various fill levels due to sway excitation. The two-phase flow was modeled using finite volumes and the Volume-Of-Fluid (VOF) technique using the commercial software FLUENT®. The fill levels were varied from 10 – 95%, and the excitation time periods were varied from 0.8 - 2.8 s for constant sway amplitude of 0.25m (peak-peak) at 1:30 scale. The proposed numerical model gives the average peak pressures and peak pressure locations required for the future experiments to be conducted in a laboratory. INTRODUCTION Sloshing of fluid in containers is considered to be a serious problem in the marine transport industry. There is a renewed interest in sloshing phenomenon because of the current global demand for larger Liquefied Natural Gas (LNG) container sizes and transportation at various fill levels. The study of sloshing in marine containers started with analytical solutions verified by experimentation. Graham and Rodriguez (1952) analyzed sloshing loads using linear potential flow theory. One of the important limitations of this classical method is that linear potential theory can only be used for small motions which are not suitable for higher magnitude complex ship motions. Abramson (1966) in his analytical study mentioned that nonlinear sloshing can be bewildering in its complexity. Faltinsen (1974) and Warnitchai and Pinkaew (1998) have

treated sloshing motions by analytical methods and validated their results with experimental data. However, their approach is based on assumption of ideal, incompressible and non-viscous fluid which may not be an exact representation of the real situation. Faltinsen (1978) modeled nonlinearities in sloshing by introducing artificial damping but found that transient flow behavior was not well predicted. Modern membrane LNG carriers have complex tank shapes which are cumbersome to model using conventional analytical methods. Numerical techniques were thus found to be appealing for modeling sloshing. Ikegawa (1974) and Washizu and Ikegawa (1974) studied nonlinear sloshing in a two-dimensional container using a finite element technique. Peric and Zorn (2005) studied the structural impact of sloshing loads caused by arbitrary motion of the tanks. The experiments were conducted for a range of time periods at a constant fill level of 20%. They quantified various nonlinear factors that affect the performance, stability and structural integrity of the tanks and floating vessels. Razaei and Ketabdari (2007) in their numerical analysis made a similar type of study to simulate sloshing waves at two filling levels in tanks with different motions. Their model sometimes over-predicted the pressure magnitude compared to experiments. A companion paper by some of the authors (von Bergheim and Thiagarajan 2008) examined characteristics of sloshing phenomenon in 2D rectangular tank for a specific filling level by varying amplitudes and frequencies of excitation through CFD methods. From a cross-section of the literature survey presented here, it can be seen that sloshing phenomenon has not been thoroughly studied at

a-xiii

various filling levels ranging from the ballast conditions to higher fill levels to quantify the pressure magnitude with which it impacts the walls of the containment which carries it. The present study is an attempt in this direction where a consolidated analysis has been made to study the phenomenon of sloshing under a range of fill levels from 10 - 95%, and for time periods from 0.8 - 2.8 s at constant sway amplitude of 0.25m (peak-peak). These conditions correspond to a 1:30 scaled model of a prototype tank. The tank is rectangular in geometry, and the two phases are air and water. The study uses the standard k-ε turbulence model to capture the turbulence features of the flow, and a volume of fluid (VOF) model to track the sloshing free surface.

NUMERICAL APPROACH BASICS The computational solution of any fluid dynamics problem requires approach into the conservation of mass and the conservation of momentum (Navier-Stokes) equations. These equations are as follows: The conservation of mass,

0i

i

ux∂

=∂

(1)

The conservation of momentum

ji i ij

j i j j i

uu u upu Ft x x x x x

ρ ρ µ⎡ ⎤⎛ ⎞∂∂ ∂ ∂∂ ∂

+ = − + +⎢ ⎥⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂⎢ ⎥⎝ ⎠⎣ ⎦ (2) where, ρ is the density of the fluid, p is the pressure acting in the direction of flow, F is the net body force acting over the domain and, ui and uj are the components of free stream velocity (u) acting in x-direction and y-direction respectively. Turbulent flows at realistic Reynolds numbers are characterised by a large range of turbulent length and time scales. The Direct Numerical Simulation (DNS) and Large Eddy Simulation (LES) of such turbulent flows would require very high computing power and hence such methods remains very expensive. Hence, it is necessary to reduce the complexity of the simulations by time averaging the turbulence effects considering mean quantities of the flow. With the introduction of time averaging procedure, the instantaneous velocity

'iii uUu += (where, iU and '

iu are mean and fluctuating components of velocity). By substitution of ui in equation (1) and (2) results in average equations (RANS) for incompressible flow as follows:

Continuity equation: 0=∂∂

i

i

xU

(3)

Momentum equation: 1 ji i i

j ijj i j j i

uU U upU Ft x x x x x

ρ ρ µ τρ

⎡ ⎤⎛ ⎞∂∂ ∂ ∂∂ ∂+ = − + + +⎢ ⎥⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂⎢ ⎥⎝ ⎠⎣ ⎦

(4) The additional term ( ijτ ) present in the equation (4) represents the Reynolds stresses and is given by,

' ' 23

jiij i j t ij

j i

uuu u kx x

τ ρ µ ρ δ⎛ ⎞∂∂

= − = + −⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠ (5)

2

tkCµµ ρε

= (6)

In the above Reynolds stress equation tµ is the turbulent viscosity. The Reynolds stresses can be calculated with the knowledge of the turbulent kinetic energy k and dissipation rate ε. The transport equations for k and ε are given as (Launder and Spalding, 1974),

( )/ij ij t k

j j j j

uk k kut x x x x

ρ ρ τ ρε µ µ σ⎡ ⎤∂∂ ∂ ∂ ∂

+ = − + +⎢ ⎥∂ ∂ ∂ ∂ ∂⎢ ⎥⎣ ⎦

(7)

( )2

1 2 /ij ij t

j j j j

uu C Ct x k x k x xε ε εε ε ε ε ερ ρ τ ρ µ µ σ

⎡ ⎤∂∂ ∂ ∂ ∂+ = − + +⎢ ⎥

∂ ∂ ∂ ∂ ∂⎢ ⎥⎣ ⎦ (8) Where, Cµ, Cε1, Cε2, σk and σε in the above equations are the model constants given by Launder et al. (1974). For Standard k- ε model the value of Cµ=0.09, Cε1=1.44, Cε2=1.92, σk =1.00 and σε=1.30. The Volume of Fluid (VOF) method is adopted to capture the free surface effects on sloshing impact pressures. The VOF formulation relies on the fact that two or more fluids (or phases) are not interpenetrating and for each additional phase a new variable, that is, the volume fraction of the corresponding phase is introduced. In each control volume (element), the volume fractions of all phases sum to unity. The fields for all variables and properties are shared by the phases and represent volume-averaged values, as long as the volume fraction of each of the phases is known at each location. Thus the variables and properties in any given cell are either purely representative of one of the phases, or representative of a mixture of the phases, depending upon the volume fraction values. Based on the local volume fraction of the qth fluid (�q), the appropriate variables and properties are assigned to each cell within the domain. The

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1300 mm

900

mm Pressure

measuring nodes

22 21

1

2

3

19

20

1300 mm1300 mm

900

mm Pressure

measuring nodes

22 21

1

2

3

19

20

900

mm Pressure

measuring nodes

22 21

1

2

3

19

20

Pressure measuring

nodes

Pressure measuring

nodes

22 21

1

2

3

19

2022 21

1

2

3

19

20

1

2

3

19

20

19

20

19

20

Figure 2. Typical grid layout with boundary conditions.

Wal

l bou

ndar

y, u

=0,

v=0

Wall boundary, u=0,

v=0

Wall boundary, u=0, v=0

Wall boundary, u=0, v=0

Wal

l bou

ndar

y, u

=0,

v=0

Wall boundary, u=0,

v=0

Wall boundary, u=0, v=0

Wall boundary, u=0, v=0

Wall boundary, u=0,

v=0

Wall boundary, u=0, v=0

Wall boundary, u=0,

v=0W

all boundary, u=0, v=0

Wall boundary, u=0, v=0

Wall boundary, u=0, v=0

tracking of the interfaces between the phases is accomplished by the solution of a continuity equation for the volume fraction of phases. For the qth phase, this equation has the following form,

0q qi

i

ut xα α∂ ∂

+ =∂ ∂

.(9)

Note that the volume fraction equation is not solved for the primary phase, but based on the following relationship

11

n

qqα

=

=∑ . (10)

For an N-phase system, the volume-fraction-averaged density takes on the following form:

1

n

q qq

ρ α ρ=

=∑ (11)

A single momentum equation is solved throughout the domain and the resulting velocity field is shared among the phases. In the case of turbulent quantities, a single set of transport equations is solved and the turbulence variables are shared by the phases throughout the field.

COMPUTATIONAL PROCEDURE The computational model representing the two phase system (i.e. air and water phase, where hw is the depth of water column) is shown in Fig. 1. The chosen dimensions correspond to a 1:30 scaled slice of a prototype tank. A structured moving-grid mesh is used to simulate 2D sloshing flow in a rectangular tank for various fill levels ranging from 10% to 95% and each fill level is simulated for parameters given in Table 1. Pressures are measured at various locations of the right side wall by dividing the wall in to 45mm strips each (Fig. 1). Pressures are monitored using vertex average method for each strip throughout the simulation. A typical grid layout with boundary conditions is depicted in Fig. 2. The dynamic mesh feature is appropriate to model moving free surface boundaries. In this methodology the mesh is updated for each set of successive iteration from the given initial conditions. The sway simulator model considered in numerical simulations is depicted in Fig. 3. Its crank shaft speed as well as piston stroke varies with sway period and amplitude. The time step for the simulations is arrived on the basis of the crank angle. Simulations were carried out for a defined time step over a period of time with error residual in the order of 10-3 maintained for all parameters. The

solver parameters used are: PISO (Pressure Implicit with Splitting of Operators) for pressure velocity coupling, PRESTO for pressure discretization, first order upwind scheme for momentum, turbulent kinetic energy and turbulent dissipation rate discretization. An under relaxation factor of 0.80 and 0.85 are considered for successive momentum and pressure calculation in their next time steps respectively. RESULTS AND DISCUSSION Grid independence To establish the credibility of the present simulation a grid independence study was conducted, along the same lines of von Bergheim and Thiagarajan (2008). The grid independence study was carried out at 10% (90 mm) filling level for pressure location 1 (i.e. 45mm above the tank base) and 29.99 rpm.

Figure 1. Computational domain with pressure locations

Table 1. Test for matrix for each fill level

Time period rpm Amplitude 0.8 74.96 0.25 0.9 66.66 0.25 1.1 54.54 0.25 1.3 46.15 0.25 1.4 42.85 0.25 1.7 35.29 0.25 2 29.99 0.25

2.8 21.42 0.25

Water hw

Air

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Table 2. Grid dependency analysis for average peak pressure

This condition correspond reasonably with the experimental conditions of Peric and Zorn (2005), whose tank was 1200 mm long at a fill level of 120 mm. The grid details are given in Table 2. Numerical simulations were carried out for grid densities of 88x80, 106x80 and 132x100 cells using VOF method along with Standard k-ε model for boundary conditions mentioned earlier. The percentage deviation between numerical and experimental average peak pressure values at pressure location 1 for Grid1, Grid2 and Grid3 are 21.11%, 17.10% and 14.16% respectively. Considering the difference between the geometry of Peric and Zorn (2005) and the present study, such a difference is considered acceptable. The error percentage of Grid1 was felt to be high and was neglected. When comparing Grid2 and Grid 3, Grid 3 seems to be slightly superior then Grid2, but has 56% higher number of cells making it computationally expensive. It was decided to perform all simulations in this study with Grid2 (106 x 80 quadrilateral cells of size 12mm x 11mm). Validation Figure 4 shows the comparison of computed pressure variation with Peric and Zorn (2005) for 8 seconds of sloshing motion corresponding to 10% fill level. The peak

pressures are recorded at the same interval as that of the published results with difference in the magnitude which can be accounted in terms of the variation of geometry, filling level and time period. The minimum pressure attained by the system during this period of sloshing shows that at some point of time the troughs of the pressure harmonics flattens down to values much lesser than those obtained in the experiments. This is because of possible piling up of the liquid at one side leaving the other side dry at some instants of time. This effect can be seen in the free surface contours shown in Fig. 5 at times corresponding to the zero-pressure regions. Influence of tank fill levels on peak pressures and pressure locations

Figure 4. Comparison of pressure variation at location1 corresponding to 10% fill level, 0.25m (pk-pk) sway amplitude at 30 rpm with Peric & Zorn(2005).

A typical time trace of pressure measured at various points on the wall is shown in Fig. 6. The first cycle corresponds to transients as the flow builds up to a fully developed condition. The average peak pressure is defined as arithmetic average of all peak pressures over the simulated time period (typically eight cycles) neglecting initial noise present in the pressure history plots. The pressure peaks considered in calculating the average are highlighted in Fig. 6. Further, peak pressure locations are described as the distance from the bottom of the tank to centre of the strip. For roof impact, the position of peak pressure is measured from the top right corner. The distance along the ceiling is appended to the ordinate length and presented in graphs below.

Grid type

Grid density

Avg. peak pressure,

Pa (Numerical

)

Avg. peak

Pressure, Pa

(Exp.)

% Devi

-ation

Grid1

88x80 2322 21.11

Grid2

106x80 2438 17.10

Grid3

132x100 2527

2944

14.16

Figure 5. Free surface contours at 10% fill level for30 rpm at time instants 1.5, 3.5, 5.5 and 7.5

θ

A

ω

L Piston Axis

Figure 3. Arrangement for generation of Moving-grid

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Typical graphs showing the variation of peak pressures and pressure locations at various filling levels corresponding to time periods of 2.0, 1.4 and 1.1s (29.99, 42.85 and 46.15 rpm) are depicted in Fig. 7. The period T=2.0s is equivalent to resonance period of tank at 20% fill level. The graph corresponding to 2.0s period clearly shows that maximum pressure of 17.5KPa occurring at 20% fill level at a distance of 425mm from the tank bottom.

This can be explained in terms of natural sloshing period of the tank at that particular filling level. Since a system with 20% filling level has a natural sloshing period of 2.02sec corresponding to 29.99 rpm it is obvious that the peak pressure of the system will be attained at this filling level. Similarly, it can be seen that for period T=1.4s which represents the tank resonance period at 60% fill level a peak pressure of 18.5KPa occurs at a distance of 920mm (i.e. maximum pressure is occurring near the tank top, since the height of tank was 900mm only) from the bottom of the tank. Running on the same lines the graph corresponding to the time period of T=1.3s records a peak pressure value of 22.9KPa at a distance of 920mm from the bottom and this time period corresponds to tank resonance period at 80% fill level. However, graph at T=1.3s shows some inconsistency in peak pressure location, this may be due to using of under-relaxation factor in numerical simulations as mentioned earlier. On the whole the proposed numerical model predicts the peak pressures as well as resonance periods of the tank for ballast condition to nearly full condition. Figure 8 gives the peak pressure values with their corresponding resonating frequencies and their respective locations at which the peaks occur. It can be seen that for the frequencies studied there is a sharp increase in the pressure location value as we move from 0.5 Hz to 0.7 Hz. This rise is mainly in the region corresponding to those filling levels which are less than 50%. As the filling level increases beyond 50% the roof top phenomenon restricts the pressure to be impacting at regions which are about 65 mm from the corner on the roof top.

Influence of tank fill level and oscillation frequency on average peak pressures The variation of average peak pressure with time period/frequency of oscillation and fill level is depicted in Fig.9. It can be seen from the graph that the peak pressure for 0.36 Hz is about 8.4 kPa. Here 0.36 Hz corresponds to the natural sloshing frequency of 10% fill level but the peak pressures are recorded at 95 %. fill level at the tank bottom instead of 10% fill level.

0

5000

10000

15000

20000

25000

0.5 0.7 0.77 0.90

200

400

600

800

1000

1200

Averaged Peak Pressure (Pa) Location (mm) Figure 8. Variation of peak pressure and pressure location for various filling levels with corresponding resonating frequency

Figure 6. Peak pressure recording of each sloshed cycle used for averaging

0

2000

4000

6000

8000

10000

12000

0 1 2 3 4 5 6 7 8 9 10

Time (s)

Pres

sure

Pressure Location (mm

)

Peak

Pre

ssur

e (P

a)

05000

100001500020000

0 20 40 60 80 1000

200

400

600

0

5000

10000

15000

20000

0 20 40 60 80 1000

200

400

600

800

1000

1200

Figure 7. Variation of peak pressure and pressure location for various filling levels at T= 2.0, 1.4 and 1.1s and sway amplitude of 0.25m (pk-pk).

0

5000

10000

15000

20000

25000

0 20 40 60 80 1000

200

400

600

800

1000

1200

Peak Pressure Peak Pressure Location

Filling Level %

a-xvii

This may be because of the hydro--static pressure at 95% fill level is greater than the impacting pressure of 10% fill level. As we increase the frequency to 0.5 Hz we register a peak pressure of 17.5 kPa obtained at 20% fill level as 0.5 Hz corresponds to the natural sloshing of 20% fill level. The frequencies 0.6 Hz, 0.7 Hz and 0.77 Hz correspond to the natural frequencies of 50%, 60% and 80% filling level respectively. We register corresponding peak pressures of 18.9 kPa, 19.1 kPa and 23kPa at those filling levels. The dip in the curve at 0.9 Hz is evident because it does not correspond to any of the natural frequencies of the system. Again the value of the impact pressure shoots up as the frequency increases to 1.1 Hz as this frequency corresponds to natural frequency of 50% to 95%. The second peak of 30 kPa in the below figure is obtained at the 80% fill level on the tank roof. Influence of tank oscillation frequency and fill level on peak pressure location Figure 10 depicts the variation of normalized peak pressure location (zp/h, where Zp=peak pressure location from tank bottom and h=height of the tank) with fill level and frequency of oscillation of the tank. As the frequency of oscillations and filling levels in the tank increases the peak pressure location in the tank shifts towards the tank roof (z/h= 0.02 to 1.07). The shifting of peak pressure location to tank top starts at 50% fill level corresponding to time period 1.70s. This can be better explained though hydraulic jump, which can form due to change of flow in the tank from supercritical to subcritical stage. The hydraulic jump in the tank raises the impact pressures (14.08 kPa) in tank as well as shifts the peak pressure location towards the tank roof. In the present study a peak pressure location is observed at z/h=1.07 or 965mm corresponding to 0.8 fractional fill level at T=0.90s. The maximum pressure observed was 30.30KPa. However for 95% fill level the peak pressure location is observed at 20 mm above the tank bottom in contrast to roof top due to dominance of hydrostatic pressure when compared with sloshing impact pressure. One more important observation found in this study was shifting of peak pressure location towards the tank roof for 50%, 60%, and 80% fill levels from period 1.71s onwards in its decreasing trend. This is because of the overlap between the higher frequency and the natural frequency of sloshing at higher filling levels (i.e. >50%). The maximum peak pressure location observed in this study was 965mm (i.e. 65 mm from the top right corner of the tank). This can be better clearly visualized through Fig. 11 and Fig. 12 given by free surface contours along

with their corresponding pressure and velocity contours at 60% fill level for period of 1.4s.

A close observation of the figure shows that the liquid hits the roof top with corresponding location in pressure contours presenting maximum pressure (Fig. 12). Similar plots are given in Fig. 13 for 80% fill level for a period of 1.3s. This figure also exhibits the formation of hydraulic jump and toping of wave after hitting of the tank roof. The velocity contours (Fig. 12) show minimum velocity with strong recirculation taking place at 60% filling level when sloshed at a frequency of 0.71 Hz. The 80% filling level does not have the strong recirculation when compared with 60% filling level.

Figure 10. Variation of peak pressure location with frequency and fill level

Fill level fraction

Freq

uenc

y (H

z)

Peak

pre

ssur

e lo

catio

n fr

om

tank

bot

tom

(zp/

h)

0.1 0.2 0.4 0.5 0.6 0.8 0.950.36

0.50

0.59

0.71

0.77

0.91

1.11

1.251.00-1.100.90-1.000.80-0.900.70-0.800.60-0.700.50-0.600.40-0.500.30-0.400.20-0.300.10-0.200.00-0.10

Figure 9. Variation of average peak pressure with fill level and frequency (Hz)

Ave

rage

pea

k pr

essu

re (K

Pa)

0.1

0.4

0.6

0.95

0.36

0.71

1.1102468

10121416182022242628303230-32

28-3026-2824-2622-2420-2218-2016-1814-1612-1410-128-106-84-62-40-2

Fractional fill level

Frequency (Hz)

a-xviii

CONCLUSIONS The following are the conclusions drawn from the present study-

• Simulated pressure time series compares

reasonably with the experimental and numerical values of Peric and Zorn (2005). Discrepancies are attributed to difference in geometries and fill levels.

• Numerical model predicts resonant and non-resonant peak pressure for all fill levels considered in this study.

• The dynamics of the flow is well predicted as the visualizations show good agreement with numerical model for some distinct flow features.

• Roof impact of the water initiates at 50% fill level corresponding to a frequency of T=1.7s.

• The maximum peak pressure (30.30 KPa) was observed on the ceiling at 65 mm from the top right corner

(z/h=1.07). This corresponds to 0.8 fractional fill level for sloshing time period of T=0.90s.

• The position and magnitude of maximum pressure at various fill levels may be used as a guide for selection and positioning of pressure transducers in a laboratory tank set up.

ACKNOWLEDGEMENTS The study was funded by the Western Australian Energy Research Alliance and Chevron Energy Technology Company, through the Advanced Energy Solutions (AES) Joint Venture. The authors acknowledge the support of Mr Tim Finnigan and Dr. Robert Seah from Chevron ETC, and Dr. Dominique Roddier from Marine Innovation & Technology. REFERENCES

Abramson, H. N, 1966, “The dynamic behavior of liquids in moving containers with applications to space vehicle technology,” National Aeronautics and Space Administration, Technical Report SP-106.

Faltinsen, O M 1974, “A nonlinear theory of sloshing in rectangular tanks,” J. Ship Research, 18, pp. 224–241.

Faltinsen, O M. A numerical nonlinear method of sloshing in tanks with 2D flow J. Ship Research, 22:193–202, 1978.

Graham, E.W., and Rodriguez, A.M., 1952, “The characteristics of fuel motion which affect airplane dynamics,” J. Applied Mechanics, pp. 381–388.

Hirt, C.W., and Nichols, B.D., 1981, “Volume of fluid method fluid (VOF) method for

Figure 13. Plots showing surface and pressure contours at 80% fill level for T=1.3s

Figure 11. Surface and pressures contours plots at 60% fill level for T=1.4s

Figure 12. Velocity contours at 60% fill level for

a-xix

the dynamics of free boundaries,” J. Computational Physics, 39, pp. 201-225.

Ikegawa, M., 1974, “Finite element analysis of fluid motion in a container,” UAH Press, Hunstsville.

Launder, B.E., and Spalding. D.B., 1974 “The numerical calculation of turbulent flows,” Computer Methods in Applied Mechanics and Engineering, 3 (2), pp. 269-289.

Peric, M., and Zorn, T., 2005, “Simulation of sloshing loads on moving tanks,” Proceedings 24th Intl. Conf. Offshore Mechanics and Arctic Engineering, Halkidiki, Greece, pp. 1-10.

von Bergheim, P and Thiagarajan, K P, 2008, “The air-water sloshing problem: parametric studies on excitation magnitude and frequency” (under review) Proceedings 27th Intl. Conf. Offshore Mechanics and Arctic Eng, Lisbon Portugal.

Washizu, K., and Ikegawa, M., 1974, “Some applications of finite element method to fluid mechanics,” Theoretical and Applied Mechanics, 22, pp. 143-154.

Warnitchai, P., and Pinkaew, T., 1998, “ Modelling of liquid sloshing in rectangular tanks with flow- dampening devices, Engineering Structures, 20, 7, pp. 593-600.

Fifth Australian Conference on Laser Diagnostics in Fluid Mechanics and Combustion The University of Western Australia 3-4 December 2008

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Appendix-3

Fifth Australian Conference on Laser Diagnostics in Fluid Mechanics and Combustion The University of Western Australia 3-4 December 2008

a-xxi

Application of nano-paints for hydrodynamic pressure visualization

D Rakshit, S Dev, and K P Thiagarajan

School of Mechanical Engineering, The University of Western Australia,

35, Stirling Highway, Crawley, WA 6009ABSTRACT Non-intrusive methods for obtaining pressure maps in a fluid flow are attractive because of enhanced accuracy as well as providing simultaneous multipoint measurements. Nano material based paints change colour in response to surface stress, when illuminated at certain wavelengths. While these techniques have been tested on aerodynamic applications, recently Pressure Sensitive Paints (PSP) based on Nano Structured Powder are being developed for application to marine industries. The present paper will elucidate various aspects of non-oxygenic PSP system and its testing procedure. The nano formulations explaining the new system along with the perquisite hardware requirements viz. camera, lighting and data acquisition systems are discussed. The technique is currently being deployed at the author’s laboratory. Up-to-date results will be presented at the conference. 1. INTRODUCTION Convenient, reliable and accurate methods for determining surface pressures, particularly of marine based applications, have been the focus of many researchers. As far as marine applications are concerned the transportation and storage of Liquefied Natural Gas (LNG) requires exigent monitoring of slamming pressure. Sloshing-induced impact pressures can make the surface of the container eroded over a period of time [Lee et al. 2004; Arswendy and Moan, 2006]. Traditional measurement techniques for acquiring slamming pressure distributions on models have utilized embedded arrays of pressure taps. Hundreds of pressure taps [Mikelis and Journee, 1984; Bunnik and Hujismans, 2007] are required to get the complete pressure distribution on a model’s surface. This requires considerable construction and set up time while producing data with limited spatial resolution. Sophisticated interpolation procedures are employed to generate the distribution between pressure tap locations. Recently, various non-intrusive pressure measurement techniques have been developed to measure fluid dynamic pressure. Liquid Crystals Measurement Technique [Fujisawa et al. 2003; Bonnet et al. 1989], Pressure Sensitive Paints (PSP) and Temperature Sensitive Paints (TSP) are some of them. The most easily accessible

Thermochromic Liquid Crystals (TLCs) have been successfully used in many non-intrusive heat transfer and fluid mechanics studies. A breakthrough in the measurement techniques came when PSP was introduced for an aerodynamic testing [McLachlan et al. 1995]. Pressure sensitive paints are generally luminescent coatings, applied on the surfaces whose pressures (static or dynamic) are to be measured. This is an optical method that can resolve colour intensities resulting from various pressure values on the surface. Traditionally PSP is based on oxygen quenching luminescence having the measurement dependent on the partial pressure of the oxygen molecules. Therefore oxygen must be present at a density that is proportional to the slamming pressure to be measured. Hence there is a constraint of oxygenic environment to make PSP function [Hamner and Joseph, 2004] which sometimes is not possible to follow up. In recent times new types of pressure sensitive luminescent coatings have been developed [Hamner and Fontaine, 2003]. These are basically nano based materials that are designed at microscopic scale to exhibit a specific macroscopic property. These coatings sense changes in pressure directly, rather than using oxygen as did previous PSPs. Therefore, these coatings can operate in virtually any test environment, e.g. water, nitrogen or air. The present paper will give an outlook of some of the salient features of these techniques with details of its principle and functionality. This will be followed by a brief introduction of PSP utilisation in studying marine sloshing. 2. PRINCIPLE The operation of the pressure sensitive paint is based on the luminescent principle. The absorption of light energy is approximated by the Beer-Lambert law, which depends on the illumination intensity, luminophore depth, effective absorption, cross-sectional area and luminophore concentration (Figure 1).

a-xxii

Fig1. Basic Pressure Sensitive Paint System in combination with its principle

(Courtesy- Innovative Scientific Solutions Inc.)

The absorption of energy takes the luminophore molecules to higher energy levels of the shells. The molecules returning to their ground state produce either fluorescence or phosphorescence or quench. The quenching process corresponds to the phenomenon of losing luminescence by transferring their absorbed energy to the oxygen molecule in terms of vibrational energy. The Pressure Sensitive Paint uses three modes for measuring the pressure. They are- Intensity based methods Time based methods Frequency based methods

Generally, measurement is done with the intensity based methods that requires a full-field optical detector and a monochromatic light source. The intensity based methods require two readings - one at a reference point of constant pressure (wind-off condition), and one at an unknown point (wind-on condition). The intensity ratio Iref/I is inversely proportional to the fluid pressure which is to be measured. By exploiting the dynamic quenching of luminescence by oxygen, Stern-Volmer relationship [McLachlan and Bell] [ ( ) ( )ref refI I A T B T P P= + ] becomes the base to correlate the light intensity with the pressure measurement. This dynamic quenching is a collisional process. Therefore, oxygen must be able to permeate throughout the paint in order to “collide” with the chromophores present and hence quench their luminescence. It is worth mentioning here that, chromophores that are pressure sensitive are also temperature sensitive. Hence conventional PSP results are affected by the temperature fluctuations which lead to erroneous pressure predictions. New formulations of PSP under development can give actual pressure data acquired using PSP without any temperature effects. By the early 2000’s alternative formulations that were true pressure sensors had been formulated. These formulations were designed to be essentially “plug and play” components in existing “paint systems” so that users could select a PSP

formulation appropriate to their test requirements without having to make additional or extensive changes to other components, such as cameras, lights, etc. These nano based paints being developed involves photochemical reactions that can be optically detected allowing direct acquisition of quantitative pressure measurements as a function of light emission intensity or as a function of excited state life time. However these new materials involve distance dependant photochemical reactions that emit multiple wavelengths of light rather than dynamic quenching by oxygen. This forms the basis of this new technology having Nano Based PSP as the pressure measurement technique. 3. BASIC PAINT SYSTEMS 3.1 Paints and excitation sources All PSP have to be excited at a certain wavelength. It is thus preferable to describe the paints together with the sources used to excite them. These can be broken down effectively into those paints that are excited in the ultraviolet and those excited in the visible. The excitation and detection systems must be spectrally separated. 3.1.1 Paints excited in the ultraviolet range Paints excited in the ultraviolet use molecules derived from pyrene or porphyrins of palladium or platinum. Pyrene based paints have a continuous emission spectrum, while those using porphyrins exhibit a line spectrum [Sant and Merienne, 2005]. They are invariably used with a reference dye for measuring the excitation intensity. This dye is generally an oxysulphide of doped rare earth. The main difficulty is to get a good separation of the PSP dye spectrum from that of the reference dye. Possible excitation sources are: – Mercury vapor or mercury-xenon vapor lamps. The power of these lamps is moderate, but adequate for small facilities. In larger installations, on the other hand, the exposure times are long (30 s). The stability is good (deviation is less than 0.5% per hour), but is not enough to do away with the reference dye. – Xenon flash lamps. They pulse at a frequency of about 30 Hz. The power may be high, but a reference dye is needed because of the inadequate stability. – Mercury–Xenon lamp for UV curing. They are extremely powerful (100 × standard Hg lamp) and nearly perfectly stable (stability below 0.1%). This is a perfect solution if the wavelength fits the requirement for the used PSP. The only one drawback is the limited choice of devoted excitation filters.

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– Nitrogen lasers. These are pulsed lasers with a maximum frequency of 100 Hz (except one Russian laser that goes up to 1000 Hz). They have high power and are very stable. They are used in systems following the lifetime method All these UV light sources are used in connection with optical fibers, which are expensive specially for industrial testing where the typical length is 20 m. 3.1.2. Paints excited in the visible range Paints excited in the visible are porphyrins or derivative molecule of ruthenium. The temperature sensitivity is generally high – of the order of 1%/K – so they are used with in situ calibration. The visible sources used for excitation are lamps and lasers. Light-emitting diodes (LED) are coming into greater and greater use because they are very stable and relatively cheap. The design of the new coatings turned to the identification of fluorescent compounds that exhibited the desired distance dependent photochemical reactions, e.g. compounds involved in the formation of excited-state dimers (excimers) or excited-state complexes (exciplexes) and those involved in fluorescence resonance energy transfer (FRET) [Hamner, 2008]. The first formulation of these pressure sensitive coatings included a system of anthracene and N,N-dimethylaniline (DMA) that forms an exciplex which was incorporated into polystyrene. This precludes use of the anthracene/aniline exciplex is excited with UV light, typically centred at about 365 nm. 3.2 Instrumentation There are two types of instrumentation: cameras and scanning systems. Cameras are scientific cameras that are cooled to minimize the reading noise. A few attempts have been made to use ordinary cameras, but they required high precision – of the order of 0.1% (100 Pa) – can only be achieved with appropriate systems (digitization better than 14 bits). 4. Experimental Details The current application of PSP is to study spatial map of pressure distribution that occurs during a sloshing event. As seen below in figure 2 the experimental setup is composed of a number of separate elements. 4.1 Tank set up: The above experimental setup consists of a rectangular tank filled with water sloshing at fixed amplitude of 50mm. The side of the tank which is subjected to the measurement of PSP is made up of glass which allows ultra violet

light of wavelength range of 350-400 nm to pass through it. The cross validation of the PSP will be done using a number of pressure transducers (Kulite) on the opposite face of the PSP applied glass.

Fig 2. Schematic figure of Experimental setup

(NTS)

4.2 Excitation Light Source: The core of the illumination sub-component is the light source. The primary consideration for the light source is the constant or repeatedly constant delivery of uniform illumination, i.e. uniform both spatially and temporally. A PSP rule-of-thumb is less than 0.1% fluctuation in intensity. A 200 Watt mercury bulb operated by stable AC supply will be used as a light source. With high intensity lamps, to prevent fracture in the optical path of the light, filters or focusing lenses are used. On the basis of the requirement of the photochemical system (PSP formulation dependant) which is excited with long-wavelength UV light, it would be advantageous to choose a mercury or mercury-xenon arc lamp for use in these lights. The spectrum for a mercury-xenon arc lamp is shown below. The relative pressures of the gases in the lamp and operation at or off the design point may change the relative intensity of the lamp’s output at different wavelengths.

Fig 3. Mercury Lamp Wavelength Characteristics

(Courtesy Cairn Research Limited)

However, the 365 nm UV line of mercury is one of the strongest outputs for these arc lamps and is compatible with the PSP selected. The use of mirrors, cold or hot mirrors depending on the set-up, to dump heat from their high power output will

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be necessary. Cold mirrors can be used to “dump” heat in the form of IR waves while reflecting other wavelengths of light, conversely hot mirrors reflect IR wavelengths while transmitting the remaining wavelengths of light. The last pieces of optics required will be the appropriate filter, possibly using UG-1 Schott glass [Geschäftsbereich Optik, 2000], and a focusing lens to spread out the radiant light over the test article. 4.3 Nano Based PSP preparation: The general method for PSP preparation is the PSP components (binder, dye(s), etc.) are dispersed in a volatile solvent (or a mixture of solvents) and then the solution is sprayed on the substrate. The first formulation of these pressure sensitive coatings includes a system of anthracene and N,N-dimethylaniline (DMA) that forms an exciplex which was incorporated into polystyrene. The molecular structure of these luminophores was modified to allow their direct incorporation into the polymer through the process of radical co-polymerization. A ratio of the luminophores (energy donor to energy acceptor molecules) is co-polymerized onto the polystyrene in a way that promotes the formation of the exciplex. The resulting coating is tested for pressure sensitivity. 4.4 Photo-detector: We are intending to use a 14 bit CCD digital camera to detect the paint emission from the model. These cameras can provide l000 time’s 1018 pixel resolution and 14-bit intensity resolution. To separate the paint emission from excitation light, a long wave pass or a band pass filter is placed on the camera object lens. The camera is operated in what is often called the "shot noise limited" regime [Iriel et al. 2008]. This takes advantage of the physics we know about CCD cameras to minimize the overall noise level. But essentially what it also does is it constrains us to work with what appears to be a small fluctuation on top of a very large bias. With regard to the present experiment PSP uses emission at 2 wavelengths, one that decreases and one that actually increases with changes in pressure (Exciplex and FRET based), hence the experiments should be setup intelligently. Although experiments will be operating in the "shot noise limited" regime to minimize noise but we may want to back off from 3/4 full well to accommodate the wavelength that increases in emission intensity with increases in pressure. 4.5 Data acquisition: Data acquisition system is being constructed around a personal computer of Windows environment. We are developing dedicated data acquisition software, which can set up camera parameters, control a shutter for illumination light, acquire images from a CCD camera through a frame graver board, and store

them on the PC hard drive. Simple image processing such as ratioing, averaging, and filtering can be carried out on site in an interactive mode. 5. Validation with CFD In the present study of sloshing to capture the slamming pressure, from the CFD analysis [Rakshit et al. 2008] it was understood that the exact location of peak impact pressure can never be obtained with more “traditional” instrumentation such as pressure transducers due to their discrete nature. The most important limitation was that the time period for which the impact pressure measurement exists is of the order of milliseconds which can never be measured accurately both in the temporal and spatial distribution. This gives birth to the requirement for a technique which reports the globalized values of pressure at each location of the wall such as the PSPs measurement technique. On the basis of initial analysis being done using CFD as a tool we obtained results for a swaying tank giving an indication for the various impacting location across the tank walls. Figure 4 shows the pressure distribution across the wall of a swaying tank having amplitude of 25 cm at 75 rpm. 6. Limitation The greatest disadvantage of PSP is the long time elapse, which can occur between the reference and data images resulting in significant changes in contamination of paint. Hence, the most important criteria in this respect are the durable physical properties of the paint such as hardness, abrasion resistance, surface finish, and thickness with correct

Fig 4. Impact Pressure on the walls of tank are to be

quantified through the Pressure Sensitive Paints

photo-physical properties like absorption and emission properties, pressure and temperature sensitivity, time response, quantum efficiency, and photo-stability.

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7. Conclusion Until recently, surface pressure measurements were made by pressure taps in any hydrodynamic environment. Pressure measurements with Pressure-Sensitive Paint are now a reality and have been shown to be successful in many tests. The enthusiasm for this technique is prompted by the considerable savings offered in terms of model instrumentation cost and model construction time, while the wealth of information that can be extracted from the images makes it a preferential investigation tool for complex flows. It has been shown in this article that PSP is relatively simple to use, but requires many precautions if a high level of precision is desired. Developments are in progress to improve the accuracy and extend the field of application. There is no doubt that Nano based PSP will be used routinely in the near future on a broad variety of configurations involving a non oxygenic environment. ACKNOWLEDGMENTS The study was funded by the Australian Research Council. The authors acknowledge the support of Dr. Marvine Hamner, Head, Lea Tech, LLC, in terms of her continuous feedbacks regarding the development of the experimental setup. REFERENCES

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