entropy generatioin study for bubble separation in pool boiling

ENTROPY GENERATION STUDY FOR BUBBLE SEPARATION

IN POOL BOILING

A Project

Presented to the

Faculty of

California State Polytechnic University, Pomona

In Partial Fulfillment

Of the Requirements for the Degree

Master of Science

Mechanical Engineering

Jeffrey William Schultz

ACKNOWLEDGEMENTS

I would like to start by thanking Dr. Hamed Khalkhali for his continued

support throughout the investigation. This work would not have been possible

without his suggestion of the problem statement. His advice and push to look at

the problem in a different light has been greatly appreciated and helped drive this

investigation to a successful conclusion.

To my wife Melissa goes my greatest appreciation for her continued

support throughout my work towards a Master of Science degree and especially

during my work on this investigation. She has helped make an extremely busy

schedule over the last two years manageable and enjoyable.

Additionally I would like to thank my parents Nancy and Charlie, sister

Kristen, mother and father in-law Peggy and Ed, sister in-law Margaret, and

brother in-law Mark for their continued support and motivation.

I would also like to thank Dr Rajesh Pendekanti and Dr Keshava Datta for

providing me with the initial motivation to pursue a Master of Science degree in

Mechanical Engineering. Throughout my progress in the program at California

State Polytechnic University, Pomona, they have provided me with advice,

support, and flexibility at work to allow me to pursue this degree.

ABSTRACT

The current entropy generation rate study of spherical bubbles undergoing

growth in nucleate pool boiling produces a novel correlation for predicting bubble

departure radii. Two models for entropy generation rate in spherical bubbles are

developed by modeling the work performed by a bubble as that of a

thermodynamic system, and as a function of the net force acting on the bubble

and the rate of bubble grow. While the derived entropy generation rate equations

fail to support the hypothesis presented in this paper, one of the two models

leads to a novel correlation which predicts published experimental data within

TABLE OF CONTENTS

Signature Page ...................................................................................................... ii

Acknowledgements .............................................................................................. iii

Abstract ................................................................................................................ iv

Table of Contents ................................................................................................. v

List of Tables ....................................................................................................... vii

List of Figures ....................................................................................................... ix

Nomenclature ....................................................................................................... xi

Introduction ........................................................................................................... 1

Previous Work ........................................................................................ 1

Problem Statement ................................................................................. 6

Methodology ........................................................................................... 6

General Assumptions ............................................................................. 8

Entropy Generation Rate Study (Pressure-Volume Method) .............................. 10

Derivation of Heat Transfer Rate .......................................................... 10

Derivation of Entropy Generation Rate ................................................. 16

Analysis of Second Order, Non-Linear Differential Equation ................ 19

Confirmation of Correlation ................................................................... 60

Summary .............................................................................................. 64

Entropy Generation Rate Study (Net Force Method) .......................................... 67

Derivation of Heat Transfer Rate .......................................................... 67

Derivation of Entropy Generation Rate ................................................. 74

Analysis of Net Force Correlation ......................................................... 80

Summary .............................................................................................. 85

Conclusions ........................................................................................................ 86

Bibliography ........................................................................................................ 90

Appendix A: Derivation of Entropy Generation Rate (Pressure Method) ............ 94

Appendix B: Defining the General Solution ...................................................... 109

Appendix C: Derivation of Entropy Generatoin Rate (Net Force Method) ........ 114

Appendix D: MatLab Programs ......................................................................... 128

LIST OF TABLES

Table 1. Departure Diameter Correlations .................................................... 1

Table 2. Forces acting on a bubble prior to separation. ................................ 5

Table 3. Values of C for the General Solution Derived from Rayleigh Equation with Experimental Data of (Van Stralen, Cole, Sluyter, & Sohal, 1975). ................................................................ 24

Table 4. Values of D for the General Solution Derived from Rayleigh Equation with Experimental Data of (Van Stralen, Cole, Sluyter, & Sohal, 1975). ................................................................ 25

Table 5. Error Analysis of Predicted Departure Radii based on Rayleigh Based General Solution. ................................................ 26

Table 6. Error Analysis of Predicted Departure Radii based on Rayleigh Based Modified General Solution. .................................. 30

Table 7. Values of C for the General Solution Derived Using Plesset-Zwick Equation with Experimental Data of (Van Stralen, Cole, Sluyter, & Sohal, 1975). ....................................................... 37

Table 8. Values of D for the General Solution Derived Using Plesset-Zwick Equation with Experimental Data of (Van Stralen, Cole, Sluyter, & Sohal, 1975). ....................................................... 38

Table 9. Error Analysis of Predicted Departure Radii based on Plesset-Zwick Based General Solution. ........................................ 39

Table 10. Error Analysis of Predicted Departure Radii based on Plesset-Zwick Based Modified General Solution. .......................... 43

Table 11. Values of C for the General Solution Derived Using MRG Equation with Experimental Data of (Van Stralen, Cole, Sluyter, & Sohal, 1975). ................................................................ 51

Table 12. Values of D for the General Solution Derived Using MRG Equation with Experimental Data of (Van Stralen, Cole, Sluyter, & Sohal, 1975). ................................................................ 52

Table 13. Error Analysis of Predicted Departure Radii based on MRG Based General Solution. ............................................................... 53

Table 14. Error Analysis of Predicted Departure Radii based on MRG Based Modified General Solution. ................................................. 58

Table 15. Comparison of Derived Equation with Experimental Data of (Cole & Shulman, 1966b) .............................................................. 61

Table 16. Comparison of Derived Equation with Experimental Data of (Ellion, 1954). ................................................................................ 63

Table 17. Alternative dimensionless scaling factors calculated from bubble departure correlations. ...................................................... 70

Table 18. Net Force Derivatives. ................................................................... 78

Table 19. Vapor Pressure Derivatives .......................................................... 80

Table 20. MRG Equation Derivatives. ........................................................... 81

LIST OF FIGURES

Figure 1. Forces Acting on a Bubble. ............................................................. 3

Figure 2. Forces Acting on Bubble. (A) Buoyancy Force, (B) Excess Pressure Force, (C) Inertia Force, (D) Surface Tension Force), (E) Drag Force. ................................................................... 4

Figure 3. Balance of Energy for First Law of Thermodynamics ...................... 7

Figure 5. Comparison of Predicted Departure Radii from Rayleigh Based Equation and Experimental Departure Radii. ..................... 27

Figure 6. Error Plot of Predicted Departure Radii from Rayleigh Based Equation. ....................................................................................... 28

Figure 7. Comparison of Predicted Departure Radii from Modified Rayleigh Based Equation with Experimental Departure Radii. ...... 31

Figure 8. Error Plot of Predicted Departure Radii using Rayleigh Based Modified Equation. ............................................................. 32

Figure 9. Entropy Generation Rate vs. Bubble Radius for Experimental Data Obtained from (Van Stralen, Cole, Sluyter, & Sohal, 1975) (A=1.924969, B=0.267915). ................................................ 35

Figure 10. Comparison of Predicted Departure Radii from Plesset-Zwick Based Equation with Experimental Departure Radii. .......... 40

Figure 11. Error Plot of Predicted Departure Radii using Plesset-Zwick Based Equation. ............................................................................ 41

Figure 12. Comparison of Predicted Departure Radii from Modified Plesset-Zwick Based Equation with Experimental Departure Radii. ............................................................................................. 44

Figure 13. Error Plot of Predicted Departure Radii using Plesset-Zwick Based Modified Equation. ............................................................. 45

Figure 14. Residual Value vs. Time for Experimental Data Obtained from (Van Stralen, Cole, Sluyter, & Sohal, 1975) (A=1.924969, B=0.267915). .......................................................... 48

Figure 15. Comparison of Predicted Departure Radii from MRG Based Equation with Experimental Departure Radii. ................................ 54

Figure 16. Error Plot of Predicted Departure Radii using MRG Based Equation ........................................................................................ 55

Figure 17. Comparison of Predicted Departure Radii from Modified MRG Based Equation with Experimental Departure Radii. ........... 59

Figure 18. Error Plot of Predicted Departure Radii using MRG Based Modified Equation ......................................................................... 59

Figure 19. Comparison of Predicted Departure Radii with Experimental Data of (Cole & Shulman, 1966b). ................................................ 62

Figure 20. Comparison of Predicted Maximum Radii with Experimental Data of (Ellion, 1954). ................................................................... 64

Figure 21. Bubble Dimensions. ...................................................................... 69

Figure 22. Entropy Generation Rate vs. Bubble Radius for Experimental Data Obtained from (Van Stralen, Cole, Sluyter, & Sohal, 1975) (A=1.924969, B=0.267915). ................................................ 83

NOMENCLATURE

General Symbols

𝐴 parameter for Rayleigh Equation

𝐴𝑟 Archimedes number

𝑏 constant for Plesset-Zwick Equation

𝐵 parameter for Plesset-Zwick Equation

𝑐𝑝 specific heat at constant pressure [J/kg-K]

𝐶 constant of general solution

𝑑 diameter [m]

𝐷 diameter [m] or constant of general solution

𝑒 internal energy per unit mass [J/kg]

𝐸 internal energy [J]

𝐸 energy change rage [W]

𝐹 force [N]

𝐹𝑏 buoyant force [N]

𝐹𝐷 drag force [N]

𝐹𝑖 inertia force [N]

𝐹𝑛𝑒𝑡 net force[N]

𝐹𝑝 excess pressure force [N]

𝐹𝜍 surface tension force [N]

𝑔 gravitational acceleration [m/s2]

𝑕 enthalpy [J/kg]

𝑕𝑓𝑔 latent heat of vaporization [J/kg]

𝐻 enthalpy [J]

𝐽𝑎 Jakob number

𝑘 thermal conductivity [W/m-K]

𝑚 bubble mass [kg]

𝑚 mass flow rate [kg/s]

𝑝 pressure [Pa]

𝑝∞ system pressure [Pa]

𝑃𝑟 Prandtl number

𝑞" heat transfer per area [W/m2]

𝑄 heat transfer [J]

𝑄 heat transfer rate [W]

𝑅 bubble radius [m]

𝑅 bubble growth rate [m/s]

𝑅 radial acceleration of bubble [m/s2]

𝑠 entropy [J/kg-K]

𝑆 entropy change rate [W/K]

𝑆 𝑔𝑒𝑛 entropy generation rate [W/K]

𝑆𝜍 dimensionless scaling factor for surface tension force

𝑇 temperature [K]

𝑇∞ uniform system temperature [K]

𝑇𝑠𝑎𝑡 (𝑝∞) saturation temperature at 𝑝∞ [K]

∆𝑇 superheat [K]

𝑡 time [s]

𝑣 specific volume of liquid [m3/kg]

𝑉 bubble volume [m3]

𝑊 work [J]

𝑊 rate of work [W]

Greek Symbols

𝛼 thermal diffusivity of liquid

𝛽 contact angle

𝜇 viscosity

𝜌 density of liquid [kg/ m3]

𝜃 subcooling factor

𝜍 surface tension [N/m]

Subscripts

𝑏 base

𝑑𝑒𝑝𝑡 departure

𝑖 interface

𝑙 liquid

𝑣 vapor

𝑤 wait

𝑤𝑎𝑙𝑙 wall

Superscripts

𝑥 modified term

+ dimensionless

∗ modified term

INTRODUCTION

Previous Work

Bubble departure diameters in nucleate pool boiling have been studied

extensively both analytically and experimentally. In 1935, Fritz developed a

correlation for bubble departure diameter in nucleate boiling by balancing

buoyancy and surface tension forces for a static bubble (Fritz, 1935). This

equation has since been expanded by other investigators. Bubble growth rate

was included in a correlation by (Staniszewski, 1959) after observing that bubble

departure diameter is dependent on the rate at which the bubble grows. Others

have expanded the range of the Fritz correlation to low pressure systems such

as (Cole & Rohsenow, 1969), while (Kocamustafaogullari, 1983) have expanded

it to fit high pressure systems. More recently, (Gorenflo, Knabe, & Bieling, 1986)

established an improved correlation for bubble departure at high heat fluxes. A

summary of bubble departure correlations is provided in Table 1.

Table 1. Departure Diameter Correlations

Source Departure Diameter Model Comments

(Fritz, 1935) 𝑑𝑑𝑒𝑝𝑡 = 0.0208𝛽

𝑔(𝜌𝑙 − 𝜌𝑣)

Correlation balances buoyancy force with surface tension force

(Staniszewski, 1959) 𝑑𝑑𝑒𝑝𝑡 = 0.0071𝛽

1 + 0.435𝑑𝐷

𝑑𝑡

Correlation includes affect of bubble growth

(Zuber, 1959) 𝑑𝑑𝑒𝑝𝑡 =

6𝑘𝑙 𝑇𝑤𝑎𝑙𝑙 − 𝑇𝑠𝑎𝑡 𝑝∞

(Ruckenstein, 1961) and (Zuber, 1964)

𝑑𝑑𝑒𝑝𝑡

= 3𝜋2𝜌𝑙𝛼𝑙

1 2 𝑔1 2 𝜌𝑙−𝜌𝑣 1 2

𝜍3 2

𝐽𝑎4 3 𝜍

𝑔 𝜌𝑙 − 𝜌𝑣

(Borishanskiy & Fokin, Heat transfer and hydrodynamics in steam generators,

𝑑𝑑𝑒𝑝𝑡 = −𝐶

+ 𝑅𝐹2

𝑅𝐹 = 𝑟𝑎𝑑𝑖𝑢𝑠 𝑓𝑟𝑜𝑚 𝐹𝑟𝑖𝑡𝑧 𝐶𝑜𝑟𝑟𝑒𝑙𝑎𝑡𝑖𝑜𝑛

𝐶 = 6

𝜌𝑙𝜌𝑙−𝜌𝑣

𝜌𝑣𝜌𝑙

𝜌𝑣𝑕𝑓𝑔

(Cole & Shulman, 1966a) 𝑑𝑑𝑒𝑝𝑡 =

𝑤𝑕𝑒𝑟𝑒 𝑝 𝑖𝑠 𝑖𝑛 𝑚𝑚𝐻𝑔

(Cole, 1967) 𝑑𝑑𝑒𝑝𝑡 = 0.04𝐽𝑎

(Cole & Rohsenow, 1969) 𝑑𝑑𝑒𝑝𝑡 = 𝐶 𝐽𝑎𝑥 5 4

𝐽𝑎𝑥 =𝑇𝑐𝑐𝑝 ,𝑙𝜌𝑙𝜌𝑣𝑕𝑓𝑔

𝐶 = 1.5𝑥10−4 for water

𝐶 = 4.65𝑥10−4 for fluids other than water

Correlation for low pressure systems

(Golorin, Kol'chugin, & Zakharova, 1978)

𝑑𝑑𝑒𝑝𝑡

=1.65𝑑∗𝜍

𝑔 𝜌𝑙 − 𝜌𝑣 +

15.6𝜌𝑙𝑔 𝜌𝑙 − 𝜌𝑣

𝛽𝑑𝑘𝑙 𝑇𝑤𝑎𝑙𝑙 − 𝑇𝑠𝑎𝑡

𝜌𝑣𝑕𝑓𝑔

𝑑∗ = 6.0𝑥10−3𝑚𝑚

𝛽𝑑 = 6.0

Correlation includes dynamic relationship

(Kutateladze & Gogonin, 1980) 𝑑𝑑𝑒𝑝𝑡 =. 25 1 + 105𝐾1

1 2 𝜍

𝐾1 = 𝐽𝑎

𝑃𝑟𝑙

𝑔𝜌𝑙 𝜌𝑙 − 𝜌𝑣

𝜇𝑙2

𝐾1 ≤ 0.06

(Borishanskiy, Danilova, Gotovskiy,

Borishanskaya, Danilova, &

Kupriyanova, 1981)

𝑑𝑑𝑒𝑝𝑡 = 5𝑥105 𝑝

𝑝𝑐 −0.46

𝑃𝑐𝑀

𝑘𝐵𝑇𝑐 −1 3

(Kocamustafaogullari, 1983) 𝑑𝑑𝑒𝑝𝑡 = 2.64𝑥10−5

𝜌𝑙 − 𝜌𝑣

𝜌𝑣

Expansion of Fritz

correlation to include high pressure systems

(Jensen & Memmel, 1986) 𝑑𝑑𝑒𝑝𝑡 = 0.19 1.8 + 105𝐾1

2 3 𝜍

Correlation is a

proposed improvement to (Kutateladze &

Gogonin, 1980)

(Gorenflo, Knabe, & Bieling, 1986) 𝑑𝑑𝑒𝑝𝑡 = 𝐶1

𝐽𝑎4𝑘𝑙2

1 + 1 +2𝜋

3𝐽𝑎

Correlation for high heat fluxes

(Stephan, 1992)

𝑑𝑑𝑒𝑝𝑡 = 0.25 𝜍

1 + 𝐽𝑎

𝑃𝑟𝑙

𝐴𝑟

Correlation valid for

5𝑥10−7 ≤ 𝐽𝑎

𝑃𝑟𝑙

𝐴𝑟≤ 0.1

(Kim & Kim, 2006) 𝑑𝑑𝑒𝑝𝑡 = 0.1649𝐽𝑎0.7

Correlation valid for high and low Jakob

numbers

An evaluation of forces acting on bubbles forming

in normal and reduced gravitational fields was performed

by (Keshock & Siegel, 1964). Five forces acting on

bubbles during growth while attached to a wall were

identified as buoyancy, excess pressure, inertia, surface

tension and drag forces; each of which acts to keep the

bubble attached to the wall or to promote separation. The

buoyancy force accounts for the difference in liquid and

vapor densities. Density differences between the vapor in the bubble and liquid

of the fluid pool promote bubble departure. Buoyancy is aided by the excess

pressure force which accounts for the vapor pressure acting on the region of wall

within the bubble base diameter. This force aids in pushing the liquid vapor

interface away from the wall. The resulting equation for this force takes the same

form as that for surface tension.

Inertia, surface tension and drag forces work to limit bubble separation.

The inertia force is exerted as the surrounding fluid pool is forced to flow in a

Figure 1. Forces Acting on a Bubble.

𝐹𝑏 + 𝐹𝑝

𝐹𝑖 + 𝐹𝜍 + 𝐹𝐷

radial direction away from the bubble boundary due to bubble growth. As the

fluid is displaced, its viscosity creates resistance to bubble growth. It can be

seen in the equations in Table 2 that the inertia force is scaled by a factor of

11/16. The scaling factor was proposed by (Han & Griffith, 1962) to approximate

mass of affected fluid around the outer surface of the bubble. The surface

tension force accounts for the force of the liquid vapor interface with the wall and

the drag force accounts for the motion of the growing bubble through the

surrounding liquid. These forces can be seen graphically in Figure 2 along wither

a list of their corresponding equations in Table 2.

𝜌𝑣

𝜌∞

𝑭𝒃

𝑃∞

𝑭𝒑

𝑃𝑣

𝑅𝑏

𝑭𝝈 𝑭𝝈

𝐹𝑆𝑢𝑟𝑓 𝑇𝑒𝑛

𝜌∞

𝑭𝒊

𝑑𝑅

𝜇∞

𝑭𝑫

𝑑𝑅

Figure 2. Forces Acting on Bubble. (A) Buoyancy Force, (B) Excess Pressure Force, (C) Inertia Force, (D) Surface Tension Force), (E) Drag Force.

(A) (B)

(C) (D) (E)

Table 2. Forces acting on a bubble prior to separation.

Force Equation

Buoyancy Force 𝐹𝑏 =

4𝜋𝑅3

3 𝜌𝑙 − 𝜌𝑣 𝑔

Excess Pressure Force 𝐹𝑝 = 𝜋𝑅𝑏𝜍 sin𝛽

Inertia Force 𝐹𝑖 =

𝑑𝑡 𝑚

𝑑𝑅

𝑑𝑡 ≅

𝑑𝑡

16𝜌

4𝜋𝑅3

𝑑𝑅

𝑑𝑡

Surface Tension Force 𝐹𝜍 = 2𝜋𝑅𝑏𝜍 sin𝛽

Drag Force 𝐹𝐷 =

4𝑎𝜇𝑙𝑅

𝑑𝑅

𝑑𝑡,𝑎 = 45

Bubble separation occurs when buoyancy and excess pressure forces

exceed the net affects of the inertia, surface tension, and drag forces. The work

of (Keshock & Siegel, 1964) demonstrated that varying system conditions

produce varying levels of influence for each of the forces associated with bubble

departure.

While extensive research has led to the development a number of

correlations for bubble departure diameter, a universal correlation is lacking. It

can be seen by analysis of the correlations provided in the Table 1 that bubble

departure is a function of many variables including contact angle, bubble growth

rate, Jakob number, thermal diffusivity, system temperatures, pressures, and a

number of others. Additionally, while most correlations are proportionate to

𝑔−1 2 , it can be seen that departure diameters determined by the correlations of

(Zuber, 1959) and (Gorenflo, Knabe, & Bieling, 1986) are proportionate to 𝑔−1 3 .

Development of a universal correlation will require a function of multiple system

and fluid properties which can be utilized to model a wide range of system

conditions.

Problem Statement

Is it possible to develop a correlation for bubble departure radius or

diameter in nucleate pool boiling by analyzing entropy generation rate during

bubble growth?

It is suspected that the rate entropy generation reaches a maximum value

at the point at which a bubble departs from a wall during nucleate pool boiling.

As demonstrated later in this paper, the entropy generation rate for a spherical

bubble in nucleate pool boiling is defined by the equation below.

𝑆 𝑔𝑒𝑛 = −1

𝑇𝑤𝑎𝑙𝑙 𝑊 + 𝐸 −

𝑑𝑡 𝑚𝑕

As the entropy generation rate reaches a maximum value, the sum of rate

of work performed by the bubble on its surroundings and the rate of change of

internal energy minus the rate of energy transfer to the bubble reaches a

minimum. It is believed that at this point, the bubble reaches a state of

equilibrium which results in departure or collapse in the case of sub-cooled

boiling. If this suspicion is correct, an entropy generation analysis of bubble

growth using the second law of thermodynamics may lead to a novel correlation

for determination of bubble departure radius.

Methodology

The maximum rate of entropy generation can be determined by taking the

derivative of entropy generation rate with respect to bubble radius and setting it

equal to zero. This method requires that the net heat transfer rate for the bubble

be substituted into the entropy generation equation. The proposed method is

accomplished by evaluation of the bubble using the first and second laws of

thermodynamics.

First Law of Thermodynamics

The first law of thermodynamics

states that energy must be conserved. By

analyzing the bubble using the first law of

thermodynamics, it is possible to determine

the rate of heat transfer. Heat transferred

to the bubble must result in changes to the

accumulated energy of the bubble, work

performed on the bubble boundary, and

energy flow at the bubble boundary. In the case of a bubble undergoing growth

at a wall, the net energy flows into the bubble. Energy flow out of the bubble is

therefore ignored. The resulting first law equation for a bubble reduces to the

following equation which can be seen graphically in Error! Reference source

not found..

𝑄 = 𝑊 + 𝐸 − 𝑑

𝑑𝑡 𝑚𝑕

𝑖𝑛

It is possible to determine the rate of heat transfer by determining the rate

of work performed, the change rate for the accumulated energy, and the rate of

net energy flow into the bubble. Given this value, it is then possible to solve for

entropy generation rate using the second law of thermodynamics.

Figure 3. Balance of Energy for First Law of Thermodynamics

dt 𝑚𝑕

𝑖𝑛

Second Law of Thermodynamics

The second law of thermodynamics is a statement to the irreversibility of a

system. It states that entropy of a system not at equilibrium will increase with

time. For a system with open boundaries such as a bubble, entropy generation

rate is a function of the rate of entropy accumulation inside a control volume, the

entropy transfer rate, and net entropy flow rate at the boundaries of the control

volume. The second law of thermodynamics can be written as follows:

𝑆 𝑔𝑒𝑛 = 𝑆 − 𝑄𝑖

𝑇𝑖𝑖

dt 𝑚𝑠

𝑖𝑛

Given the heat transfer rate determined by the first law of

thermodynamics, it is possible to determine entropy generation rate using the

second law of thermodynamics.

General Assumptions

The following chapters cover the derivation of two novel correlations for

bubble departure radius in nucleate pool boiling. These derivations will be made

based on the assumptions listed below.

Bubble maintains spherical shape during growth.

State of vapor flowing into the bubble is at the same state as vapor

accumulated within the bubble.

The state of the fluid pool is constant and uniform with no thermal

boundary layer around bubble surface or wall.

Bubble radius can be accurately modeled by the (Mikic, Rohsenow, &

Griffith, 1970) (MRG) correlation during both inertia and heat-diffuse

stages of bubble growth.

Quasi equilibrium

Additional assumptions will be introduced throughout the derivation of the

correlations for the purpose of simplifying equations.

Vapor pressure is constant and equal to the saturation pressure of the

bulk liquid pool.

ENTROPY GENERATION RATE STUDY (PRESSURE-VOLUME METHOD)

A novel correlation is derived for bubble departure radius using the second

law of thermodynamics. In this chapter, work performed by the bubble is

modeled as the integral of the system pressure multiplied by the rate of change

in bubble volume. All steps of the following work are shown in Appendix A.

Derivation of Heat Transfer Rate

Solution of the second law of thermodynamics requires an understanding

of the heat transfer rate for the system. This is accomplished by solving the first

law of thermodynamics. Equations will be derived for the rate of work performed

by a bubble, the energy change rate, and the energy transfer rate.

Rate or Work

In this chapter, the rate of work performed by a bubble is modeled using

the equation for work done by a thermodynamic system. This equation is a

function of the driving pressure and the change in system volume.

𝑊 = 𝑃𝑑𝑉𝑉2

For a bubble undergoing growth in a pool, the driving pressure is

equivalent to the difference between vapor pressure within the bubble and the

interface pressure of the fluid surrounding the bubble. For the purposes of this

investigation, the interface pressure is assumed equivalent to the bulk fluid

pressure. Furthermore, the bubble is assumed to maintain a spherical shape

which allows for the change in volume to be replaced by the following

relationship.

𝑑𝑉 = 4𝜋𝑅2𝑑𝑅

Application of these relationships leads to the following equation for work

performed by the bubble on the surrounding fluid.

𝑊 = 4𝜋 𝑝𝑣 − 𝑝∞ 𝑅2𝑑𝑅

In the above equation, vapor pressure is a function of bubble radius.

Successive integration by parts is therefore required to solve for the work done

by a bubble on its surroundings. The resulting equation is shown below.

𝑊 =4𝜋

3𝑅3 𝑝𝑣 − 𝑝∞ −

𝑑𝑝𝑣𝑑𝑡

𝑅 +1

𝑑2𝑝𝑣𝑑𝑡2

𝑅2 −1

𝑅3 + ⋯

The rate at which work is done by a bubble on its surrounding is

determined by taking the derivative of the above equation with respect to time.

Doing so results in the following relationship.

𝑊 =4𝜋𝑅2

3 3 𝑝𝑣 − 𝑝∞ 𝑅

+ 𝑅 − 1 −𝑑𝑝𝑣𝑑𝑡

𝑅 +1

𝑅2 −1

𝑅3 +1

𝑅4 + ⋯

It can be seen in the equation above that the rate of work performed by a

spherical bubble is a function of the rate of bubble growth and the rate at which

vapor pressure changes. It is possible to reduce this equation to a function of

constant fluid properties and bubble growth rate by utilization of the Young-

Laplace equation or the equation of motion for a spherical bubble.

𝑝𝑣 = 𝑝𝑖 +2𝜍𝑙𝑅

𝑝𝑣 = 𝑝∞ +2𝜍𝑙𝑅

+ 𝜌𝑙 3

2 𝑑𝑅

𝑑𝑡

+ 𝑅𝑑2𝑅

𝑑𝑡2

For the purposes of this derivation, the rate of work performed by a bubble

will be maintained as a function of the rate of bubble growth and rate of vapor

pressure change.

If vapor pressure is assumed constant and equivalent to the saturation

pressure of the bulk liquid pool through the life of the bubble, the equation can be

reduced to the following.

𝑊 ≅ 4𝜋 𝑝𝑠𝑎𝑡 𝑇∞ − 𝑝∞ 𝑅 𝑅2

This assumption will not accurately model the rate of work performed by a

bubble growth within the inertia controlled region as this region is characterized

by rapidly changing vapor pressures. However, it is believed to be an acceptable

model for bubbles undergoing growth in the heat diffuse region in which the rate

of vapor pressure change is minimal.

Energy Change Rate

The Internal energy of a system is a measure of its total kinetic and

potential energy. In the case of a bubble, internal energy can be determined by

multiplying bubble vapor mass by the energy per unit mass at a given state.

𝐸 = 𝑒𝑣𝑚𝑣 = 𝑒𝑣𝑚𝑣 R

= 4π 𝜌𝑣𝑒𝑣

𝑅2𝑑𝑅

As all variables in the equation above are functions of bubble radius,

integration must be completed using successive integration by parts. Doing so

leads to the following series for internal energy.

𝐸 =4𝜋𝑅3

3 𝜌𝑣𝑒𝑣 −

4 𝜌𝑣

𝑑𝑒𝑣𝑑𝑡

+ 𝑒𝑣𝑑𝜌𝑣𝑑𝑡

𝑅 +1

20 𝜌𝑣

𝑑2𝑒𝑣𝑑𝑡2

+ 2𝑑𝑒𝑣𝑑𝑡

𝑑𝜌𝑣𝑑𝑡

+ 𝑒𝑣𝑑2𝜌𝑣𝑑𝑡2

120 𝜌𝑣

+ 3𝑑𝜌𝑣𝑑𝑡

𝑑2𝜌𝑣𝑑𝑡2

𝑅3 + ⋯

The rate at which the internal energy of a system changes can be

determined by taking the derivative of the internal energy with respect to time.

𝐸 =4𝜋𝑅2

3 3𝜌𝑣𝑒𝑣𝑅

+ 𝑅 − 1 − 𝜌𝑣𝑑𝑒𝑣𝑑𝑡

4 𝜌𝑣

20 𝜌𝑣

120 𝜌𝑣

+ 6𝑑2𝜌𝑣𝑑𝑡2

𝑅4 + ⋯

If the state of the vapor within the bubble is again assumed constant and

equal to the saturation pressure of the bulk liquid pool, the above equation is

simplified to the following form.

𝐸 = 4𝜋𝑒𝑣𝜌𝑣𝑅2𝑅

Energy Transfer Rate

The energy transfer across the bubble boundary is defined as derivative

with respect to time of the total vapor mass flowing across the boundary

multiplied by the enthalpy per unit mass of the transferred vapor. For the

purposes of this analysis, the state of the vapor entering the bubble is assumed

to equivalent to that of the vapor within the bubble. This implies that enthalpy of

the vapor flowing in is the same as the enthalpy of the vapor in the bubble.

𝑑𝑡 𝑚 𝑕𝑖𝑛

By performing successive integration by parts and taking the derivative of

the resulting series, the following equation for energy transfer rate is derived.

𝑑𝑡 𝑚𝑕𝑖𝑛 =

4𝜋𝑅2

3 3𝜌𝑣𝑕𝑣𝑅

+ 𝑅 − 1 − 𝜌𝑣𝑑𝑕𝑣𝑑𝑡

+ 𝑕𝑣𝑑𝜌𝑣𝑑𝑡

4 𝜌𝑣

𝑑2𝑕𝑣𝑑𝑡2

+ 2𝑑𝑕𝑣𝑑𝑡

+ 𝑕𝑣𝑑2𝜌𝑣𝑑𝑡2

20 𝜌𝑣

120 𝜌𝑣

𝑅4 + ⋯

If the state of the vapor is assumed constant and equal to the saturation

pressure of the bulk liquid pool, the energy transfer rate reduces to a function of

bubble growth rate.

𝑑𝑡 𝑚𝑕𝑖𝑛 = 4𝜋𝑕𝑣𝜌𝑣𝑅

Heat Transfer

Substitution of the equations derived above into the first law of

thermodynamics produce the following equation for heat transfer rate.

𝑄 =4𝜋𝑅2

3 3 𝑝𝑣 − 𝑝∞ 𝑅 + 3𝜌𝑣 𝑒𝑣 − 𝑕𝑣 𝑅

𝑅 +1

𝑅2 −1

𝑅3 +1

+ −𝑑𝜌𝑣𝑑𝑡

𝑅 +1

𝑅2 −1

𝑅3 +1

𝑅4 𝑒𝑣 − 𝑕𝑣

+ −𝜌𝑣𝑅 +1

𝑅2 −3

𝑅3 +1

𝑅4 𝑑𝑒𝑣𝑑𝑡

−𝑑𝑕𝑣𝑑𝑡

4𝜌𝑣𝑅

2 −3

𝑅3 +1

𝑅4 𝑑2𝑒𝑣𝑑𝑡2

−𝑑2𝑕𝑣𝑑𝑡2

+ −1

20𝜌𝑣𝑅

120𝜌𝑣𝑅

4 𝑑4𝑒𝑣𝑑𝑡4

This equation can be further reduced application of the definition of

enthalpy.

𝑒𝑣 − 𝑕𝑣 = −𝑝𝑣𝑣𝑣 = −𝑝𝑣𝜌𝑣

Substitution of the above equation and its derivatives allows the heat

transfer rate equation for a spherical bubble to be reduced.

𝑄 = −4𝜋𝑝∞𝑅2𝑅

It is noted that this solution is identical to the solution derived by applying

the assumption of constant vapor pressure. The rate of heat transfer for a

spherical bubble is a function of bulk pressure and radial growth behavior of the

bubble. The assumption that vapor pressure is constant is acceptable for

determination of heat transfer rate. However, the rate at which vapor pressure

changes may still have a significant influence on the rate of work, rate of

accumulated energy, and rate of energy transfer for a spherical bubble

undergoing growth in the inertia controlled region.

Derivation of Entropy Generation Rate

With heat transfer rate defined, it is possible to determine the rate of

entropy generation. Like determination of heat transfer rate, this requires

relationships for the rate of entropy accumulation, entropy transfer rate, and the

net entropy flow rate.

Entropy Accumulation Rate

Entropy accumulation rate within the bubble is determined by taking the

derivative of the total entropy accumulated with respect to time.

𝑆 =𝑑

𝑑𝑡 𝑆𝑣 =

𝑑𝑡 𝑠𝑣𝑚𝑣

𝑑𝑡 4π 𝜌𝑣𝑠𝑣𝑅

2𝑑𝑅R

The total entropy accumulated can be solved for by successive integration

by parts of the entropy per unit mass multiplied by the rate of mass change.

𝑆𝑣 =4𝜋

3𝑅3 𝜌𝑣𝑠𝑣 −

𝑑 𝜌𝑣𝑠𝑣

𝑑𝑡𝑅 +

𝑑2 𝜌𝑣𝑠𝑣

𝑑𝑡2𝑅2 −

𝑑𝑡3𝑅3 + ⋯

Taking the derivative with respect to time of the total accumulated entropy

leads to the following equation.

𝑆 =4𝜋

3𝑅2 3𝜌𝑣𝑠𝑣𝑅

+ 𝑅 − 1 −𝑑 𝜌𝑣𝑠𝑣

𝑑𝑡𝑅 +

𝑑𝑡2𝑅2 −

𝑑𝑡3𝑅3

𝑑𝑡4𝑅4 + ⋯

By applying the assumption of constant vapor properties at the saturation

point of the bulk liquid pool, this equation reduces to the following form.

𝑆 = 4𝜋𝜌𝑣𝑠𝑣𝑅2𝑅

Entropy Transfer Rate

The entropy transfer rate for a bubble growing on a wall is determined by

dividing the heat transfer rate by the wall temperature. By substitution of the

derived heat transfer rate equation, the following equation is defined.

𝑇𝑤𝑎𝑙𝑙= −

𝑇𝑤𝑎𝑙𝑙𝑝∞𝑅

Net Entropy Flow Rate

The net entropy flow rate is defined as follows.

𝑑𝑡 𝑆𝑣

𝑖𝑛

𝑑𝑡 4π 𝜌𝑣𝑠𝑣

𝑅2𝑑𝑅

Since the state of the vapor flowing into the bubble is assumed to be

equivalent to the state of the vapor accumulated within the bubble, the equation

for net entropy flow rate takes the same form as that derived for the entropy

change rate.

Entropy Generation Rate

The rate of entropy generation is determined by substitution of the derived

equations into the second law of thermodynamics. As it was previously noted,

the net entropy flow rate and the entropy transfer rate are equivalent and

therefore cancel. The resulting entropy generation rate equation is a function of

only the heat transfer rate.

𝑆 𝑔𝑒𝑛 =4𝜋

If entropy generation rate reaches a maximum value at the point of bubble

departure as hypothesized, the bubble departure radius can be determined by

taking the derivative of entropy generation rate with respect to bubble radius and

setting it equivalent to zero.

𝑑𝑆 𝑔𝑒𝑛𝑑𝑅

= 0 =𝑑

𝑑𝑅

𝑇𝑤𝑝∞𝑅

2𝑅 =4𝜋

𝑇𝑤𝑝∞

𝑑𝑡 𝑅2𝑅

𝑑𝑡

𝑑𝑅

This reduces to the following equation.

0 =4𝜋

𝑇𝑤𝑎𝑙𝑙𝑝∞𝑅 2𝑅 + 𝑅

Rearranging of the equation produces the following second order, non-

linear differential equation; the solution to which should describe the departure

radius if the hypothesis is true.

0 = 𝑅𝑅 + 2𝑅 2

By utilization of substitution methods, it can be shown that the general

solution to the second order, non-linear differential equation takes the following

𝑅𝑑𝑒𝑝𝑡 = −3𝑒−2𝐶𝑡 + 3𝐷 1 3

For this solution to be useful, variables 𝐶 and 𝐷 must be defined. This

requires the application of two boundary conditions. The first boundary condition

can be determined by evaluation of experimental data for bubble departure radii.

Comparison of the rate of change for both the general solution and the

experimental bubble at departure can be used to satisfy the second boundary

condition.

Analysis of Second Order, Non-Linear Differential Equation

Analysis of the second order, non-linear differential equation requires an

understanding of growth behavior of bubbles during pool boiling. Bubble

behavior has been described by a number of researchers including (Rayleigh,

1917), (Plesset & Zwick, 1954), and (Mikic, Rohsenow, & Griffith, 1970). In the

following sections, the equations derived by these researches will be utilized to

solve the second order, non-linear differential equation.

Experimental data published by (Van Stralen, Cole, Sluyter, & Sohal,

1975) for bubbles undergoing growth in superheated water at sub-atmospheric

pressures will be utilized for comparison and refinement of the second order,

non-linear differential equation. Application of the equations for bubble growth

requires an understanding of both fluid and vapor properties. For the purposes

of this analysis, bulk liquid pool properties are assumed uniform and constant,

and effects of thermal boundary layers and the liquid-vapor interface are ignored.

Furthermore, the state of vapor within the bubble may be estimated by utilizing

the saturation point of the bulk liquid pressure. While the vapor pressure within a

bubble is highly dynamic, it approaches the bulk liquid pressure as growth

transitions from an inertia controlled region to heat diffuse region. As described

by (Lien, 1969), the following liquid properties will be utilized to solve for the

Jakob number of the system as well as additional system constants for use in the

growth equations.

Thermal Conductivity of Liquid Saturated liquid at 𝑇∞

Surface Tension of Liquid Saturated liquid at 𝑇∞

Specific Heat of Liquid Saturated liquid at 𝑇∞

Density of Liquid Saturated liquid at 𝑇∞

Latent Heat of Vaporization Saturated liquid at 𝑃∞

Density of Vapor Saturated liquid at 𝑃∞

Vapor Pressure Saturated liquid at 𝑇∞

The liquid and vapor properties listed above will be determined by

utilization of equations defined by the International Association for the Properties

of Water and Steam (Revised release on the IAPWS Industrial Formulation of

1997 for the thermodynamic properties of water and steam, 2007) (IAPWS

release on surface tension of ordinary water substance, 1994)

Analysis Using Rayleigh Equation

Bubble growth is defined by two distinct regions. Initial bubble growth is

described as inertia controlled growth in which high internal pressures produce

rapid growth of the bubble. Growth in this region is limited by the amount of

momentum available to displace the surrounding fluid. As internal pressures

drop and the effect of inertia becomes negligible, bubbles transition to heat

diffuse growth in which bubble growth is driven primarily by heat transfer.

Correlations have been developed for each of these regions to describe the

bubbles growth characteristics.

In 1917, Rayleigh derived an equation of motion for the flow of and

incompressible fluid around spherical bubble. The equation takes the following

𝑅𝑑2𝑅

𝑑𝑡2+

2 𝑑𝑅

𝑑𝑡

𝜌𝑙 𝑝𝑣 − 𝑝∞ −

It was shown by Rayleigh that this equation can be reduced to the

following form by utilization of a linearirzed Clausis-Clapeyron equation.

𝑅𝑑2𝑅

𝑑𝑡2+

2 𝑑𝑅

𝑑𝑡

=𝑕𝑓𝑔𝜌𝑣

𝜌𝑙 𝑇∞ − 𝑇𝑠𝑎𝑡

𝑇𝑠𝑎𝑡

Integration of the above equation leads to the Rayleigh equation for

bubble growth

𝑅 = 2

𝑕𝑓𝑔𝜌𝑣𝜌𝑙

𝑇∞ − 𝑇𝑠𝑎𝑡

𝑇𝑠𝑎𝑡

This equation is commonly written as follows.

𝑅 = 𝐴𝑡

𝑤𝑕𝑒𝑟𝑒

𝐴 = 𝑏𝑕𝑓𝑔𝜌𝑣𝜌𝑙

𝑇∞ − 𝑇𝑠𝑎𝑡

𝑇𝑠𝑎𝑡

, 𝑏 =2

From the relationship above, it is possible to determine the radial velocity

and acceleration of a growing bubble by taking the first and second derivatives

with respect to time.

𝑑𝑅

𝑑𝑡= 𝐴

𝑑2𝑅

𝑑𝑡2= 0

Utilization of the bubble growth equations defined above, the second

order, non-linear differential equation derived in the section above may be solved

by direction substitution. If the hypothesis that entropy generation reaches a

maximum value at the point of bubble departure, the solution to the equation

bellow describes the departure radius for a bubble undergoing pool boiling on a

𝑅𝑅 + 2𝑅 2 = 0

Substitution of the Rayleigh equations into the equation above produces

the following relationship.

2𝐴2 = 0

By observation, it can be seen that the above equation is invalid for any

non-zero value of 𝐴. Furthermore, the equation is not a function of bubble radius.

Substitution of the Rayleigh equation into the second order, non-linear differential

equation does not produce a departure radius for a spherical bubble.

While direct substitution of the Rayleigh equation and it derivative into the

second order, non-linear differential equation does not produce a departure

radius, utilization of the general solution may provide improved results. Earlier in

this chapter a general solution was determined for the derived second order,

non-linear differential equation. This general solution takes the following form.

𝑅 = −3𝑒−𝐶𝑡 + 𝐷 1 3

𝑊𝑕𝑒𝑟𝑒 𝑡𝑕𝑒 𝑑𝑒𝑟𝑖𝑣𝑎𝑡𝑖𝑣𝑒 𝑖𝑠 𝑑𝑒𝑓𝑖𝑛𝑒𝑑 𝑎𝑠:

𝑑𝑅

𝑑𝑡= −𝑒−𝐶 −3𝑒−𝐶𝑡 + 𝐷 −1 3

Utilization of the general solution requires that constants 𝐶 and 𝐷 be

determined. This is accomplished by applying boundary conditions. For the

purposes of this analysis the boundary conditions will be defined at the time of

bubble departure. At departure, the radius defined by the Rayleigh equation will

be set equal to the radius defined by the general solution. Additionally, the slope

of both equations will be assumed perpendicular at this time.

𝑅𝑅𝑎𝑦𝑙𝑒𝑖𝑔 𝑕 𝑡=𝑡𝑑𝑒𝑝𝑡= 𝑅𝐺𝑒𝑛 𝑆𝑜𝑙𝑢𝑡𝑖𝑜𝑛 𝑡=𝑡𝑑𝑒𝑝𝑡

𝑑𝑅

𝑑𝑡 𝑅𝑎𝑦𝑙𝑒𝑖𝑔 𝑕

𝑡=𝑡𝑑𝑒𝑝𝑡

= − 𝑑𝑅

𝑑𝑡 𝐺𝑒𝑛 𝑆𝑜𝑙𝑢𝑡𝑖𝑜𝑛 −1

By substation of the appropriate equations into the boundary conditions

defined above, a system of equations is created. This system of equations is

reduced to define the constant 𝐶. The derivation of this is located in Appendix B.

𝐶 = −𝑙𝑛 𝐴𝑡𝑑𝑒𝑝𝑡2

This equation is rewritten in terms of bubble departure radius by utilization

of the Rayleigh equation.

𝐶 = −𝑙𝑛 1

𝐴𝑅𝑑𝑒𝑝𝑡

Solving for constant 𝐶 requires experimental data including system

conditions and the departure radius. By averaging results for experimental data

sets, a value for constant 𝐶 can be defined.

𝐶 = −𝑙𝑛

1𝐴𝑖

𝑅𝑒𝑥𝑝 ,𝑖2

𝑖=1

To define the constant 𝐶, experimental data published by (Van Stralen,

Cole, Sluyter, & Sohal, 1975) is utilized. Results of this analysis are shown in

Table 3.

Table 3. Values of C for the General Solution Derived from Rayleigh Equation with Experimental Data of (Van Stralen, Cole, Sluyter, & Sohal, 1975).

Bubble Number

Experimental Departure Radius of (Van Stralen, Cole, Sluyter, &

Sohal, 1975)

𝑅𝑒𝑥𝑝 , m

Ja 𝐴 𝐶

1 0.00092 64.4322 6.032567 15.779446

2 0.0079 124.4618 2.572322 10.626594

3 0.0119 200.1375 2.549390 9.798288

4 0.0136 385.8247 2.411500 9.475620

5 0.0268 895.6793 2.309636 8.075797

6 0.0415 2038.6934 1.924969 7.019034

Average 10.129130

It is possible to solve for constant 𝐷 by substitution of constant 𝐶 into the

general solution and rearranging.

𝐷 = 𝑅𝑒𝑥𝑝 ,𝑖

3 + 3𝑒−𝐶

𝐴𝑖 𝑅𝑒𝑥𝑝 ,𝑖

𝑖=1

Evaluation of the equation above is again accomplished by utilizing

experimental data published by (Van Stralen, Cole, Sluyter, & Sohal, 1975) and

the average constant 𝐶 derived above. Results are shown in Table 4.

Table 4. Values of D for the General Solution Derived from Rayleigh Equation with Experimental Data of (Van Stralen, Cole, Sluyter, & Sohal, 1975).

Bubble Number

Sohal, 1975)

Ja 𝐴 𝐷

1 0.00092 64.4322 6.032567 1.90337E-08

2 0.0079 124.4618 2.572322 8.60658E-07

3 0.0119 200.1375 2.549390 2.2439E-06

4 0.0136 385.8247 2.411500 3.19052E-06

5 0.0268 895.6793 2.309636 2.06378E-05

6 0.0415 2038.6934 1.924969 7.4054E-05

Average 1.683431E-05

Substitution of these constants into the general solution produces a linear

relationship for bubble radius that satisfies the second order, non-linear

differential equation.

𝑅 = −1

8354.181454𝑡 + 1.683431E − 05

At departure, this equation will be equivalent to the Rayleigh equation.

Setting the general solution equal to the Rayleigh equation produces a function

of the departure time. In order to solve for bubble departure radius, the

departure time is replaced by utilizing the Rayleigh equation.

0 = 𝑅𝑑𝑒𝑝𝑡3 +

3𝑒−𝐶

𝐴𝑅𝑑𝑒𝑝𝑡 − 𝐷

Substitution of the constants 𝐶 and 𝐷 results in the following equation.

8354.181454 ∗ 𝐴 𝑅𝑑𝑒𝑝𝑡 − 1.683431E − 05

The above equation has three possible solutions for the departure radius.

The exact solution corresponding to the bubble departure must be real, positive

and should be in the scale of expected results. Evaluation of experimental data

from (Van Stralen, Cole, Sluyter, & Sohal, 1975) with the equation above

produces the predicted departure radii presented in Table 5.

Table 5. Error Analysis of Predicted Departure Radii based on Rayleigh Based General Solution.

Bubble Number Ja

Sohal, 1975)

Predicted Departure Radius

𝑅𝑑𝑒𝑝𝑡 , m

% Error

1 64.4322 0.00092 0.025371 2657.7113

2 124.4618 0.0079 0.025024 216.7582

3 200.1375 0.0119 0.025018 110.2391

4 385.8247 0.0136 0.024984 83.7026

5 895.6793 0.0268 0.024955 -6.8839

6 2038.6934 0.0415 0.024821 -40.1915

Analysis of the table indicates that predicted departure radii fail to

consistently fit with experimental data. This is seen graphically in Figure 4 and

Figure 5.

Figure 4. Comparison of Predicted Departure Radii from Rayleigh Based Equation and Experimental Departure Radii.

Figure 5. Error Plot of Predicted Departure Radii from Rayleigh Based Equation.

The large error associated with predicted bubble departure radii is

associated with the average values of constants 𝐶 and 𝐷. Results are improved

by modifying constants 𝐶 and 𝐷 to be functions of system values 𝐴 and/or 𝐽𝑎.

While values 𝐶 and 𝐷 are now variable from system to system, they are constant

for a given a given boiling condition. By comparison of the calculated values of 𝐶

presented in Table 3 with system constant 𝐴, it is determined that 𝐶 is

approximated by the following equation.

𝐶 = 7.459635𝑙𝑛(𝐴) + 2.607226

This equation fits the values of 𝐶 presented in Table 3 with a 𝑅2 value of

0.9579. Comparison of the constant 𝐶 with Jakob numbers for the experimental

systems fails to produce a satisfactory curve fit. The modified values of

calculated constant 𝐶 are now used to calculate modified values for constant 𝐷.

By again comparing the modified values of constant 𝐷 with system values

𝐴 and 𝐽𝑎, a relationship is determined. Constant 𝐷 is best estimated with a 𝑅2

value of 0.9971 by the following equation.

𝐷 = 2.278040 −11 𝐽𝑎2 + 6.485067𝐸 −09 𝐽𝑎 − 3.367751𝐸(−07)

Comparison of constant 𝐷 with constant 𝐴 fails to create an equally good

curve fit.

The derived equations for constants 𝐶 and 𝐷 are substituted into the

general solution to create a new correlation. The modified general solution takes

the following form.

𝑅 = −3𝑒− 7.459635 𝑙𝑛 (𝐴)+2.607226 𝑡 + 2.278040 −11 𝐽𝑎2 + 6.485067𝐸 −09 𝐽𝑎

− 3.367751𝐸(−07) 1 3

By setting this equation equivalent to the Rayleigh equation, the following

relationship is derived.

3𝑒− 7.459635 𝑙𝑛 (𝐴)+2.607226

𝐴 𝑅𝑑𝑒𝑝𝑡 − 2.278040 −11 𝐽𝑎2

− 6.485067𝐸 −09 𝐽𝑎 + 3.367751𝐸(−07)

This equation takes the same form as that previously derived using the

Rayleigh equation. However, the equation is now a function of the system values

𝐴 and 𝐽𝑎 defined in the Rayleigh equation. Analysis of experimental data from

(Van Stralen, Cole, Sluyter, & Sohal, 1975) using the modified general solution is

presented in Table 6.

Table 6. Error Analysis of Predicted Departure Radii based on Rayleigh Based Modified General Solution.

Bubble Number

Ja 𝐶 𝐷

Experimental Departure Radius of

(Van Stralen, Cole,

Sluyter, & Sohal, 1975)

Predicted Departure

Radius

% Error

1 64.4322 16.01348 1.44669E-07 0.00092 0.005597 508.3716

2 124.4618 9.655156 7.92277E-07 0.0079 0.006802 -13.8934

3 200.1375 9.588356 1.84262E-06 0.0119 0.010174 -14.5016

4 385.8247 9.173562 5.52545E-06 0.0136 0.015300 12.5015

5 895.6793 8.851611 2.37162E-05 0.0268 0.026591 -0.7785

6 2038.6934 7.492615 0.000107535 0.0415 0.041509 0.0223

The results obtained from the modified general solution derived using the

Rayleigh equation demonstrate an improved fit with experimental data. This is

seen graphically in Figure 6 and Figure 7.

Figure 6. Comparison of Predicted Departure Radii from Modified Rayleigh Based Equation with Experimental Departure Radii.

Figure 7. Error Plot of Predicted Departure Radii using Rayleigh Based Modified Equation.

The departure radii predicted using the Rayleigh based modified general

solution demonstrates greatly improved fit with experimental data of (Van

Stralen, Cole, Sluyter, & Sohal, 1975). For bubbles having a Jakob number

greater than 100, experimental departure radii are predicted within 15% results

obtained experimentally. Results improve as the Jakob number for the system

grows.

Analysis Using Plesset-Zwick Equation

The previous section evaluated the use of the Rayleigh solution to provide

a departure radius for a bubble growing on a wall in pool boiling. It was noted

that the Rayleigh equation is only effective for modeling bubble growth occurring

within the inertia controlled growth region. To better understand the growth

behavior of a bubble, another equation is required.

In 1954, Plesset and Zwick developed an equation to describe bubble

growth occurring in the heat diffuse region. The derived equation is a function of

the Jakob number of the system and the thermal diffusivity of the surrounding

liquid.

𝑅 = 𝐽𝑎 12𝛼𝑙

𝑡1 2

The equation is commonly written as follows.

𝑅 = 𝐵𝑡1 2

𝐵 = 𝐽𝑎 12𝛼𝑙

𝐽𝑎 =𝜌𝑙𝑐𝑝 ,𝑙

𝜌𝑣𝑕𝑓𝑔 𝑇∞ − 𝑇𝑠𝑎𝑡

In the case of a bubble growing on a wall, the variable 𝐵 and the Jakob

number are rewritten as follows.

𝐵 = 𝐽𝑎∗ 12𝛼𝑙

𝐽𝑎∗ =𝜌𝑙𝑐𝑝 ,𝑙

𝜌𝑣𝑕𝑓𝑔 𝑇𝑤𝑎𝑙𝑙 − 𝑇𝑠𝑎𝑡

The Plesset-Zwick equation is utilized to determine the radial velocity and

acceleration of a bubble by taking its first and second derivatives.

𝑑𝑅

𝑑𝑡=

2𝐵𝑡−1 2

𝑑2𝑅

𝑑𝑡2= −

4𝐵𝑡−3 2

The Plesset-Zwick equations defined above is used to solve the second

order, non-linear differential equation derived in this chapter by direct

substitution. Doing so results in the following equation.

4𝐵2𝑡−1 = 0

By observation, it is seen that there are only two possible solutions to the

equation above; either 𝐵 is equal to zero or 𝑡 is equal to infinity. The variable 𝐵

must be a non-zero value for the Plesset-Zwick equation to model bubble growth.

This implies that bubble departure will only occur at a time equal to infinity.

Substitution of the Plesset-Zwick equation into the derived second order, non-

linear differential equation is not a suitable method for determining the radius of a

bubble at departure. Furthermore, it indicates that the suspicion that entropy

generation reaches a maximum value at bubble departure may be invalid. This

is confirmed by plotting the calculated entropy generation rate against the bubble

radius for on experimental data set from (Van Stralen, Cole, Sluyter, & Sohal,

1975).

Figure 8. Entropy Generation Rate vs. Bubble Radius for Experimental Data Obtained from (Van Stralen, Cole, Sluyter, & Sohal, 1975) (A=1.924969, B=0.267915).

As seen in Figure 8, the calculated entropy generation rate does not reach

a maximum value. This failure to reach a maximum entropy generation rate may

be associated with the inability to effectively model bubble radius and vapor

properties within the bubble.

While direct substitution fails to produce a reasonable solution and

identifies a failure of the calculated entropy generation rate to reach a maximum

value, utilization of the Plesset-Zwick equation to solve general solution to the

second order, non-linear differential equation may result in a correlation which

predicts departure radii of bubbles undergoing nucleate pool boiling. As

previously shown, the general solution takes the following form.

𝑅 = −3𝑒−𝐶𝑡 + 𝐷 1 3

𝑤𝑕𝑒𝑟𝑒 𝑖𝑡𝑠 𝑑𝑒𝑟𝑖𝑣𝑎𝑡𝑖𝑣𝑒 𝑖𝑠

𝑑𝑅

𝑑𝑡= −𝑒−𝐶 −3𝑒−𝐶𝑡 + 𝐷 −1 3

Utilization of the general solution requires that constants 𝐶 and 𝐷 be

solved. This is accomplished by applying boundary conditions. For the purposes

of this analysis the boundary conditions are defined at the time of bubble

departure. At departure, the radius defined by the Plesset-Zwick equation is set

equal to the radius defined by the general solution. Additionally, the slope of

both equations is assumed perpendicular at this time.

𝑅𝑃𝑙𝑒𝑠𝑠𝑒𝑡 −𝑍𝑤𝑖𝑐𝑘 𝑡=𝑡𝑑𝑒𝑝𝑡 = 𝑅𝐺𝑒𝑛 𝑆𝑜𝑙𝑢𝑡𝑖𝑜𝑛 𝑡=𝑡𝑑𝑒𝑝𝑡

𝑑𝑅

𝑑𝑡 𝑃𝑙𝑒𝑠𝑠𝑒𝑡 −𝑍𝑤𝑖𝑐𝑘

= − 𝑑𝑅

arranged to solve for the constant 𝐶.

𝐶 = −𝑙𝑛 2𝐵𝑡𝑑𝑒𝑝𝑡3 2

The time at departure is replaced using the Plesset-Zwick equation.

𝐶 = −𝑙𝑛 2𝑅𝑑𝑒𝑝𝑡

Solving for constant 𝐶 requires experimental data including environmental

conditions and the radius at bubble departure. By averaging results for

experimental data sets, a value for constant 𝐶 is defined.

𝐶 =

−𝑙𝑛 2𝑅𝑑𝑒𝑝𝑡 ,𝑖

𝐵𝑖2

𝑖=1

1975) is used to determine a value for 𝐶. Results of this analysis are shown in

Table 7

Table 7. Values of C for the General Solution Derived Using Plesset-Zwick Equation with Experimental Data of (Van Stralen, Cole, Sluyter, & Sohal, 1975).

Bubble Number

Sohal, 1975)

Ja 𝑩 𝑪

1 0.00092 64.4322 0.009769 11.023181

2 0.0079 124.4618 0.017661 5.756737

3 0.0119 200.1375 0.028018 5.450687

4 0.0136 385.8247 0.053010 6.325360

5 0.0268 895.6793 0.120425 5.931457

6 0.0415 2038.6934 0.267915 6.218867

Average 6.784382

With constant 𝐶 defined, constant 𝐷 is solved for. By substitution of the

constant 𝐶 into the general solution, a solution for constant 𝐷 is determined.

𝐷 =

𝑅𝑑𝑒𝑝𝑡 ,𝑖3 +

3𝑒−𝐶

𝐵𝑖2 𝑅𝑑𝑒𝑝𝑡 ,𝑖

𝑖=1

Experimental data of (Van Stralen, Cole, Sluyter, & Sohal, 1975) is again

utilized to evaluate this equation. Results of this evaluation are shown in Table

Table 8. Values of D for the General Solution Derived Using Plesset-Zwick Equation with Experimental Data of (Van Stralen, Cole, Sluyter, & Sohal, 1975).

Bubble Number

Sohal, 1975)

Ja 𝑩 𝐷

1 1.37057E-09 64.4322 0.009769 3.010152E-05

2 3.02747E-06 124.4618 0.017661 6.795799E-04

3 4.50537E-06 200.1375 0.028018 6.139248E-04

4 2.93983E-06 385.8247 0.053010 2.259056E-04

5 1.9709E-05 895.6793 0.120425 1.873373E-04

6 7.16988E-05 2038.6934 0.267915 1.529070E-04

The resulting general solution to the second order, non-linear differential

equation after substitution of the defined constants is defined as follows.

𝑅 = 3.393921𝐸(−03)𝑡 + 3.149594E(−04) 1 3

At departure, the equation is set equivalent to the Plesset-Zwick equation.

Setting the equations equal produces a function of the departure time. The

equation can be re-written by replacing departure time using the Plesset-Zwick

equation.

3𝑒−𝐶

𝐵2 𝑅𝑑𝑒𝑝𝑡

2 − 𝐷

Substitution of the constants 𝐶 and 𝐷 results in the following equation.

294.644440 ∗ 𝐵2 𝑅𝑑𝑒𝑝𝑡

2 − 3.149594𝐸(−04)

The above equation has three possible solutions for the departure radius.

The solution related to the departure radius of a bubble must be real, positive

and should be in the scale of expected results. Analysis of experimental data

using the equation above is shown in Table 9.

Table 9. Error Analysis of Predicted Departure Radii based on Plesset-Zwick Based General Solution.

Bubble Number Ja

Sohal, 1975)

% Error

1 64.4322 0.00092 0.002976 223.4550

2 124.4618 0.0079 0.005379 -31.9130

3 200.1375 0.0119 0.008527 -28.3459

4 385.8247 0.0136 0.016043 17.9597

5 895.6793 0.0268 0.034263 27.8470

6 2038.6934 0.0415 0.055387 33.4620

The predicted radii from the general solution derived using the Plesset-

Zwick equation is an improved fit with experimental data when compared to

predicted values obtained using the averaged constants version of the Rayleigh

based equation. This is seen graphically in Figure 9 and Figure 10.

Figure 9. Comparison of Predicted Departure Radii from Plesset-Zwick Based Equation with Experimental Departure Radii.

Figure 10. Error Plot of Predicted Departure Radii using Plesset-Zwick Based Equation.

While the predicted departure radii are a better fit with experimental data,

an improved fit will be achieved solving for constants 𝐶 and 𝐷 and functions of

system properties. Like the solution derived using the Rayleigh equation, error is

introduced by determined constants 𝐶 and 𝐷 to be averages over a range of

experimental data points. Results are improved by comparison of values for

constant 𝐶 with system properties 𝐴, 𝐵, and the Jakob number. Doing so leads

to the following relationship for constant 𝐶.

𝐶 = −0.894132 𝑙𝑛 𝐵 + 4.010944

This equation is a poor fit with values of 𝐶 presented in Table 7 with a 𝑅2

value of 0.2758. However new values of constant 𝐷 will be calculated and

compensate for the error in this curve fit. By again comparing the modified

values of constant 𝐷 with system values 𝐴, 𝐵 , and Jakob number, a relationship

is determined. Constant 𝐷 is estimated with a 𝑅2 value of 0.9832 by use of the

following equation.

𝐷 = 4.127304 −03 𝑒𝑥𝑝1.036544∗𝐴

The derived equations for constants 𝐶 and 𝐷 are substituted into the

general solution to create a new relationship. The modified general solution

takes the following form.

𝑅 = −3𝑒− −0.894132 𝑙𝑛 𝐵 +4.010944 𝑡 + 4.127304 −03 𝑒𝑥𝑝1.036544∗𝐴 1 3

By setting this equation equivalent to the Plesset-Zwick equation, the

following equation is derived.

3𝑒 0.894132 𝑙𝑛 𝐵 −4.010944

2 − 4.127304 −03 𝑒𝑥𝑝1.036544𝐴

This equation takes the same form as that previously derived using the

Plesset-Zwick equation. However, the equation is now a function of the system

value 𝐴 defined in the Rayleigh equation and the system value 𝐵 defined in the

Plesset-Zwick equation. Analysis of experimental data from (Van Stralen, Cole,

Sluyter, & Sohal, 1975) using the modified general solution is presented in Table

Table 10. Error Analysis of Predicted Departure Radii based on Plesset-Zwick Based Modified General Solution.

Bubble Number Ja

Sohal, 1975)

% Error

1 64.4322 0.00092 0.000935 1.6524

2 124.4618 0.0079 0.007791 -1.3751

3 200.1375 0.0119 0.010167 -14.5669

4 385.8247 0.0136 0.015478 13.8098

5 895.6793 0.0268 0.025268 -5.7164

6 2038.6934 0.0415 0.044919 8.2390

The results obtained using the modified constants are presented

graphically in Figure 11 and Figure 12.

Figure 11. Comparison of Predicted Departure Radii from Modified Plesset-Zwick Based Equation with Experimental Departure Radii.

Figure 12. Error Plot of Predicted Departure Radii using Plesset-Zwick Based Modified Equation.

It is seen that the Plesset-Zwick based modified general solution shows

significantly improved fit with experimental data of (Van Stralen, Cole, Sluyter, &

Sohal, 1975). The error associated with predicted values is within 15% of

experimental values for the full range of Jakob numbers.

Analysis Using MRG Equation

The Rayleigh and Plesset-Zwick equations describe growth of a bubble in

specific regions. However, neither equation fully describes the growth of bubble

throughout all regions including the transition region from inertia controlled

growth to heat-diffuse controlled growth.

In 1970, Mikic, Rohsenow, and Griffith (MRG) developed an equation

which spans all regions of growth. This was accomplished by writing both the

Rayleigh and Plesset-Zwick equations in terms of 𝑇𝑣−𝑇𝑠𝑎𝑡

∆𝑇. They solved for this

term by rearranging the Plesset-Zwick equation and substituted into the Rayleigh

equation. The result was an equation which describes the growth through all

regions for a bubble growing either on a wall or in an infinite body of liquid. This

resulting dimensionless equation is a function of the variable 𝐴 introduced in the

Rayleigh equation, the variable 𝐵 introduced in the Plesset-Zwick equation,

dimensionless waiting time, and a scaling factor 𝜃 which relates the wall

superheat to the pool superheat.

𝑑𝑅+

𝑑𝑡+= 𝑡+ + 1 + 𝜃

𝑡+ + 𝑡𝑤+

− 𝑡+ 1 2

𝑅+ =𝐴

𝐵2𝑅

𝑡+ =𝐴2

𝐵2𝑡

𝜃 =𝑇𝑊 − 𝑇∞𝑇𝑊 − 𝑇𝑠𝑎𝑡

𝑡𝑤+ = 𝑑𝑖𝑚𝑒𝑛𝑠𝑖𝑜𝑛𝑙𝑒𝑠𝑠 𝑤𝑎𝑖𝑡 𝑡𝑖𝑚𝑒

If the wait time is assumed to be very large, the equation reduces to the

following form.

𝑅+ =2

3 𝑡+ + 1 3 2 − 𝑡+ 3 2 − 1

Changing the equation back to its dimensional form produces the following

equation.

𝑅 =2𝐵2

𝐵2𝑡 + 1

− 𝐴2

𝐵2𝑡

The radial velocity and acceleration of the bubble during its growth are

determined by taking the first and second derivatives of the equation above.

𝑑𝑅

𝑑𝑡= 𝐴

𝐵2𝑡 + 1

− 𝐴2

𝐵2𝑡

𝑑2𝑅

𝑑𝑡2=

2𝐵2

𝐵2𝑡 + 1

−1 2

− 𝐴2

𝐵2𝑡

−1 2

Given the equations above for bubble growth behavior, the second order,

non-linear differential equation derived in this chapter can be solved by direct

substitution. Substituting the MRG equations into the second order, non-linear

differential equation results in the following.

2𝐵2

𝐵2𝑡𝑑𝑒𝑝𝑡 + 1

− 𝐴2

𝐵2𝑡𝑑𝑒𝑝𝑡

− 1 𝐴3

2𝐵2

−1 2

− 𝐴2

−1 2

+ 2 𝐴 𝐴2

− 𝐴2

By observation, it is seen that a solution to the equation above is not

easily achieved. Plotting the left side of the equation above shows that this

relationship only holds true at 𝑡𝑑𝑒𝑝𝑡 = ∞. This departure time is not feasible

solution to the problem as it implies that the radius of departure is infinitely large.

A plot of the value of the left side of the equation above (defined as residual)

versus time is provided in Figure 13 for a set of experimental data obtained from

(Van Stralen, Cole, Sluyter, & Sohal, 1975).

Figure 13. Residual Value vs. Time for Experimental Data Obtained from (Van Stralen, Cole, Sluyter, & Sohal, 1975) (A=1.924969, B=0.267915).

As shown above, direct substitution of the MRG equation and its

derivatives into the second order, non-linear differential equation fails to produce

a predicted departure radius. Additionally, it further supports that fact that the

calculated entropy generation rate fails to reach a maximum value.

While the analysis above is further evidence that the derived entropy

generation rate equation fails to reach a maximum value, utilization of the MRG

equation in the general solution may still produce a bubble departure radius

correlation. It was shown in earlier in this chapter that a general solution exists

for the second order, non-linear differential equation. This general solution takes

the following form.

𝑅 = −3𝑒−𝐶𝑡 + 𝐷 1 3

The derivative of the general solution takes the following form.

𝑑𝑅

𝑑𝑡= −𝑒−𝐶 −3𝑒−𝐶𝑡 + 𝐷 −1 3

Solution of the general solution requires that constants 𝐶 and 𝐷 be solved.

This is accomplished by applying boundary conditions. For the purposes of this

analysis the boundary conditions are defined at the time of bubble departure. At

the time, the radius defined by the MRG equation is set equal to the radius

defined by the general solution. Additionally, the slope of both equations is

assumed perpendicular at this time.

𝑅𝑀𝑅𝐺 𝑡=𝑡𝑑𝑒𝑝𝑡 = 𝑅𝐺𝑒𝑛 𝑆𝑜𝑙𝑢𝑡𝑖𝑜𝑛 𝑡=𝑡𝑑𝑒𝑝𝑡

𝑑𝑅

𝑑𝑡𝑀𝑅𝐺

𝑡=𝑡𝑑𝑒𝑝 𝑡

= − 𝑑𝑅

By substitution of the appropriate equations into the boundary conditions

utilized to solve for constant 𝐶.

𝐶 = −𝑙𝑛

4𝐵4

9𝐴3

𝐵2 𝑡𝑑𝑒𝑝𝑡 + 1 3 2

− 𝐴2

𝐵2 𝑡𝑑𝑒𝑝𝑡 3 2

𝐵2 𝑡𝑑𝑒𝑝𝑡 + 1 1 2

− 𝐴2

If the radial velocity is known at the point of departure, the constant 𝐶 may

be more easily solve using the following form.

𝐶 = −𝑙𝑛 𝑅𝑑𝑒𝑝𝑡

𝑑𝑅𝑑𝑡 𝑑𝑒𝑝𝑡

Unlike the analysis using the Rayleigh and Plesset-Zwick equation, the

MRG equation cannot be rearranged to provide the time of bubble departure as

function of departure radius. Determination of the departure time requires

numerical analysis of the MRG equation.

Solving for constant 𝐶 requires experimental data including environmental

conditions and the radius at bubble departure. By averaging results for multiple

experimental data sets, a value for constant 𝐶 is defined.

𝐶 =

−𝑙𝑛

4𝐵𝑖

9𝐴𝑖3

𝐴𝑖

𝐵𝑖2 𝑡𝑑𝑒𝑝𝑡 ,𝑖 + 1

− 𝐴𝑖

𝐵𝑖2 𝑡𝑑𝑒𝑝𝑡 ,𝑖

𝐴𝑖

− 𝐴𝑖

𝑖=1

1975) is utilized to solve for this constant. Results of this analysis are shown in

Table 11.

Table 11. Values of C for the General Solution Derived Using MRG Equation with Experimental Data of (Van Stralen, Cole, Sluyter, & Sohal, 1975).

Bubble Number

Experimental Departure

Radius of (Van Stralen, Cole,

Departure Time from

Mikic Equation

𝑡𝑑𝑒𝑝𝑡 , s

Ja 𝑨 𝑩 𝑪

1 0.00092 9.070000E-03 64.4322 6.032567 0.009769 11.012151

2 0.0079 2.041900E-01 124.4618 2.572322 0.017661 5.746493

3 0.0119 1.866000E-01 200.1375 2.549390 0.028018 5.433665

4 0.0136 7.331000E-02 385.8247 2.411500 0.053010 6.269883

5 0.0268 6.485000E-02 895.6793 2.309636 0.120425 5.786335

6 0.0415 5.179000E-02 2038.6934 1.924969 0.267915 5.751597

Average 6.666687

With constant 𝐶 defined, constant 𝐷 is solved using the general solution.

The resulting constant 𝐷 is defined by the following equation.

𝐷 =

2𝐵𝑖

3𝐴𝑖

𝐴𝑖

− 𝐴𝑖

𝐵𝑖2 𝑡𝑑𝑒𝑝𝑡

+ 3𝑒−𝐶𝑡𝑑𝑒𝑝𝑡 ,𝑖

𝑖=1

The experimental data of (Van Stralen, Cole, Sluyter, & Sohal, 1975) is

again utilized to solve for constant 𝐷 as shown in Table 12.

Table 12. Values of D for the General Solution Derived Using MRG Equation with Experimental Data of (Van Stralen, Cole, Sluyter, & Sohal, 1975).

Bubble Number

Departure Time from

Mikic Equation

𝑡𝑑𝑒𝑝𝑡 , s

Ja 𝐴 𝐵 𝐷

1 0.00092 9.070000E-03 64.4322 6.032567 0.009769 3.462842E-05

2 0.0079 2.041900E-01 124.4618 2.572322 0.017661 7.800542E-04

3 0.0119 1.866000E-01 200.1375 2.549390 0.028018 7.140906E-04

4 0.0136 7.331000E-02 385.8247 2.411500 0.053010 2.823998E-04

5 0.0268 6.485000E-02 895.6793 2.309636 0.120425 2.668351E-04

6 0.0415

5.179000E-02 2038.693

4 1.924969 0.267915 2.691968E-04

At departure, the fully defined general solution equation is equivalent to

the MRG equation. As previously stated, the resulting relationship cannot be

changed to a function of departure radius. However, the general solution may be

rearranged to define departure time as a function of departure radius.

Substitution of this rearranged general solution in the MRG equation produces

the following function of bubble departure radius.

0 =2𝐵2

3𝐴 −

𝐵23𝑒−𝐶𝑅𝑑𝑒𝑝𝑡

3 +𝐴2𝐷

𝐵23𝑒−𝐶+ 1

− −𝐴2

𝐵23𝑒−𝐶𝑅𝑑𝑒𝑝𝑡

3 +𝐴2𝐷

𝐵23𝑒−𝐶

− 1 −𝑅𝑑𝑒𝑝𝑡

Substitution of the constants 𝐶 and 𝐷 produces the following equation.

0 =2𝐵2

3𝐴 −

𝐵23𝑒−6.666687𝑅𝑑𝑒𝑝𝑡

3 +1.000685E − 03𝐴2

𝐵23𝑒−6.666687+ 1

− −𝐴2

𝐵23𝑒−6.666687𝑅𝑑𝑒𝑝𝑡

3 +1.000685E − 03𝐴2

𝐵23𝑒−6.666687

− 1 −𝑅𝑑𝑒𝑝𝑡

Numerical evaluation of the equation above results in the predicted

departure radii shown in Table 13.

Table 13. Error Analysis of Predicted Departure Radii based on MRG Based General Solution.

Bubble Number Ja

Sohal, 1975)

% Error

1 64.4322 0.00092 0.003116 238.7446

2 124.4618 0.0079 0.005572 -29.4677

3 200.1375 0.0119 0.008758 -26.4000

4 385.8247 0.0136 0.016121 18.5371

5 895.6793 0.0268 0.032843 22.5476

6 2038.6934 0.0415 0.050305 21.2171

It is seen in that table above that the general solution derived using the

MRG equation predicts bubble departure radii within 30% for Jakob numbers

greater than 100. This is seen graphically in Figure 14.

Figure 14. Comparison of Predicted Departure Radii from MRG Based Equation with Experimental Departure Radii.

Figure 15. Error Plot of Predicted Departure Radii using MRG Based Equation

While the predicted departure radii represent a reasonable prediction of

experimental data, results can be further improved. Like with results obtained

using the Rayleigh and Plesset-Zwick equations, solving for constants 𝐶 and 𝐷

as functions of constant system variables will improve results. Forcing constants

𝐶 and 𝐷 to be functions of system values such as 𝐴, 𝐵, or the Jakob number will

further improve results.

Analysis of the values for 𝐶 in Table 11 indicate a relationship with system

constant 𝐴. Constant 𝐶 is estimated with a 𝑅2 value of 0.9498 by the following

equation.

𝐶 = 5.814845𝐸 −02 𝐴2 + 8.891619𝐸(−01)𝐴 + 3.399097

The improved value for 𝐶 is used to generate new values for constant 𝐷.

The new values of D are again compared to the system constants 𝐴, 𝐵, and 𝐽𝑎.

Analysis indicates that the constant D is modeled with a 𝑅2 value of 0.9943 by

the following equation.

𝐷 = −1.957951𝐸 −12 𝐵4

𝐴2+ 1.124843𝐸 −07

𝐴+ 7.128086𝐸 −04

The modified general solution now takes the form shown below.

𝑅 = −3𝑒− 5.814845𝐸 −02 𝐴2+8.891619𝐸(−01)𝐴+3.399097 𝑡 − 1.957951𝐸 −12 𝐵4

+ 1.124843𝐸 −07 𝐵2

𝐴+ 7.128086𝐸 −04

By rearranging the general solution above to solve for time, it may be

substituted into the MRG equation to generate of a function of departure radius.

The solution to the following equation results in the radius of departure for a

bubble.

=2𝐵2

−𝐴2

𝐵23𝑒− 5.814845𝐸 −02 𝐴2+8.891619𝐸(−01)𝐴+3.399097 𝑅𝑑𝑒𝑝𝑡

+ −1.957951𝐸 −12

𝐴2 + 1.124843𝐸 −07 𝐵2

𝐴+ 7.128086𝐸 −04 𝐴2

𝐵23𝑒− 5.814845𝐸 −02 𝐴2+8.891619𝐸(−01)𝐴+3.399097 + 1

− −𝐴2

𝐵23𝑒− 5.814845𝐸 −02 𝐴2+8.891619𝐸(−01)𝐴+3.399097 𝑅𝑑𝑒𝑝𝑡

+ −1.957951𝐸 −12

𝐴2 + 1.124843𝐸 −07 𝐵2

𝐴+ 7.128086𝐸 −04 𝐴2

𝐵23𝑒− 5.814845𝐸 −02 𝐴2+8.891619𝐸(−01)𝐴+3.399097

−𝑅𝑑𝑒𝑝𝑡

The predicted departure radii determined from the equation above are

presented in Table 14.

Table 14. Error Analysis of Predicted Departure Radii based on MRG Based Modified General Solution.

Bubble Number

Ja 𝐶 𝐷

Predicted Departure

Radius

% Error

1 64.4322 5.862968 7.452335E-05 0.00092 0.033729 3566.1467

2 124.4618 5.381753 1.507299E-03 0.0079 0.005585 -29.3044

3 200.1375 5.378633 1.057462E-03 0.0119 0.008660 -27.2248

4 385.8247 5.359889 8.078969E-04 0.0136 0.014679 7.9313

5 895.6793 5.346063 7.306733E-04 0.0268 0.028679 7.0112

6 2038.6934 5.294014 7.158238E-04 0.0415 0.037803 -8.9093

These results are seen graphically in Figure 16 and Figure 17.

Figure 16. Comparison of Predicted Departure Radii from Modified MRG Based Equation with Experimental Departure Radii.

Figure 17. Error Plot of Predicted Departure Radii using MRG Based Modified Equation

As shown above, the MRG based modified general solution has improved

predicted departure radii for bubbles with higher Jakob numbers when compared

with experimental data of (Van Stralen, Cole, Sluyter, & Sohal, 1975). The

predicted departure radii fit experimental data within 30% for systems is a Jakob

number larger than approximately 100. Additionally, it can be seen that the error

is further reduced for Jakob numbers of approximately 300 and larger.

Confirmation of Correlation

Comparison of the models generated in this chapter indicates that the

general solution to the second order, non-linear differential equation is capable of

predicting departure radii of (Van Stralen, Cole, Sluyter, & Sohal, 1975) within

15%. This is achieved by determining the real, non-negative solution to the

following system property dependant; third order equation derived using the

modified Plesset-Zwick equation for bubble growth.

3𝑒 0.894132 𝑙𝑛 𝐵 −4.010944

2 − 4.127304 −03 𝑒𝑥𝑝1.036544𝐴

Confirmation of the equation is performed by analysis of additional

experimental data sets. Data published by (Cole & Shulman, 1966b) for bubbles

growing in sub atmospheric pressure water is utilized. This data set differs from

that of (Van Stralen, Cole, Sluyter, & Sohal, 1975) in that the fluid temperature is

maintained at saturation temperature rather than superheated temperatures.

Results of the analysis are presented in Table 16.

Table 15. Comparison of Derived Equation with Experimental Data of (Cole & Shulman, 1966b)

Bubble Jakob Number

Experimental Departure Radius

of (Cole & Shulman, 1966b)

% Error

1 89.2283 0.00900 0.003734 -58.5149

2 0.00775 -51.8237

3 0.00650 -42.559

Average 0.00775 -51.8237

4 191.9251 0.00925 0.007453 -19.4321

5 0.00800 -6.84342

Average 0.008625 -13.5933

6 296.9101 0.01900 0.015787 -16.913

7 0.01500 5.243592

8 0.01275 23.81599

9 0.01300 21.43491

10 0.00925 70.66528

11 0.01275 23.81599

12 0.01175 34.35352

13 0.01100 43.51399

14 0.01025 54.01501

15 0.00950 66.17409

Average 0.012425 27.08554

16 1993.5703 0.02075 0.02701 30.17266

17 0.02000 35.05414

18 0.01900 42.16225

Average 0.019917 35.61723

These results are presented graphically in Figure 18.

Figure 18. Comparison of Predicted Departure Radii with Experimental Data of (Cole & Shulman, 1966b).

Evaluation of the data indicates that the equation derived for departure

radii is capable of estimated experimental departure radii within approximately

50% for system undergoing boiling at saturated conditions.

Further analysis of the equation is performed by comparison with

experimental data of (Ellion, 1954) for bubbles undergoing growth in sub cooled

water at atmospheric pressure. The experimental data utilized for this analysis

represents the average maximum radii reached during sub cooled boiling at

specific system conditions. These predicted radii vary from departure radii in that

the bubbles do not depart from the heating surface. Rather the bubbles reach a

maximum radius at which point they begin to collapse.

Results of the analysis are summarized in Table 16.

Table 16. Comparison of Derived Equation with Experimental Data of (Ellion, 1954).

Bubble Jakob Number

Average Experimental

Maximum Radius of (Ellion, 1954)

𝑅𝑚𝑎𝑥 , m

Predicted Maximum Radius

𝑅𝑚𝑎𝑥, m

% Error

1 63.81317 0.000559 0.000956 71.0993

2 79.72192 0.000495 0.000763 54.0291

3 83.96245 0.004700 0.000718 52.8118

4 93.35631 0.000470 0.000630 34.1205

5 94.98782 0.000462 0.000616 33.1931

6 95.18446 0.000495 0.000616 24.2931

7 100.0765 0.000445 0.000575 29.2549

8 100.599 0.000437 0.000574 31.3915

9 106.478 0.000376 0.000533 41.7945

10 108.5569 0.000351 0.000518 47.9145

These results are shown graphically in Figure 19.

Figure 19. Comparison of Predicted Maximum Radii with Experimental Data of (Ellion, 1954).

The analysis above indicates that the derived equation is capable of

predicting maximum radii from the experimental data of (Ellion, 1954) within

approximately 70% for systems with Jakob numbers ranging from approximately

60 to 110.

Summary

A novel correlation for bubble departure radii is derived by performing an

entropy generation study on a spherical bubble undergoing nucleate pool boiling.

The entropy generation study results in a second order, non-linear differential

which is described by a general solution. This is achieved by modeling the rate

of work performed by the bubble as that of a thermodynamic system. Rayleigh,

Plesset-Zwick, and MRG equations for bubble growth are utilized to solve the

second order, non-linear differential equation and its general solution.

Direct substitution of the three equations into the differential equation is

unsuccessful in producing predicted departure radii. Furthermore, results of this

analysis, along with analysis of the derived entropy generation rate equation

shown below, indicates that calculated entropy generation rates do not reach a

maximum value.

Utilization of the Rayleigh, Plesset-Zwick, and MRG equations to

solve for the constants 𝐶 and 𝐷 of the general solution results in varying ability to

predict departure radii. Most accurate predicted departure radii are achieved by

utilization of the Plesset-Zwick equation. The resulting equation is presented

below.

3𝑒 0.894132 𝑙𝑛 𝐵 −4.010944

2 − 4.127304 −03 𝑒𝑥𝑝1.036544𝐴

Further analysis of additional experimental data fails to reproduce the

same level of accuracy achieved for superheated boiling. The inability to

accurately predict departure radii for these data sets is likely associated with the

inability of accurately consider all system properties in the derived equation.

Evaluation of the equation above indicates that it is not a function of all variables

commonly associated with bubble departure correlations including gravity,

contact, angle, water superheat and many others. Each of the data sets utilized

for the analysis have been obtained at various levels of liquid superheat. This

distinction is not accounted for in the derived equation. Furthermore, the derived

equation relies on constant values which have been obtained by evaluation of

experimental data of (Van Stralen, Cole, Sluyter, & Sohal, 1975). As a result, the

equation is biased towards accurately predicting departure radii from this set of

experimental data.

While the derived correlation does not consider all variables commonly

associated with bubble departure, the correlation and the method utilized to

derive it are valuable. Utilizing the methods presented in this chapter, the

correlation above can be adapted to more accurately predict departure radii for a

given heater system. This ability allows for the prediction of bubble departure

radii within this given system at operating conditions other than those specifically

tested.

ENTROPY GENERATION RATE STUDY (NET FORCE METHOD)

In Chapter 2, an entropy generation rate study was performed by

modeling the work performed by the bubble as that of a thermodynamic system.

While the resulting equating fails to read a maximum value, it derivative led to a

correlation for bubble departure radii for bubbles undergoing growth on a wall

during nucleate pool boiling. It has been shown that the resulting correlation is

capable of predicting departure radii within 15% of experimental data of (Van

Stralen, Cole, Sluyter, & Sohal, 1975) but is less capable of predicting

experimental data sets for bubble growing in saturated and sub-cooling boiling.

In this chapter, a novel correlation for bubble departure radius is determined

using the second law of thermodynamics. The rate of work performed by the

bubble on its surrounding will be calculated using the net force acting on a

bubble during growth on a wall. All steps of the following work are shown in

Appendix C.

Derivation of Heat Transfer Rate

Determination of the heat transfer rate requires relationships for the rate of

work performed by a bubble, the rate of energy accumulation within the bubble,

and the rate at which energy is transferred across the bubble boundary.

Rate or Work

In this section rate of work performed by the bubble is modeled as a

function of the net forces acting on it. It is assumed that the net force acting on

the bubble results in purely radial growth of the bubble. In this case, the total

work done by a bubble is determined by integrating the product of the net force

acting on the bubble and the rate of radial growth of the bubble.

𝑊 = 𝐹𝑛𝑒𝑡 𝑑𝑅𝑅

As the net force is a function of bubble radius, the total work performed is

solved by performing successive integration by parts of the equation above.

Doing so results in the following equation.

𝑊 = 𝑅 𝐹𝑛𝑒𝑡 −1

𝑑𝐹𝑛𝑒𝑡𝑑𝑡

𝑅 +1

𝑑2𝐹𝑛𝑒𝑡𝑑𝑡2

𝑅2 −1

𝑅3 + ⋯

The rate at which work is done by a bubble on its surrounding is found by

taking the derivative of the above equation with respect to time.

𝑊 = 𝐹𝑛𝑒𝑡𝑅 + 𝑅 − 1 −𝑑𝐹𝑛𝑒𝑡𝑑𝑡

𝑅 +1

𝑅2 −1

𝑅3 +1

𝑅4 + ⋯

If the influence of the net force derivatives is neglected, the equation for

work rate is reduced to the first term of the equation above.

𝑊 = 𝐹𝑛𝑒𝑡𝑅

Solution of the rate of work equation requires a definition for net force.

The net force acting on a bubble is modeled by evaluation of the forces

described by (Keshock & Siegel, 1964).

𝐹𝑛𝑒𝑡 = 𝐹𝑏 + 𝐹𝑝 − 𝐹𝑖 − 𝐹𝜍 − 𝐹𝐷

Substitution of the appropriate relationships for each force (defined in

Table 2) into the equation leads to the following.

𝐹𝑛𝑒𝑡 =4𝜋𝑅3

3 𝜌𝑙 − 𝜌𝑣 𝑔 + 𝜋𝑅𝑏𝜍 sin𝛽 −

𝑑𝑡

16𝜌𝑙

4𝜋𝑅3

3 𝑅 − 2𝜋𝑅𝑏𝜍𝑙 sin𝛽

−𝜋

4𝑎𝜇𝑙𝑅𝑅

It is noted that the equations for excess pressure, surface tension, and the

term representing sum of the two forces are functions of the bubble base radius.

Since this value may not be known or modeled, it is suggested that a

dimensionless scaling factor 𝑆𝜍 be introduced.

𝑆𝜍 =𝑅𝑏

𝑅sin𝛽

The proposed dimensionless scaling

factor is a ratio of the bubble base radius,

bubble radius, and contact angle. Each of

these dimensions is seen graphically in Figure

20. This dimensionless scaling factor allows

for the excess pressure, surface tensions, and

their sum to be calculated in terms of the

bubble radius, allowing for consistency of

variables throughout the model.

While this scaling term requires the bubble base radius to calculate, it may

be approximated by evaluation of previous bubble departure diameter

correlations. For the sake of simplicity, the Fritz equation is utilized for

demonstration.

2𝑅𝑑𝑒𝑝𝑡 = 0.0208𝛽 𝜍𝑙

Figure 20. Bubble Dimensions.

𝑅𝑏

By simple mathematical analysis, the Fritz equation is rearranged in the

form of a balance of the buoyant and surface tension forces at the point of bubble

departure.

3𝑅𝑑𝑒𝑝𝑡

3𝑔(𝜌𝑙 − 𝜌𝑣) = 2𝜋𝑅𝑑𝑒𝑝𝑡 𝜍𝑙 0.0208𝛽 2

Comparison of the surface tension term of the Fritz correlation with the

proposed form of the surface tension force equation leads to the following

definition for the dimensionless scaling factor 𝑆𝜍 .

𝑆𝜍 = 0.0208𝛽 2

Analysis of other bubble departure diameter correlations leads to

additional definitions for the dimensionless scaling factor. Several scaling factors

are provided in Table 17.

Table 17. Alternative dimensionless scaling factors calculated from bubble departure correlations.

Correlation Derived Dimensionless Scaling Factor, 𝑺𝝈

(Fritz, 1935) 𝑆𝜍 =

0.0208𝛽 2

(Cole, 1967) 𝑆𝜍 =

6 0.04𝐽𝑎 2

(Cole & Rohsenow, 1969) 𝑆𝜍 =

𝐶2 𝐽𝑎𝑥 5 2

𝐽𝑎x =𝑇𝑐𝑐𝑝 ,𝑙𝜌𝑙𝜌𝑣𝑕𝑙𝑣

𝐶 = 1.5𝑥10−4 for water

𝐶 = 4.65𝑥10−4 for fluids other than water

Rearranging the net force equation and applying the non-dimensional

scaling factor results in the following equation.

𝐹𝑛𝑒𝑡 =4𝜋

3𝑔 𝜌𝑙 − 𝜌𝑣 𝑅

3 − 𝜋𝜍𝑙𝑆𝜍 𝑅 −11𝜋

12𝜌𝑙 3𝑅

2𝑅 2 + 𝑅3𝑅 −𝜋

4𝑎𝜇𝑙 𝑅𝑅

Energy Change Rate

A relationship for rate of energy change was derived in Chapter 2. This

function is utilized for the work presented in this section.

𝐸 =4𝜋𝑅2

4 𝜌𝑣

20 𝜌𝑣

120 𝜌𝑣

𝑅4 + ⋯

If the state of the vapor within the bubble is again assumed constant and

equivalent to the saturation pressure of the bulk liquid, the above equation can

be simplified to the following form.

𝐸 = 4𝜋𝑒𝑣𝜌𝑣𝑅2𝑅

Energy Transfer Rate

As described in Chapter 2, the rate of energy transfer across the boundary

of a spherical bubble is defined by the following series.

𝑑𝑡 𝑚𝑕𝑖𝑛 =

4𝜋𝑅2

3 3𝜌𝑣𝑕𝑣𝑅

4 𝜌𝑣

20 𝜌𝑣

120 𝜌𝑣

𝑅4 + ⋯

If the state of the vapor is assumed constant and equal to the saturation

pressure of the bulk liquid, the energy transfer rate reduces to a function of

bubble growth rate.

𝑑𝑡 𝑚𝑕𝑖𝑛 = 4𝜋𝑕𝑣𝜌𝑣𝑅

Heat Transfer

By substitution of the equations derived above into the first law of

thermodynamics, the heat transfer rate for the bubble is defined.

𝑄 = 𝐹𝑛𝑒𝑡𝑅 + 𝑅 − 1 −𝑑𝐹𝑛𝑒𝑡𝑑𝑡

𝑅 +1

𝑅2 −1

𝑅3 +1

𝑅4 + ⋯

+4𝜋𝑅2

3 3𝜌𝑣 𝑒𝑣 − 𝑕𝑣 𝑅

+ 𝑅

− 1 −𝑑𝜌𝑣𝑑𝑡

𝑅 +1

𝑅2 −1

𝑅3 +1

+ −𝜌𝑣𝑅 +1

𝑅2 −3

𝑅3 +1

4𝜌𝑣𝑅

2 −3

𝑅3 +1

+ −1

20𝜌𝑣𝑅

120𝜌𝑣𝑅

This equation is further reduced application of the definition of enthalpy.

𝑒𝑣 − 𝑕𝑣 = −𝑝𝑣𝑣𝑣 = −𝑝𝑣𝜌𝑣

Substitution of this relationship results in the following equation.

𝑅 +1

𝑅2 −1

𝑅3 +1

𝑅4 + ⋯

+4𝜋𝑅2

3 −3𝑝𝑣𝑅

+ 𝑅 − 1 𝑅𝑑𝑝𝑣𝑑𝑡

4𝑅2

20𝑅3

120𝑅4

If vapor pressure is assumed constant and equivalent to the saturation

pressure of the bulk liquid, and the affects of net force derivative are ignored, the

equation reduces to the following.

𝑄 = 𝐹𝑛𝑒𝑡𝑅 − 4𝜋𝑝𝑠𝑎𝑡 𝑇∞ 𝑅2𝑅

Unlike the equation derived in Chapter 2, the equations for heat transfer

derived using the net force method are functions of bubble radius, net force, and

vapor pressure.

Derivation of Entropy Generation Rate

With heat transfer rate defined, it is possible to determine the rate of

entropy generation. Like determination of heat transfer rate, this requires

relationships for the rate of entropy accumulation, entropy transfer rate, and the

net entropy flow rate.

Entropy Accumulation Rate

The entropy accumulation rate for a spherical bubble is determined the

same way as described in Chapter 2. The resulting equation for the rate of

entropy accumulation is as follows.

𝑆 =4𝜋

𝑑𝑡𝑅 +

𝑑𝑡2𝑅2 −

𝑑𝑡3𝑅3

𝑑𝑡4𝑅4 + ⋯

By applying the assumption of constant vapor properties at the saturation

point of the bulk liquid, this equation reduces to the following form.

𝑆 = 4𝜋𝜌𝑣𝑠𝑣𝑅2𝑅

Entropy Transfer Rate

The entropy transfer rate for a bubble growing on a wall is determined by

dividing the heat transfer rate by the wall temperature.

𝑇𝑤𝑎𝑙𝑙=

𝐹𝑛𝑒𝑡𝑇𝑤𝑎𝑙𝑙

+ 𝑅 − 1

𝑇𝑤𝑎𝑙𝑙 −

𝑅 +1

𝑅2 −1

𝑅3 +1

𝑅4 + ⋯

+4𝜋𝑅2

3𝑇𝑤𝑎𝑙𝑙 −3𝑝𝑣𝑅

4𝑅2

20𝑅3

120𝑅4

If the vapor pressure within the bubble is assumed constant, the equation

can be reduced to the following.

+ 𝑅 − 1

𝑅 +1

𝑅2 −1

𝑅3 +1

𝑅4 + ⋯

−4𝜋

𝑇𝑤𝑎𝑙𝑙𝑝𝑠𝑎𝑡 𝑇∞ 𝑅

If the affects of the net force derivatives are neglected, the rate of heat

transfer is further simplified.

𝑅 −4𝜋

Net Entropy Flow Rate

Since the state of the vapor flowing into the bubble is assumed to be

equivalent to the state of the vapor accumulated within the bubble, the equation

for net entropy flow rate takes the same form as that derived for the entropy

change rate.

Entropy Generation Rate

The rate of entropy generation is determined by substitution of the

equations derived above in the second law of thermodynamics. As noted in

Chapter 2, the influence of the rate of entropy accumulation is canceled by the

influence of the rate of entropy transfer at the bubble boundary.

𝑆 𝑔𝑒𝑛 = −𝐹𝑛𝑒𝑡𝑇𝑤𝑎𝑙𝑙

− 𝑅 − 1

𝑅 +1

𝑅2 −1

𝑅3 +1

𝑅4 + ⋯

−4𝜋𝑅2

4𝑅2

20𝑅3

120𝑅4

If the vapor pressure is assumed constant and equivalent to the saturation

pressure of the bulk liquid, and the net force derivatives are neglected, the

equation is reduced to the following form.

𝑅 +4𝜋

If entropy generation rate reaches a maximum at the point of bubble

departure as hypothesized, the bubble departure radius is determined by taking

the derivative of entropy generation rate with respect to radius and setting

equivalent to zero. The resulting equations are shown below.

0 = −𝑅 𝐹𝑛𝑒𝑡 + 𝑅𝑅 + 𝑅 2 − 2𝑅 𝑑𝐹𝑛𝑒𝑡𝑑𝑡

+ 2𝑅𝑅 −1

2𝑅2𝑅 − 𝑅𝑅 2 − 𝑅

6𝑅3𝑅 +

2𝑅2𝑅 2 +

2𝑅2 − 𝑅2𝑅

3𝑅3𝑅 −

24𝑅4𝑅 −

6𝑅3𝑅 2 −

6𝑅3

24𝑅4 −

24𝑅4𝑅

−4𝜋

3 −6𝑅𝑅 2 − 3𝑅2𝑅 𝑝𝑣 + −6𝑅2𝑅 + 3𝑅2𝑅 2 + 𝑅3𝑅

+ −𝑅3 + 2𝑅3𝑅 − 𝑅3𝑅 2 −1

4𝑅4𝑅

4𝑅4 −

2𝑅4𝑅 +

5𝑅4𝑅 2 +

20𝑅5𝑅

+ −1

20𝑅5 +

20𝑅5𝑅 −

20𝑅5𝑅 2 −

120𝑅6𝑅

120𝑅6 −

120𝑅6𝑅

If the affects of the net force derivatives are neglected and the vapor

pressure is assumed constant, the rate of heat transfer is simplified.

0 = −𝑅 𝐹𝑛𝑒𝑡 + 4𝜋 2𝑅𝑅 2 + 𝑅2𝑅 𝑝𝑠𝑎𝑡 𝑇∞

Rearranging the equation above allows it to be rewritten as a modified

force balance equation.

𝐹𝑛𝑒𝑡 = 4𝜋 2𝑅𝑅 2

𝑅 + 𝑅2 𝑝𝑠𝑎𝑡 𝑇∞

The equation above modifies the force balance analysis performed by

(Keshock & Siegel, 1964) by implying that departure of a bubble undergoing

nucleate pool boiling on a wall occurs at a value other than 0.

Both the full equation and the simplified equations must be solved

numerically by substitution of relationships for net force and vapor pressure. The

net force has already been defined in this chapter. A relationship for vapor

pressure was introduced in Chapter 2. Both equations and their first five

derivatives are provided in Table 18 and Table 19.

Table 18. Net Force Derivatives.

Order of Derivative

Equation

0 𝐹𝑛𝑒𝑡 =

3 − 𝜋𝜍𝑙𝑆𝜍𝑅 −11𝜋

12𝜌𝑙 3𝑅2𝑅 2 + 𝑅3𝑅 −

1 𝑑𝐹𝑛𝑒𝑡𝑑𝑡

=4𝜋

3𝑔 3 𝜌𝑙 − 𝜌𝑣 𝑅

2𝑅 − 𝑅3𝑑𝜌𝑣𝑑𝑡

− 𝜋𝜍𝑙𝑆𝜍𝑅 −11𝜋

12𝜌𝑙 6𝑅𝑅 3 + 9𝑅2𝑅 𝑅 + 𝑅3𝑅

−𝜋

4𝑎𝜇𝑙 𝑅𝑅 + 𝑅 2

2 𝑑𝐹𝑛𝑒𝑡𝑑𝑡

= 4𝜋𝑔 6𝑅𝑅 2 + 3𝑅2𝑅 𝜌𝑙 − 𝜌𝑣 − 6𝑅2𝑅 𝑑𝜌𝑣𝑑𝑡

− 𝑅3𝑑2𝜌𝑣𝑑𝑡2

− 𝜋𝜍𝑙𝑆𝜍𝑅

−11𝜋

12𝜌𝑙 6𝑅 4 + 36𝑅𝑅 2𝑅 + 9𝑅2𝑅 2 + 12𝑅2𝑅 𝑅 + 𝑅3

𝑑4𝑅

𝑑𝑡4

−𝜋

4𝑎𝜇𝑙 3𝑅 𝑅 + 𝑅𝑅

Order of Derivative

Equation

3 𝑑3𝐹𝑛𝑒𝑡𝑑𝑡3

=4𝜋

3𝑔 6𝑅 3 + 18𝑅𝑅 𝑅 + 3𝑅2𝑅 𝜌𝑙 − 𝜌𝑣 − 18𝑅𝑅 2 + 9𝑅2𝑅

− 9𝑅2𝑅 𝑑2𝜌𝑣𝑑𝑡2

− 𝑅3 𝑑3𝜌𝑣𝑑𝑡3

− 𝜋𝜍𝑙𝑆𝜍 𝑅

−11𝜋

12𝜌𝑙 60𝑅 3𝑅 + 90𝑅𝑅 𝑅 2 + 60𝑅𝑅 2𝑅 + 30𝑅2𝑅 𝑅 + 15𝑅2𝑅

𝑑4𝑅

𝑑𝑡4

+ 𝑅3𝑑5𝑅

𝑑𝑡5 −

4𝑎𝜇𝑙 3𝑅

2 + 4𝑅 𝑅 + 𝑅𝑑4𝑅

𝑑𝑡4

=4𝜋

3𝑔 36𝑅 2𝑅 + 18𝑅𝑅 2 + 24𝑅𝑅 𝑅 + 3𝑅2

𝑑4𝑅

𝑑𝑡4 𝜌𝑙 − 𝜌𝑣

− 24𝑅 3 + 72𝑅𝑅 𝑅 + 12𝑅2𝑅 𝑑𝜌𝑣𝑑𝑡

− 36𝑅𝑅 2 + 18𝑅2𝑅 𝑑2𝜌𝑣𝑑𝑡2

− 12 𝑅2𝑅 𝑑3𝜌𝑣𝑑𝑡3

− 𝜋𝜍𝑙𝑆𝜍 𝑑4𝑅

𝑑𝑡4

−11𝜋

12𝜌𝑙 270𝑅 2𝑅 2 + 120𝑅 3𝑅 + 90𝑅𝑅 3 + 360𝑅𝑅 𝑅 𝑅 + 30𝑅2𝑅 2

+ 45𝑅2𝑅 𝑑4𝑅

𝑑𝑡4+ 90𝑅𝑅 2

𝑑4𝑅

𝑑𝑡4+ 18𝑅2𝑅

𝑑5𝑅

𝑑𝑡5+ 𝑅3

𝑑6𝑅

𝑑𝑡6

−𝜋

4𝑎𝜇𝑙 10𝑅 𝑅 + 5𝑅

𝑑4𝑅

𝑑𝑡4+ 𝑅

𝑑5𝑅

𝑑𝑡5

=4𝜋

3𝑔 90𝑅 𝑅 2 + 60𝑅 2𝑅 + 60𝑅𝑅 𝑅 + 30𝑅𝑅

𝑑4𝑅

𝑑𝑡4+ 3𝑅2

𝑑5𝑅

− 180𝑅 2𝑅 + 90𝑅𝑅 2 + 120𝑅𝑅 𝑅 + 15𝑅2𝑑4𝑅

𝑑𝑡4 𝑑𝜌𝑣𝑑𝑡

− 60𝑅 3 + 180𝑅𝑅 𝑅 + 30𝑅2𝑅 𝑑2𝜌𝑣𝑑𝑡2

− 60𝑅 2 + 30𝑅2𝑅 𝑑3𝜌𝑣𝑑𝑡3

𝑑𝑡5

−11𝜋

12𝜌𝑙 630𝑅 𝑅 3 + 1260𝑅 2𝑅 𝑅 + 120𝑅 3

𝑑4𝑅

𝑑𝑡4+ 630𝑅𝑅 2𝑅 + 420𝑅𝑅 𝑅 2

+ 630𝑅𝑅 𝑅 𝑑4𝑅

𝑑𝑡4+ 90𝑅 3

𝑑4𝑅

𝑑𝑡4+ 105𝑅2𝑅

𝑑4𝑅

𝑑5𝑅

𝑑𝑡5 + 126𝑅𝑅 2

𝑑5𝑅

𝑑𝑡5

+ 21𝑅2𝑅 𝑑6𝑅

𝑑𝑡6+ 𝑅3

𝑑7𝑅

𝑑𝑡7 −

4𝑎𝜇𝑙 10𝑅 3 + 15𝑅

𝑑4𝑅

𝑑𝑡4+ 6𝑅

𝑑5𝑅

𝑑𝑡5+ 𝑅

𝑑6𝑅

𝑑𝑡5

Table 19. Vapor Pressure Derivatives

Order of Derivative

Equation

0 𝑝𝑣 = 𝑝∞ +

2𝜍𝑙𝑅

+ 𝜌𝑙 3

2 𝑑𝑅

𝑑𝑡

+ 𝑅𝑑2𝑅

𝑑𝑡2

1 𝑑𝑝𝑣𝑑𝑡

= −2𝜍𝑙𝑅2

𝑅 + 𝜌𝑙 3𝑅 𝑅 + 𝑅 𝑅 + 𝑅𝑅

2 𝑑2𝑝𝑣𝑑𝑡2

=4𝜍𝑙𝑅3

𝑅 2 −2𝜍𝑙𝑅2

𝑅 + 𝜌𝑙 4𝑅 2 + 5𝑅 𝑅 + 𝑅𝑑4𝑅

𝑑𝑡4

= −12𝜍𝑙𝑅4

𝑅 3 +12𝜍𝑙𝑅3

𝑅 𝑅 −2𝜍𝑙𝑅2

𝑅 + 𝜌𝑙 13𝑅 𝑅 + 6𝑅 𝑑4𝑅

𝑑𝑡4+ 𝑅

𝑑5𝑅

𝑑𝑡5

=48𝜍𝑙𝑅5

𝑅 4 −72𝜍𝑙𝑅4

𝑅 2𝑅 +12𝜍𝑙𝑅3

𝑅 2 +4𝜍𝑙𝑅3

𝑑4𝑅

𝑑𝑡4

+ 𝜌𝑙 13𝑅 2 + 19𝑅 𝑑4𝑅

𝑑𝑡4+ 7𝑅

𝑑5𝑅

𝑑𝑡5+ 𝑅

𝑑6𝑅

𝑑𝑡6

= −240𝜍𝑙𝑅6

𝑅 5 +480𝜍𝑙𝑅5

𝑅 3𝑅 −252𝜍𝑙𝑅4

𝑅 𝑅 2 +28𝜍𝑙𝑅3

𝑅 2𝑅 +10𝜍𝑙𝑅3

𝑅 𝑑4𝑅

𝑑𝑡4−

2𝜍𝑙𝑅2

𝑑5𝑅

𝑑𝑡5

+ 𝜌𝑙 45𝑅 𝑑4𝑅

𝑑𝑡4+ 26𝑅

𝑑5𝑅

𝑑𝑡5+ 8𝑅

𝑑6𝑅

𝑑𝑡6+ 𝑅

𝑑7𝑅

𝑑𝑡7

Analysis of Net Force Correlation

Two equations have been derived for predicting departure radii by

modeling the work performed with the net force acting on the bubble. The

equations derived are based on the following sets of assumptions.

Constant vapor pressure and negligible net force derivatives

Vapor pressure and net force derivatives considered

Solution of each equation requires a model for the growth behavior of a

bubble in nucleate pool boiling. For this analysis, the MRG equation derived by

(Mikic, Rohsenow, & Griffith, 1970) is utilized as it accurately models bubbles

growing in both the inertia and heat-diffuse regions. This equation and its

derivatives are listed in Table 20.

Table 20. MRG Equation Derivatives.

Order of Derivative

Equation

0 𝑅 =

2𝐵2

𝐵2𝑡 + 1

− 𝐴2

𝐵2𝑡

1 𝑑𝑅

𝑑𝑡= 𝐴

𝐵2𝑡 + 1

− 𝐴2

𝐵2𝑡

2 𝑑2𝑅

𝑑𝑡2=

2𝐵2

𝐵2𝑡 + 1

−1 2

− 𝐴2

𝐵2𝑡

−1 2

3 𝑑3𝑅

𝑑𝑡3= −

4𝐵4

𝐵2𝑡 + 1

−3 2

− 𝐴2

𝐵2𝑡

−3 2

4 𝑑4𝑅

𝑑𝑡4=

3𝐴7

8𝐵6

𝐵2𝑡 + 1

−5 2

− 𝐴2

𝐵2𝑡

−5 2

5 𝑑5𝑅

𝑑𝑡5= −

15𝐴9

16𝐵8

𝐵2𝑡 + 1

−7 2

− 𝐴2

𝐵2𝑡

−7 2

6 𝑑6𝑅

𝑑𝑡6=

105𝐴11

32𝐵10

𝐵2𝑡 + 1

−9 2

− 𝐴2

𝐵2𝑡

−9 2

7 𝑑7𝑅

𝑑𝑡7= −

945𝐴13

64𝐵12

𝐵2𝑡 + 1

−11 2

− 𝐴2

𝐵2𝑡

−11 2

Solution of the three equations is performed numerically using the MatLab

code provided in Appendix D.

Constant Vapor Pressure and Negligible Net Force Derivatives

A simplified equation has been derived by assuming that the influence of

the rate of change of net force is negligible and that vapor pressure is assumed

constant and equivalent to the saturation pressure of the bulk liquid. The

resulting equation is as follows.

𝐹𝑛𝑒𝑡 = −4𝜋 2𝑅𝑅 2

𝑅 + 𝑅2 𝑝𝑠𝑎𝑡 𝑇∞

Substitution of the net force equation allows it to be reduced to the

following.

0 = 𝑅3 + 3 𝑝𝑠𝑎𝑡 𝑇∞

𝑔 𝜌𝑙 − 𝜌𝑣 2

𝑅𝑅 2

𝑅 + 𝑅2 −

3𝑆𝜍4

𝜍𝑙

𝑔 𝜌𝑙 − 𝜌𝑣 𝑅

𝜌𝑙𝑔 𝜌𝑙 − 𝜌𝑣

3𝑅2𝑅 2 + 𝑅3𝑅 −3𝑎

𝜇𝑙𝑔 𝜌𝑙 − 𝜌𝑣

𝑅𝑅

Numerical analysis of this equation is performed using scaling factors

derived from the correlations of (Fritz, 1935), (Cole, 1967), and (Cole &

Rohsenow, 1969). Results of the numerical analysis indicate that the derived

equation is unable to predict departure radii from experimental data. This is an

indication that the calculated entropy generation rate does not reach a maximum

value. Analysis of the derived entropy generation equation shown below

confirms this.

𝑅 +4𝜋

By plotting the equation above, it is confirmed that the calculated entropy

generation rate does not reach a maximum value (Figure 21).

Figure 21. Entropy Generation Rate vs. Bubble Radius for Experimental Data Obtained from (Van Stralen, Cole, Sluyter, & Sohal, 1975) (A=1.924969, B=0.267915).

Vapor Pressure and Net Force Derivatives Considered

The derived equation is improved by including affects of net force

derivatives and changes in vapor pressures. By considering all variables, the

equation expands to the following.

0 = −𝑅

𝑅 𝐹𝑛𝑒𝑡 + −2 + 𝑅 + 𝑅

𝑅 𝑑𝐹𝑛𝑒𝑡𝑑𝑡

+ 2𝑅 − 𝑅𝑅 −𝑅

𝑅 −

2𝑅2

𝑅 𝑑2𝐹𝑛𝑒𝑡𝑑𝑡2

+ −𝑅2 +1

2𝑅2𝑅 +

𝑅 +

6𝑅3

3𝑅3 −

6𝑅3𝑅 −

𝑅 −

24𝑅4

+ −1

24𝑅4 +

+4𝜋𝑅

3 6𝑅 + 3𝑅

𝑅 𝑝𝑣 + 6𝑅 − 3𝑅𝑅 − 𝑅2

𝑅 𝑑𝑝𝑣𝑑𝑡

+ −2𝑅2 + 𝑅2𝑅 +𝑅2

𝑅 +

4𝑅3

𝑅 𝑑2𝑝𝑣𝑑𝑡2

2𝑅3 −

4𝑅3𝑅 −

𝑅 −

20𝑅4

+ −1

10𝑅4 +

20𝑅4𝑅 +

𝑅 +

120𝑅5

120𝑅5 −

+ ⋯ + ⋯

In the equation above, net force is now a function of the rate of change of

vapor properties and the vapor pressure now varies with bubble size. As with the

simplified equation, the equation above must be solved numerically.

Numerical analysis of the equation above indicates that it is also unable to

predict departure radii of experimental data.

Summary

An equation for entropy generation rate is derived by studying the rate of

entropy generation for a bubble undergoing nucleate pool boiling. In the

derivation of this equation, work has been modeled as a function of the net forces

acting on the bubble and the rate at which the bubble grows. A derivative of the

derived equation was taken in an attempt to solve for a departure radii. Attempts

to do so were unsuccessful and it has been confirmed that the derived entropy

generation rate equation does not reach a maximum value.

CONCLUSIONS

Entropy generation studies of spherical bubbles undergoing growth in

nucleate pool boiling have resulted in a novel correlation for bubble departure

radii.

Two equations for entropy generation rate have been derived for spherical

bubbles undergoing growth on a wall in nucleate pool boiling. These equations

have been derived by modeling the work performed by the bubble as that of a

thermodynamic system, and as a function of the net forces acting on the bubble

and the rate at which the bubble grows.

When work performed by a spherical bubble is modeled using the

equation for a thermodynamic system, the entropy generation rate takes the

following form.

The derivative of the above equation results in a separable second order,

non-linear differential equation. Evaluation of this equation indicates that direct

substitution of the Rayleigh, Plesset-Zwick, and MRG equations fails to predict

departure radii for bubbles undergoing growth in nucleate pool boiling. Further

investigation indicates that this is caused by failure of the calculated entropy

generation rate to reach a maximum value.

Analysis of the general solution to the second order, non-linear differential

equation produces a novel correlation for predicting departure radii. By setting

the general solution of the differential equation equivalent to the Plesset-Zwick

equation for bubble growth, a third order equation is derived which allows for the

prediction of bubble departure radii.

3𝑒 0.894132 𝑙𝑛 𝐵 −4.010944

2 − 4.127304 −03 𝑒𝑥𝑝1.036544𝐴

The real, non-negative solution to the above equation estimates the

departure radius of a spherical bubble undergoing growth in nucleate pool

boiling. Predicted departure radii derived using this equation compare well with

data of (Van Stralen, Cole, Sluyter, & Sohal, 1975) but is less capable of

predicting experimental data of (Cole & Shulman, 1966b) and (Ellion, 1954) for

systems undergoing saturated and sub-cooled boiling. The ability to accurately

predict experimental departure data of (Van Stralen, Cole, Sluyter, & Sohal,

1975) can be attributed that the method by which the equation is derived.

Modeling the work performed by a bubble as the integral of the product of

the net force acting on the bubble and the rate of growth of the bubble produces

a complex equation for entropy generation rate.

− 𝑅 − 1

𝑅 +1

𝑅2 −1

𝑅3 +1

𝑅4 + ⋯

−4𝜋𝑅2

4𝑅2

20𝑅3

120𝑅4

The derivative of the above equation and a reduced form of it fail to result

in predicted departure radii for experimental data. Analysis of the equation

above and its simplified form indicate that both fail to reach a maximum value at

bubble departure.

The failure of both entropy generation rate equations to reach a maximum

value may be associated with the equation(s) utilized to model the growth of the

bubble. The Rayleigh, Plesset-Zwick, and MRG equations are intended to

approximate the growth a bubble through specific regions of growth, or in the

case of the MRG equation, through the life of the bubble. Analysis of all

equations indicates that each will model the growth of the bubble to an infinitely

large radius. This is not representative of real bubbles which reach a maximum

radius at, or near, departure prior to shrinking. The utilization of these equations

in the development of an entropy generation rate model likely introduces some

level of error near the point of departure.

Furthermore, the equations derived for heat transfer rate indicates that

heat transfer rate is always positive and growing. This cannot be true for a

bubble departing from a heated surface. In this case, the heat is supplied to the

bubble by means of a superheated surface and, potentially, a superheated liquid.

Once the temperature within the bubble exceeds the temperature of the liquid,

thermal energy is only transferred to the bubble by the wall. At the point of

bubble departure, this heat transfer rate disappears, or is greatly diminishes.

This is not consistent with results of the derived equation.

The findings of this entropy generation study do not disprove the

hypothesis. However, they do indicate that the derived equations for entropy

generation rate do not accurately demonstrate the behavior of bubbles at, or

near, the point of bubble departure. The development of improved entropy

generation rate models may lead to additional novel correlations for the

prediction of bubble departure radii.

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pool boiling. Applied Mechanics Reviews, 17 (9), 663-672.

APPENDIX A: DERIVATION OF ENTROPY GENERATION RATE (PRESSURE

METHOD)

The first law of thermodynamics is utilized to derive a relationship

for the rate of heat transfer to a spherical bubble undergoing growth in pool

boiling.

𝑄 = 𝑊 + 𝐸 − d

dt 𝑚𝑕

𝑖𝑛

The rate of work performed by the bubble is defined at a given radius 𝑅 as

follows.

𝑊 = 4𝜋 𝑝𝑣 − 𝑝∞ 𝑅2𝑑𝑅

Integration by successive parts produces the following.

𝑊 =4𝜋

3𝑅3 𝑝𝑣 − 𝑝∞ −

𝑅 +1

𝑅2 −1

𝑅3 + ⋯

The rate of work performed by the bubble at a radius 𝑅 is determined by

taking the derivative of the series above with respect to time.

𝑑𝑊

𝑑𝑡=

3𝑅2𝑅 3 𝑝𝑣 − 𝑝∞ −

𝑅 +3

𝑅2 −1

𝑅3 + ⋯

+4𝜋

3𝑅3

𝑅 −1

𝑅 +1

𝑅2 +1

𝑅𝑅

𝑅3 −1

𝑅2𝑅 + ⋯

Grouping common terms in the equation above results in the following

equation for rate of work performed by a bubble at a radius of 𝑅.

𝑊 =4𝜋𝑅2

3 3 𝑝𝑣 − 𝑝∞ 𝑅

𝑅 +1

𝑅2 −1

𝑅3 +1

𝑅4 + ⋯

The total internal energy of a spherical bubble is defined by the following

equation.

𝐸 = 𝑒𝑣𝑚𝑣 = 4π 𝜌𝑣𝑒𝑣

𝑅2𝑑𝑅

Successive integration by parts leads to the following equation.

𝐸 =4𝜋𝑅3

4 𝜌𝑣

𝑅 +1

20 𝜌𝑣

120 𝜌𝑣

𝑅3 + ⋯

The rate at which the internal energy changes at any radius 𝑅 is

determined by taking the derivative of the above equation with respect to time.

𝑑𝐸

𝑑𝑡=

33𝑅2𝑅 𝜌𝑣𝑒𝑣 −

4 𝜌𝑣

20 𝜌𝑣

120 𝜌𝑣

𝑅3 + ⋯

+4𝜋𝑅3

3 𝜌𝑣

4 𝜌𝑣

20 𝜌𝑣

10 𝜌𝑣

𝑅𝑅

120 𝜌𝑣

+ 6𝑑2𝑒𝑣𝑑𝑡2

40 𝜌𝑣

𝑅2𝑅 + ⋯

Mathematical manipulation of the above equation allows it to be reduced

to the following.

𝐸 =4𝜋

3𝑅2 3𝜌𝑣𝑒𝑣𝑅

4 𝜌𝑣

20 𝜌𝑣

120 𝜌𝑣

𝑅4 + ⋯

The rate of energy flow to a spherical bubble is defined as follows.

𝑑𝑡 𝑚𝑕

𝑖𝑛

𝑑𝑡 𝑚 𝑕𝑖𝑛

𝑑𝑡 4π 𝜌𝑣𝑕𝑖𝑛

𝑅2𝑑𝑅

Successive integration by parts leads to the following equation.

𝑑𝑡 𝑚𝑕

𝑖𝑛

𝑑𝑡

4𝜋𝑅3

4 𝜌𝑣

𝑑𝑕𝑣𝑑𝑡

20 𝜌𝑣

120 𝜌𝑣

𝑅3 + ⋯

The rate of energy flow is solved by calculating the derivative with respect

to time.

𝑑𝑡 𝑚𝑕

𝑖𝑛

=4𝜋

33𝑅2𝑅 𝜌𝑣𝑕𝑣 −

4 𝜌𝑣

20 𝜌𝑣

120 𝜌𝑣

𝑅3 + ⋯

+4𝜋𝑅3

3 𝜌𝑣

4 𝜌𝑣

20 𝜌𝑣

10 𝜌𝑣

𝑅𝑅

120 𝜌𝑣

+ 6𝑑2𝑕𝑣𝑑𝑡2

40 𝜌𝑣

𝑅2𝑅 + ⋯

Simple mathematical manipulation of the above equation allows it to be

reduced to the following.

𝑑𝑡 𝑚𝑕

𝑖𝑛

=4𝜋

3𝑅2 3𝜌𝑣𝑕𝑣𝑅

4 𝜌𝑣

20 𝜌𝑣

120 𝜌𝑣

𝑅4 + ⋯

The heat transfer rate is solved for by substitution of the equations derived

above in the first law of thermodynamics.

𝑄 =4𝜋𝑅2

3 3 𝑝𝑣 − 𝑝∞ 𝑅

𝑅 +1

𝑅2 −1

𝑅3 +1

𝑅4 + ⋯

+ +3𝜌𝑣𝑒𝑣𝑅

4 𝜌𝑣

20 𝜌𝑣

120 𝜌𝑣

𝑅4 + ⋯ − 3𝜌𝑣𝑕𝑣𝑅

− 𝑅 − 1 − 𝜌𝑣𝑑𝑕𝑣𝑑𝑡

4 𝜌𝑣

20 𝜌𝑣

120 𝜌𝑣

𝑅4 + ⋯

Grouping of common terms allows the equation to be simplified.

𝑄 =4𝜋𝑅2

3 3 𝑝𝑣 − 𝑝∞ 𝑅 + 3𝜌𝑣 𝑒𝑣 − 𝑕𝑣 𝑅

𝑅 +1

𝑅2 −1

𝑅3 +1

− 𝜌𝑣𝑑𝑒𝑣𝑑𝑡

− 𝜌𝑣𝑑𝑕𝑣𝑑𝑡

− 𝑕𝑣𝑑𝜌𝑣𝑑𝑡

4 𝜌𝑣

− 𝜌𝑣𝑑2𝑕𝑣𝑑𝑡2

− 2𝑑𝑕𝑣𝑑𝑡

− 𝑕𝑣𝑑2𝜌𝑣𝑑𝑡2

20 𝜌𝑣

− 3𝑑𝜌𝑣𝑑𝑡

120 𝜌𝑣

− 6𝑑2𝑕𝑣𝑑𝑡2

By further regrouping, the equation is reduced to the following form.

𝑄 =4𝜋𝑅2

3 3 𝑝𝑣 − 𝑝∞ 𝑅 + 3𝜌𝑣 𝑒𝑣 − 𝑕𝑣 𝑅

𝑅 +1

𝑅2 −1

𝑅3 +1

+ −𝑑𝜌𝑣𝑑𝑡

𝑅 +1

𝑅2 −1

𝑅3 +1

+ −𝜌𝑣𝑅 +1

𝑅2 −3

𝑅3 +1

4𝜌𝑣𝑅

2 −3

𝑅3 +1

+ −1

20𝜌𝑣𝑅

120𝜌𝑣𝑅

Solution of the above equation requires the relationships derived below.

These relationships begin with the definition of enthalpy.

𝐻𝑣 = 𝐸𝑣 + 𝑝𝑣𝑉𝑣

The definition above may also be written in terms of per unit mass.

𝑕𝑣 = 𝑒𝑣 + 𝑝𝑣𝑣𝑣 = 𝑒𝑣 +𝑝𝑣𝜌𝑣

This equation is rearranged for easy substitution into the derived equation

for heat transfer rate.

𝑒𝑣 − 𝑕𝑣 = −𝑝𝑣𝜌𝑣

The relationship above is used to replace additional terms in the heat

transfer rate equation by taking its derivative. The first four derivatives are

provided below.

𝑑𝑡 𝑒𝑣 − 𝑕𝑣 =

𝜌𝑣2

𝑝𝑣 −1

𝜌𝑣

𝑑𝑡2 𝑒𝑣 − 𝑕𝑣

𝜌𝑣2

− 21

𝜌𝑣3 𝑑𝜌𝑣𝑑𝑡

𝑝𝑣 + 21

𝜌𝑣2

+ −1

𝜌𝑣 𝑑2𝑝𝑣𝑑𝑡2

𝜌𝑣2

− 61

𝜌𝑣3

𝑝𝑣

𝜌𝑣2

− 61

𝜌𝑣2

+ −1

= −81

𝜌𝑣3

𝜌𝑣2

− 241

2 𝑑2𝜌𝑣𝑑𝑡2

− 61

𝜌𝑣3 𝑑2𝜌𝑣𝑑𝑡2

𝑝𝑣

𝜌𝑣2

− 241

𝜌𝑣3

𝜌𝑣2

− 121

𝜌𝑣2

+ −1

Substitution of the above relationships into the heat transfer rate results in

the following simplified equation.

𝑄 = −4𝜋𝑝∞𝑅2𝑅

To solve for entropy generation, a definition for rate of entropy

accumulation is required. This begins by defining the total entropy at a given

radius 𝑅.

𝑆 = 𝑠𝑣𝑚𝑣 = 4π 𝜌𝑣𝑠𝑣

𝑅2𝑑𝑅

By successive integration by parts, the following solution is determined.

𝑆 =4𝜋

3𝑅3 𝜌𝑣𝑠𝑣 −

𝑑𝑡𝑅 +

𝑑𝑡2𝑅2 −

𝑑𝑡3𝑅3 + ⋯

The rate at which entropy accumulation occurs at a given radius 𝑅 is

determined by taking the derivative of the equation above.

𝑆 =4𝜋

3𝑅2 3𝜌𝑣𝑠𝑣𝑅 −

𝑑𝑡𝑅𝑅 +

𝑑𝑡2𝑅2𝑅 −

𝑑𝑡3𝑅3𝑅 + ⋯

+4𝜋

3𝑅2

𝑑𝑡𝑅 −

𝑑𝑡2𝑅2 −

𝑑𝑡𝑅𝑅 +

𝑑𝑡3𝑅3

𝑑𝑡2𝑅2𝑅 −

𝑑𝑡4𝑅4 −

𝑑𝑡3𝑅3𝑅 + ⋯

This equation is simplified by grouping of common terms.

𝑆 =4𝜋

𝑑𝑡𝑅 +

𝑑𝑡2𝑅2 −

𝑑𝑡3𝑅3

𝑑𝑡4𝑅4 + ⋯

With the rate of entropy accumulation defined, it is necessary to define the

entropy transfer rate. This values defined by the following equation.

𝑇𝑤= −

𝑇𝑤𝑝∞𝑅

The final piece of the second law of thermodynamics required to solve for

the entropy generation rate is the net entropy flow rate at the boundaries of the

bubble.

𝑖𝑛

𝑑𝑡 4π 𝜌𝑣𝑠𝑣

𝑅2𝑑𝑅

By observation of the above equation, it is seen that it is takes the same

form as the equation defining the rate of entropy accumulation within the bubble.

If the state of vapor entering the bubble is assumed to be at the same state as

the vapor accumulated within the bubble, the equation becomes identical to that

for entropy accumulation rate.

𝑖𝑛

=4𝜋

𝑑𝑡𝑅 +

𝑑𝑡2𝑅2 −

𝑑𝑡3𝑅3

𝑑𝑡4𝑅4 + ⋯

Substitution of the equations defined above into the second law of

thermodynamics allows for the entropy generation rate to be defined.

𝑆 gen =4𝜋

𝑑𝑡𝑅 +

𝑑𝑡2𝑅2 −

𝑑𝑡3𝑅3

𝑑𝑡4𝑅4 + ⋯ +

𝑇𝑤𝑝∞𝑅

−4𝜋

𝑑𝑡𝑅 +

𝑑𝑡2𝑅2 −

𝑑𝑡3𝑅3

𝑑𝑡4𝑅4 + ⋯

Removal of common terms allows the equation to reduce to the following.

𝑆 gen =4𝜋

𝑇𝑤𝑝∞𝑅

If the hypothesis is true, bubble departure occurs when the rate of entropy

generation reaches a maximum. The maximum occurs when the derivative of

entropy generation rate reaches zero. Therefore, a derivative with respect to 𝑅 is

taken of the entropy generation rate equation above and set equal to zero.

= 0 =4𝜋

𝑇𝑤𝑝∞

𝑑𝑅 𝑅2𝑅 =

𝑇𝑤𝑝∞

𝑑𝑡 𝑅2𝑅 ∗

𝑑𝑡

𝑑𝑅

The resulting equation is as follows.

0 =4𝜋𝑅

𝑇𝑤𝑝∞ 2𝑅 + 𝑅

By removal of common terms, the equation reduces to the following non-

linear, second order differential equation.

𝑅𝑅 + 2𝑅 2 = 0

The solution to the above non-linear second order differential equation is

determined by substitution. The following variables are defined for the

substitution

𝑑𝑅

𝑑𝑡= 𝑢

𝑑2𝑅

𝑑𝑡2=

𝑑𝑡 𝑑𝑅

𝑑𝑡 =

𝑑𝑅 𝑑𝑅

𝑑𝑡 ∗

𝑑𝑅

𝑑𝑡= 𝑢𝑢′

By substitution of the above defined variables, the non-linear second order

differential equation is simplified.

𝑅𝑢𝑢′ + 2𝑢2 = 0

This is further simplified by removal of common terms.

𝑅𝑢′ = −2𝑢

By applying the definition of 𝑢′ , the above equation is separable.

𝑑𝑢

𝑢= −2

𝑑𝑅

Integration of the separated equation results in the following.

𝑙𝑛 𝑢 = −2𝑙𝑛 𝑅 − 𝐶

This equation is rewriten as follows by taking the exponent of both sides.

𝑢 = 𝑒−2𝑙𝑛 𝑅 −2𝐶 = 𝑅−2𝑒−𝐶

By applying the definition of 𝑢, the equation is rewritten again.

𝑅2𝑑𝑅 = 𝑒−𝐶𝑑𝑡

This equation is solved by integration

3𝑅3 = 𝑒−2𝐶𝑡 + 𝐷

Rearranging the equation results in the following equation for 𝑅.

𝑅 = 3𝑒−𝐶𝑡 + 𝐷 1 3

The following solution also works.

𝑅 = −3𝑒−𝐶𝑡 + 𝐷 1 3

APPENDIX B: DEFINING THE GENERAL SOLUTION

The general solution derived for the second order, non-linear differential

equation is fully defined by utilization of the Rayleigh, Plesset-Zwick, and MRG

Equations.

𝑅 = −3𝑒−𝐶𝑡 + 𝐷 1 3

The derivative of the general solution takes the following form.

𝑑𝑅

𝑑𝑡= −𝑒−𝐶 −3𝑒−𝐶𝑡 + 𝐷 −1 3

For the Rayleigh solution, boundary conditions will be defined as follows.

𝑅𝑅𝑎𝑦𝑙𝑒𝑖𝑔 𝑕 𝑡=𝑡𝑑𝑒𝑝𝑡= 𝑅𝐺𝑒𝑛 𝑆𝑜𝑙𝑢𝑡𝑖𝑜𝑛 𝑡=𝑡𝑑𝑒𝑝𝑡

𝑑𝑅

𝑑𝑡 𝑅𝑎𝑦𝑙𝑒𝑖𝑔 𝑕

= − 𝑑𝑅

By substitution of the appropriate equations into the boundary conditions

defined above, the following system of equation is created.

𝐴𝑡𝑑𝑒𝑝𝑡 = −3𝑒−𝐶𝑡𝑑𝑒𝑝𝑡 + 𝐷 1 3

𝐴 = 𝑒𝐶 −3𝑒−𝐶𝑡𝑑𝑒𝑝𝑡 + 𝐷 2 3

Both equations are rearranged to define constant 𝐷.

𝐷 = 𝐴3𝑡𝑑𝑒𝑝𝑡3 + 3𝑒−𝐶𝑡𝑑𝑒𝑝𝑡

𝐷 = 𝐴𝑒−𝐶 3 2 + 3𝑒−𝐶𝑡𝑑𝑒𝑝𝑡

By setting the two equations equal to each other, the constant 𝐷 is

eliminated.

𝐴3𝑡𝑑𝑒𝑝𝑡3 = 𝐴𝑒−𝐶 3 2

Rearranging the equation allows for the solution of constant 𝐶.

𝐶 = −𝑙𝑛 𝐴𝑡𝑑𝑒𝑝𝑡2

The time of bubble departure is replaced by the Rayleigh equation.

𝑡𝑑𝑒𝑝𝑡 =𝑅𝑑𝑒𝑝𝑡

Substitution into the equation for constant 𝐶 results in the following

equation.

𝐶 = −𝑙𝑛 1

By averaging results for multiple experimental data sets, a value for

constant 𝐶 is defined.

𝐶 = −𝑙𝑛

1𝐴𝑖

𝑅𝑒𝑥𝑝 ,𝑖2

𝑖=1

Constant 𝐷 is solved by substitution of constant 𝐶 into the equation below.

𝐷 = 𝐴3𝑡𝑑𝑒𝑝𝑡3 + 3𝑒−𝐶𝑡𝑑𝑒𝑝𝑡

This equation is rewritten as a function of departure radius by utilizing the

Rayleigh equation.

𝐷 = 𝑅𝑑𝑒𝑝𝑡3 +

3𝑒−𝐶

A single constant 𝐷 is determined by taking the average of multiple

experimental data sets.

𝐷 = 𝑅𝑒𝑥𝑝 ,𝑖

3 + 3𝑒−𝐶

𝐴𝑖 𝑅𝑒𝑥𝑝 ,𝑖

𝑖=1

The same procedure is utilized for the Plesse-Zwick equation. Boundary

conditions are defined as follows.

𝑅𝑃𝑙𝑒𝑠𝑠𝑒𝑡 −𝑍𝑤𝑖𝑐𝑘 𝑡=𝑡𝑑𝑒𝑝𝑡 = 𝑅𝐺𝑒𝑛 𝑆𝑜𝑙𝑢𝑡𝑖𝑜𝑛 𝑡=𝑡𝑑𝑒𝑝𝑡

𝑑𝑅

𝑑𝑡 𝑃𝑙𝑒𝑠𝑠𝑒𝑡 −𝑍𝑤𝑖𝑐𝑘

= − 𝑑𝑅

defined above, the following system of equations is created.

𝐵𝑡𝑑𝑒𝑝𝑡1 2 = −3𝑒−𝐶𝑡𝑑𝑒𝑝𝑡 + 𝐷

2𝐵𝑡𝑑𝑒𝑝𝑡

−1 2 = 𝑒𝐶 −3𝑒−𝐶𝑡𝑑𝑒𝑝𝑡 + 𝐷 2 3

These equations are rewritten to solve for constant 𝐷.

𝐷 = 𝐵3𝑡𝑑𝑒𝑝𝑡3 2 + 3𝑒−𝐶𝑡𝑑𝑒𝑝𝑡

𝐷 = 1

2𝐵𝑒−𝐶

𝑡𝑑𝑒𝑝𝑡−3 4 + 3𝑒−𝐶𝑡𝑑𝑒𝑝𝑡

The system of equations is combined.

𝐵3𝑡𝑑𝑒𝑝𝑡3 2 =

2𝐵𝑒−𝐶

𝑡𝑑𝑒𝑝𝑡−3 4

By rearranging the equation above, it is possible to solve for the constant

𝐶 = −𝑙𝑛 2𝐵𝑡𝑑𝑒𝑝𝑡3 2

The time at departure is replaced using the Plesset-Zwick equation.

𝑡𝑑𝑒𝑝𝑡 = 𝑅𝑑𝑒𝑝𝑡

The resulting equation for the constant 𝐶 is a function of variable 𝐵 and

departure radius.

𝐶 = −𝑙𝑛 2𝑅𝑑𝑒𝑝𝑡

By averaging results for multiple experimental data sets, a value for

constant 𝐶 is defined.

𝐶 =

−𝑙𝑛 2𝑅𝑑𝑒𝑝𝑡 ,𝑖

𝐵𝑖2

𝑖=1

Substitution of the constant 𝐶 into the equation below leads to the defining

of constant 𝐷.

𝐷 = 𝐵3𝑡𝑑𝑒𝑝𝑡3 2 + 3𝑒−𝐶𝑡𝑑𝑒𝑝𝑡

Like before, this equation is rewritten by replacing departure time using

the Plesset-Zwick equation.

𝐷 = 𝑅𝑑𝑒𝑝𝑡3 +

3𝑒−𝐶

𝐵2𝑅𝑑𝑒𝑝𝑡

Averaging results of experimental data results in the following equation.

𝐷 =

𝑅𝑑𝑒𝑝𝑡 ,𝑖3 +

3𝑒−𝐶

𝐵𝑖2 𝑅𝑑𝑒𝑝𝑡 ,𝑖

𝑖=1

Finally, this method is utilized to determine the value of the constants for

the general solution using the MRG equation. This begins by again defining the

boundary conditions.

𝑅𝑀𝑅𝐺 𝑡=𝑡𝑑𝑒𝑝𝑡 = 𝑅𝐺𝑒𝑛 𝑆𝑜𝑙𝑢𝑡𝑖𝑜𝑛 𝑡=𝑡𝑑𝑒𝑝𝑡

𝑑𝑅

𝑑𝑡𝑀𝑅𝐺

= − 𝑑𝑅

defined above, the following system of equations is created.

2𝐵2

− 𝐴2

− 1 = −3𝑒−𝐶𝑡𝑑𝑒𝑝𝑡 + 𝐷 1 3

𝐴 𝐴2

− 𝐴2

= 𝑒𝐶 −3𝑒−𝐶𝑡𝑑𝑒𝑝𝑡 + 𝐷 2 3

These equations is rewritten to solve for constant 𝐷.

𝐷 = 2𝐵2

− 𝐴2

+ 3𝑒−𝐶𝑡𝑑𝑒𝑝𝑡

𝐷 = 𝐴𝑒−𝐶 3 2 𝐴2

− 𝐴2

The system of equations is combined.

2𝐵2

− 𝐴2

= 𝐴𝑒−𝐶 3 2 𝐴2

− 𝐴2

This relationship is rewritten as follows.

𝑅𝑑𝑒𝑝𝑡3 = 𝑒−3𝐶 2

𝑑𝑅

𝑑𝑡 𝑑𝑒𝑝𝑡

By rearranging the equation above, it is possible to solve for the constant

𝐶 = −𝑙𝑛

4𝐵4

9𝐴3

𝐵2 𝑡𝑑𝑒𝑝𝑡 + 1 3 2

− 𝐴2

𝐵2 𝑡𝑑𝑒𝑝𝑡 + 1 1 2

− 𝐴2

If the radial velocity is known at the point of departure, the constant 𝐶 may

be more easily solve using the following equation

𝐶 = −𝑙𝑛 𝑅𝑑𝑒𝑝𝑡

𝑑𝑅𝑑𝑡 𝑑𝑒𝑝𝑡

Because departure time cannot be isolated in the MRG equation, the

value of constant 𝐶 must be determined using time. Averaging results from

analysis using experimental data sets results in the following equation.

𝐶 =

−𝑙𝑛

4𝐵𝑖

9𝐴𝑖3

𝐴𝑖

− 𝐴𝑖

𝐴𝑖

− 𝐴𝑖

𝑖=1

The constant 𝐷 is determined by substitution of constant 𝐶 into the

equation below.

𝐷 = 2𝐵2

− 𝐴2

Averaging results for multiple experimental data sets results in the

following equation.

𝐷 =

2𝐵𝑖

3𝐴𝑖

𝐴𝑖

− 𝐴𝑖

𝐵𝑖2 𝑡𝑑𝑒𝑝𝑡

+ 3𝑒−𝐶𝑡𝑑𝑒𝑝𝑡 ,𝑖

𝑖=1

APPENDIX C: DERIVATION OF ENTROPY GENERATOIN RATE (NET

FORCE METHOD)

The first law of thermodynamics is utilized to derive a relationship for the

Rate of Heat Transfer to a spherical bubble undergoing growth in pool boiling.

𝑄 −𝑊 = 𝐸 − d

dt 𝑚𝑕

𝑖𝑛

The rate of work performed by the bubble requires a definition for total

work at a given radius 𝑅.

𝑊 = 𝐹𝑛𝑒𝑡 𝑑𝑅𝑅

Integration by successive parts leads to the following.

𝑊 = 𝑅 𝐹𝑛𝑒𝑡 −1

𝑅 +1

𝑅2 −1

𝑅3 + ⋯

If the effect of changes in net force are ignored the equation is reduced to

the following.

The rate of work performed by the bubble at a radius 𝑅 is determined by

taking the derivative of the series above with respect to time.

𝑊 = 𝐹𝑛𝑒𝑡𝑅 + 𝑅 − 1 −𝑑𝐹𝑛𝑒𝑡𝑑𝑡

𝑅 +1

𝑅2 −1

𝑅3 +1

𝑅4 + ⋯

The assumption that changes in net force are negligible leads to the

following equation

The net force can be defined as the sum of the following forces as

described in Chapter 1.

𝐹𝑛𝑒𝑡 = 𝐹𝑏 + 𝐹𝑝 − 𝐹𝑖 − 𝐹𝜍 − 𝐹𝐷

Substitution of the appropriate equations results in the following.

𝐹𝑛𝑒𝑡 =4𝜋𝑅3

3 𝜌𝑙 − 𝜌𝑣 𝑔 + 𝜋𝑅𝑏𝜍 sin𝛽 −

𝑑𝑡

16𝜌𝑙

4𝜋𝑅3

3 𝑅 − 2𝜋𝑅𝑏𝜍 sin𝛽

−𝜋

A non-dimensional scaling factor is introduced to replace base radius and

contact angle.

𝑆𝜍 =𝑅𝑏

𝑅sin𝛽

Substitution of the non-dimensional scaling factor and execution of the

derivative within the net force equation lead to the following equation for net

force.

𝐹𝑛𝑒𝑡 =4𝜋

3 − 𝜋𝜍𝑙 𝑆𝜍𝑅 −11𝜋

12𝜌𝑙 3𝑅

2𝑅 2 + 𝑅3𝑅 −𝜋

4𝑎𝜇𝑙 𝑅𝑅

Solving for the rate of work requires the derivative with respect to time of

the net force equation. The first five derivatives are shown below.

=4𝜋

3𝑔 3 𝜌𝑙 − 𝜌𝑣 𝑅

2𝑅 −𝑑𝜌𝑣𝑑𝑡

𝑅3 − 𝜋𝜍𝑙𝑆𝜍 𝑅 −11𝜋

12𝜌𝑙 6𝑅𝑅

3 + 9𝑅2𝑅 𝑅 + 𝑅3𝑅

−𝜋

4𝑎𝜇𝑙 𝑅

2 + 𝑅𝑅

=4𝜋

3𝑔 6𝑅𝑅 2 + 3𝑅2𝑅 𝜌𝑙 − 𝜌𝑣 − 6𝑅2𝑅

−11𝜋

12𝜌𝑙 6𝑅

4 + 36𝑅𝑅 2𝑅 + 9𝑅2𝑅 2 + 12𝑅2𝑅 𝑅 + 𝑅3𝑑4𝑅

𝑑𝑡4

−𝜋

4𝑎𝜇𝑙 3𝑅 𝑅 + 𝑅𝑅

=4𝜋

3𝑔 6𝑅 3 + 18𝑅𝑅 𝑅 + 3𝑅2𝑅 𝜌𝑙 − 𝜌𝑣 − 18𝑅𝑅 2 + 9𝑅2𝑅

−11𝜋

12𝜌𝑙 60𝑅 3𝑅 + 90𝑅𝑅 𝑅 2 + 60𝑅𝑅 2𝑅 + 30𝑅2𝑅 𝑅 + 15𝑅2𝑅

𝑑4𝑅

𝑑𝑡4

+ 𝑅3𝑑5𝑅

𝑑𝑡5 −

4𝑎𝜇𝑙 3𝑅

2 + 4𝑅 𝑅 + 𝑅𝑑4𝑅

𝑑𝑡4

=4𝜋

3𝑔 36𝑅 2𝑅 + 18𝑅𝑅 2 + 24𝑅𝑅 𝑅 + 3𝑅2

𝑑4𝑅

− 24𝑅 3 + 72𝑅𝑅 𝑅 + 12𝑅2𝑅 𝑑𝜌𝑣𝑑𝑡

− 36𝑅𝑅 2 + 18𝑅2𝑅 𝑑2𝜌𝑣𝑑𝑡2

− 12 𝑅2𝑅 𝑑3𝜌𝑣𝑑𝑡3

𝑑𝑡4

−11𝜋

12𝜌𝑙 270𝑅 2𝑅 2 + 120𝑅 3𝑅 + 90𝑅𝑅 3 + 360𝑅𝑅 𝑅 𝑅 + 30𝑅2𝑅 2

+ 45𝑅2𝑅 𝑑4𝑅

𝑑𝑡4+ 90𝑅𝑅 2

𝑑4𝑅

𝑑5𝑅

𝑑𝑡5+ 𝑅3

𝑑6𝑅

𝑑𝑡6

−𝜋

4𝑎𝜇𝑙 10𝑅 𝑅 + 5𝑅

𝑑4𝑅

𝑑𝑡4+ 𝑅

𝑑5𝑅

𝑑𝑡5

=4𝜋

3𝑔 90𝑅 𝑅 2 + 60𝑅 2𝑅 + 60𝑅𝑅 𝑅 + 30𝑅𝑅

𝑑4𝑅

𝑑𝑡4+ 3𝑅2

𝑑5𝑅

− 180𝑅 2𝑅 + 90𝑅𝑅 2 + 120𝑅𝑅 𝑅 + 15𝑅2𝑑4𝑅

𝑑𝑡4 𝑑𝜌𝑣𝑑𝑡

− 60𝑅 3 + 180𝑅𝑅 𝑅 + 30𝑅2𝑅 𝑑2𝜌𝑣𝑑𝑡2

− 60𝑅 2 + 30𝑅2𝑅 𝑑3𝜌𝑣𝑑𝑡3

𝑑𝑡5

−11𝜋

12𝜌𝑙 630𝑅 𝑅 3 + 1260𝑅 2𝑅 𝑅 + 120𝑅 3

𝑑4𝑅

𝑑𝑡4+ 630𝑅𝑅 2𝑅

+ 420𝑅𝑅 𝑅 2 + 630𝑅𝑅 𝑅 𝑑4𝑅

𝑑𝑡4+ 90𝑅 3

𝑑4𝑅

𝑑𝑡4+ 105𝑅2𝑅

𝑑4𝑅

𝑑5𝑅

𝑑𝑡5

+ 126𝑅𝑅 2𝑑5𝑅

𝑑6𝑅

𝑑𝑡6+ 𝑅3

𝑑7𝑅

𝑑𝑡7

−𝜋

4𝑎𝜇𝑙 10𝑅 3 + 15𝑅

𝑑4𝑅

𝑑𝑡4+ 6𝑅

𝑑5𝑅

𝑑𝑡5+ 𝑅

𝑑6𝑅

𝑑𝑡5

The rate of internal energy was determined in Chapter 2. The resulting

equation is as follows.

𝐸 =4𝜋

3𝑅2 3𝜌𝑣𝑒𝑣𝑅

4 𝜌𝑣

20 𝜌𝑣

120 𝜌𝑣

𝑅4 + ⋯

The rate of energy flow to a spherical bubble is defined in Chapter 2 and is

defined by the following equation.

𝑑𝑡 𝑚𝑕

𝑖𝑛

=4𝜋

3𝑅2 3𝜌𝑣𝑕𝑣𝑅

4 𝜌𝑣

20 𝜌𝑣

120 𝜌𝑣

𝑅4 + ⋯

The heat transfer rate is solved by substitution of the equations derived

above into the first law of thermodynamics.

𝑄 = 𝑊 = 𝐹𝑛𝑒𝑡𝑅

+ 𝑅 − 1 −𝑑𝐹𝑛𝑒𝑡𝑑𝑡

𝑅 +1

𝑅2 −1

𝑅3 +1

𝑅4 + ⋯

+4𝜋𝑅2

4 𝜌𝑣

20 𝜌𝑣

120 𝜌𝑣

𝑅4 + ⋯ − 3𝜌𝑣𝑕𝑣𝑅

− 𝑅 − 1 − 𝜌𝑣𝑑𝑕𝑣𝑑𝑡

4 𝜌𝑣

20 𝜌𝑣

120 𝜌𝑣

𝑅4 + ⋯

Grouping of common terms allows the equation to be simplified.

𝑅 +1

𝑅2 −1

𝑅3 +1

𝑅4 + ⋯

+4𝜋𝑅2

− 𝜌𝑣𝑑𝑕𝑣𝑑𝑡

− 𝑕𝑣𝑑𝜌𝑣𝑑𝑡

4 𝜌𝑣

20 𝜌𝑣

120 𝜌𝑣

− 6𝑑2𝑕𝑣𝑑𝑡2

By further regrouping, the equation is reduced to the following form.

𝑅 +1

𝑅2 −1

𝑅3 +1

𝑅4 + ⋯

+4𝜋𝑅2

+ 𝑅

− 1 −𝑑𝜌𝑣𝑑𝑡

𝑅 +1

𝑅2 −1

𝑅3 +1

+ −𝜌𝑣𝑅 +1

𝑅2 −3

𝑅3 +1

4𝜌𝑣𝑅

2 −3

𝑅3 +1

+ −1

20𝜌𝑣𝑅

120𝜌𝑣𝑅

Solution of the above equation requires the relationships defined below.

These relationships begin with the definition of enthalpy.

𝐻𝑣 = 𝐸𝑣 + 𝑝𝑣𝑉𝑣

The definition above may also be written in terms of per unit mass.

𝑕𝑣 = 𝑒𝑣 + 𝑝𝑣𝑣𝑣 = 𝑒𝑣 +𝑝𝑣𝜌𝑣

This equation is rearranged for easy substitution into the derived equation

for heat transfer rate.

𝑒𝑣 − 𝑕𝑣 = −𝑝𝑣𝜌𝑣

The relationship above is used to replace additional terms in the heat

transfer rate equation.

𝑑𝑡 𝑒𝑣 − 𝑕𝑣 =

𝜌𝑣2

𝑝𝑣 −1

𝜌𝑣

𝜌𝑣2

− 21

𝑝𝑣 + 21

𝜌𝑣2

+ −1

𝜌𝑣2

− 61

𝜌𝑣3

𝑝𝑣

𝜌𝑣2

− 61

𝜌𝑣2

+ −1

= −81

𝜌𝑣3

𝜌𝑣2

− 241

2 𝑑2𝜌𝑣𝑑𝑡2

− 61

𝜌𝑣3 𝑑2𝜌𝑣𝑑𝑡2

𝑝𝑣

𝜌𝑣2

− 241

𝜌𝑣3

𝜌𝑣2

− 121

𝜌𝑣2

+ −1

Substitution of the above relationships into the heat transfer rate results in

the following simplified equation.

𝑅 +1

𝑅2 −1

𝑅3 +1

𝑅4 + ⋯

+4𝜋𝑅2

3 −3𝑝𝑣𝑅

4𝑅2

20𝑅3

120𝑅4

If vapor pressure is assumed constant and the affects of changes in net

force are neglected, the equation above reduces to the following.

𝑄 = 𝐹𝑛𝑒𝑡𝑅 − 4𝜋𝑝𝑣𝑅2𝑅

To solve for entropy generation, a definition for rate of entropy

accumulation is required. This begins by defining the total entropy at a given

radius 𝑅. The work performed to derive a relationship for rate of entropy

accumulation in Chapter 2 lead to the development of the following relationship.

𝑆 =4𝜋

𝑑𝑡𝑅 +

𝑑𝑡2𝑅2 −

𝑑𝑡3𝑅3

𝑑𝑡4𝑅4 + ⋯

The entropy transfer rate is defined using the following equation.

𝑇𝑤=

𝐹𝑛𝑒𝑡𝑇𝑤

𝑅 + 𝑅 − 1

𝑇𝑤 −

𝑅 +1

𝑅2 −1

𝑅3 +1

𝑅4 + ⋯

+4𝜋𝑅2

3𝑇𝑤 −3𝑝𝑣𝑅

4𝑅2

20𝑅3

120𝑅4

The final piece of the second law of thermodynamics required to solve for

the entropy generation rate is the net entropy flow rate at the boundaries of the

bubble. This was previously defined in Chapter 2 with the following equation.

𝑖𝑛

=4𝜋

𝑑𝑡𝑅 +

𝑑𝑡2𝑅2 −

𝑑𝑡3𝑅3

𝑑𝑡4𝑅4 + ⋯

Substitution of the equations defined above into the second law of

thermodynamics allows for the entropy generation rate to be defined.

𝑆 gen =4𝜋

𝑑𝑡𝑅 +

𝑑𝑡2𝑅2 −

𝑑𝑡3𝑅3

𝑑𝑡4𝑅4 + ⋯ −

𝐹𝑛𝑒𝑡𝑇𝑤

− 𝑅 − 1

𝑇𝑤 −

𝑅 +1

𝑅2 −1

𝑅3 +1

𝑅4 + ⋯

−4𝜋𝑅2

4𝑅2

20𝑅3

120𝑅4

−4𝜋

𝑑𝑡𝑅 +

𝑑𝑡2𝑅2 −

𝑑𝑡3𝑅3

𝑑𝑡4𝑅4 + ⋯

Removal of common terms reduces the equation to the following.

𝑆 gen = −𝐹𝑛𝑒𝑡𝑇𝑤

− 𝑅 − 1

𝑇𝑤 −

𝑅 +1

𝑅2 −1

𝑅3 +1

𝑅4 + ⋯

−4𝜋𝑅2

4𝑅2

20𝑅3

120𝑅4

If vapor pressure is assumed constant and the affects of changes in net

force are neglected, the equation above reduces to the following.

𝑆 gen = −𝐹𝑛𝑒𝑡𝑇𝑤

𝑅 −4𝜋

𝑇𝑤𝑝𝑣𝑅

The derivative of entropy generation rate with respect to bubble radius is

shown below.

= 0 = −𝑑

𝑑𝑡 𝐹𝑛𝑒𝑡𝑇𝑤

𝑅 𝑑𝑅

𝑑𝑡

−𝑑

𝑑𝑡 𝑅 − 1

𝑇𝑤 −

𝑅 +1

𝑅2 −1

𝑅3 +1

+ ⋯ 𝑑𝑅

𝑑𝑡

−𝑑

𝑑𝑡

4𝜋𝑅2

4𝑅2

20𝑅3

120𝑅4

+ ⋯ 𝑑𝑅

𝑑𝑡

Executing the derivatives and simplifying the resulting equation leads to

the following equation.

𝑑𝑆 𝑔𝑒𝑛

𝑑𝑅= 0 =

𝑇𝑤 −

𝑅 𝐹𝑛𝑒𝑡 + −2 + 𝑅 + 𝑅

𝑅 𝑑𝐹𝑛𝑒𝑡

𝑑𝑡+ 2𝑅 − 𝑅𝑅 −

𝑅 −

2𝑅2 𝑅

𝑅 𝑑2𝐹𝑛𝑒𝑡

𝑑𝑡 2 + −𝑅2 +1

2𝑅2𝑅 +

𝑅 +

6𝑅3 𝑅

𝑑𝑡 3 + 1

3𝑅3 −

6𝑅3𝑅 −

𝑅 −

24𝑅4 𝑅

𝑑𝑡 4 + −1

24𝑅4 +

𝑑𝑡 5 + ⋯ +4𝜋𝑅

3𝑇𝑤 6𝑅 + 3𝑅

𝑅 𝑝𝑣 + 6𝑅 − 3𝑅𝑅 −

𝑅2 𝑅

𝑅 𝑑𝑝𝑣

𝑑𝑡+ −2𝑅2 + 𝑅2𝑅 +

𝑅 +

4𝑅3 𝑅

𝑅 𝑑2𝑝𝑣

𝑑𝑡 2 + 1

2𝑅3 −

4𝑅3𝑅 −

𝑅 −

20𝑅4 𝑅

𝑅 𝑑3𝑝𝑣

𝑑𝑡 3 +

10𝑅4 +

20𝑅4𝑅 +

𝑅 +

120𝑅5 𝑅

𝑅 𝑑4𝑝𝑣

𝑑𝑡 4 + 1

120𝑅5 −

𝑅 𝑑5𝑝𝑣

𝑑𝑡 5 + ⋯

APPENDIX D: MATLAB PROGRAMS

The following MatLab program has been developed to predict bubble

departure radius using the equation derived by modeling rate of work using both

the pressure method and net force method.

function Bubble()

%Define Variables of Analysis

%**********************************************************************

%Define Input Variables

%A Variable Defined in Rayleigh Equation

%B Variable Defined in Plesset-Zwick Equation

%equation Defines correlation equation

%method Defines Scaling Factor Method

%model Defines model used for analysis

%Rexp Departure Radius from Experimental Data

%Define Constants

%a Scaling Constant for Net Force Method

%b System constant

%g Gravitational Acceleration [m/s^2]

%n Number of experimental data points for analysis

%m Number of points used in analysis

%tmax Maximum time value for analysis

%Define Calculated Variables

%aL Thermal Diffusivity of Liquid

%C C Constant for Pressure Method Solution

%CpL Specific Heat of Liquid (Constant Pressure)

%CvL Specific Heat of Liquid (Constant Volume)

%D D Constant for Pressure Method Solution

%DenL Liquid Density [kg/m^3]

%DenV Vapor Density [kg/m^3]

%DenWork Expanded Vapor Density Vector [kg/m^3]

%dDenV Derivative of Density Vapor [kg/s*m^3]

%d2DenV 2nd Derivative of Density Vapor [kg/s^2*m^3

%d3DenV 3rd Derivative of Density Vapor [kg/s^3*m^3]

%d4DenV 4th Derivative of Density Vapor [kg/s^4*m^3]

%d5DenV 5th Derivative of Density Vapor [kg/s^5*m^3]

%ErrR Percent error of predicted radius

%F Net Force Acting on Bubble [N]

%dF Rate of Change of Net Force [N/s]

%d2F 2nd Derivative of Net Force [N/s^2]

%d3F 3rd Derivative of Net Force [N/s^3]

%d4F 4th Derivative of Net Force [N/s^4]

%d5F 5th Derivative of Net Force [N/s^5]

%hfg Specific Enthalpy of Vaporization

%Ja Jakob Number

%Jastar Modified Jakob Number

%kL Thermal Conductivity of Liquid [W/m-K]

%Pbulk Bulk Liquid Pressure [MPa]

%Pvap Constnat Vapor Pressure [MPa]

%P Variable Vapor Pressure [MPa]

%dP Rate of Change of Vapor Pressure [MPa/s]

%d2P 2nd Derivative of Vapor Pressure [MPa/s^2]

%d3P 3rd Derivative of Vapor Pressure [MPa/s^3]

%d4P 4th Derivative of Vapor Pressure [MPa/s^4]

%d5P 5th Derivative of Vapor Pressure [MPa/s^5]

%dr Radius Interval for Analysis

%R Bubble Radius [m]

%dR Radial Velocity of Bubble Boundary [m/s]

%d2R Radial Acceleration of Bubble Boundary [m/s^2]

%d3R 3rd Derivative of Bubble Radius [m/s^3]

%d4R 4th Derivative of Bubble Radius [m/s^4]

%Res Defines Residual of equation

%Rpre Predicted Radius

%Rworking Radius for use within program

%S Scaling Factor

%StL Surface Tension of Liquid [N/m]

%t Time matrix for analysis

%dt Interval size for Time matrix

%tMikic Departure Time Predicted from Solution Derived using

Mickic

%Tbulk Bulk Liquid Temperature [K]

%Tsupw Wall Superheat [K]

%Tvap Vapor Temperature [K]

%Twall Wall Temperature [K]

%VisL Viscosity of Liquid

%Define Time Interval for Analysis

%**********************************************************************

n=1001;

tmax=0.3000;

dt=tmax/(n-1);

t=0:dt:tmax;

t(1)=dt/100;

%rmax=1.000;

%dr=rmax/(n-1);

%r=0:dr:rmax;

%Define Constants for Anlaysis

%**********************************************************************

g=9.81;

%Define Experimental Data Sets

%**********************************************************************

Data=menu('Select Data Set for Analysis','Van Stralen, Cole, Sluyter,

and Sohal (1975)','Ellion (1954)','Cole and Schulman (1966)');

if Data==1

%Data of Van Stralen, Cole, Sluyter, and Sohal

Pbulk=[.1013,.02672,0.02028,0.01321,0.00788,0.00408];

Tbulk=[373.517,340.808,334.31,325.411,315.274,304.754];

Twall=[394.617,351.808,348.71,344.211,342.674,337.354];

Rexp=[0.00092,0.0079,0.0119,0.0136,0.0268,0.0415];

b=pi/7;

elseif Data==2

%Experimental Data of Ellion

Pbulk=.101325*ones(1,10);

Tbulk=[289.8166667,298.7055556,317.5944444,325.3722222,328.7055556,353.

7055556,330.3722222,330.3722222,330.3722222,330.3722222];

Twall=[408.1243,407.5687444,405.9020778,404.2354111,403.6798556,394.235

4111,399.2354111,400.6243,404.2354111,405.9020778];

Rexp=[0.00035052,0.00037592,0.00043688,0.0004953,0.0004699,0.0005588,0.

0004953,0.0004699,0.00046228,0.0004445];

b=pi/7;

elseif Data==3

%Experimental Data of Cole and Schulmen

Pbulk=[.047996,.047996,.047996,.025998,.025998,.013066,.013066,.013066,

.013066,.013066,.013066,.013066,.013066,.013066,.013066,.006666,.006666

,.006666];

Tbulk=[353.4512,353.4512,353.4512,338.9906,338.9906,324.2882,324.2882,3

24.2882,324.2882,324.2882,324.2882,324.2882,324.2882,324.2882,324.2882,

311.2438,311.2438,311.2438];

Twall=[368.4512,368.4512,368.4512,357.3239,357.3239,339.2882,339.2882,3

39.2882,339.2882,339.2882,339.2882,339.2882,339.2882,339.2882,339.2882,

365.1327,365.1327,365.1327];

Rexp=[.009,.00775,.0065,.00925,.008,.019,.015,.01275,.013,.00925,.01275

,.01175,.011,.01025,.0095,.02075,.02,.019];

b=pi/7;

%Initialize System Property Vectors for Analysis

%**********************************************************************

m=max(size(Pbulk));

Tvap=zeros(1,m);

Pvap=zeros(1,m);

P=zeros(m,n);

dP=zeros(m,n);

d2P=zeros(m,n);

d3P=zeros(m,n);

d4P=zeros(m,n);

d5P=zeros(m,n);

CpL=zeros(1,m);

CvL=zeros(1,m);

DenL=zeros(1,m);

DenV=zeros(m,n);

dDenV=zeros(m,n);

d2DenV=zeros(m,n);

d3DenV=zeros(m,n);

d4DenV=zeros(m,n);

d5DenV=zeros(m,n);

DenWork=zeros(m,n+6);

kL=zeros(1,m);

St=zeros(1,m);

aL=zeros(1,m);

VisL=zeros(1,m);

hfg=zeros(1,m);

Tsupw=zeros(1,m);

Ja=zeros(1,m);

A=zeros(1,m);

B=zeros(1,m);

Jastar=zeros(1,m);

S=zeros(1,m);

Rc=zeros(1,m);

R=zeros(m,n);

dR=zeros(m,n);

d2R=zeros(m,n);

d3R=zeros(m,n);

d4R=zeros(m,n);

d5R=zeros(m,n);

d6R=zeros(m,n);

d7R=zeros(m,n);

Rworking=zeros(m,n);

Res=zeros(m,n);

Rint=zeros(m,3);

F=zeros(m,n);

dF=zeros(m,n);

d2F=zeros(m,n);

d3F=zeros(m,n);

d4F=zeros(m,n);

d5F=zeros(m,n);

Rpre=zeros(1,m);

ErrR=zeros(1,m);

Sgen=zeros(m,n);

%Calculate for System Properties Using IAPWS Equations

%**********************************************************************

for j=1:m

%Define Vapor State

Tvap(j)=SatTemp(Pbulk(j));

Pvap(j)=SatPress(Tbulk(j));

%Define Bulk Liquid Properties

[CpL(j),CvL(j)]=SpecHeatLiq(Tbulk(j),Pvap(j)*1E-9);

CpL(j)=1E3*CpL(j);

CvL(j)=1E3*CvL(j);

[DenL(j)]=DenLiq(Tbulk(j),Pvap(j));

[kL(j)]=ThermCond(Tbulk(j),Pvap(j))/1000;

[St(j)]=SurfTen(Tbulk(j));

aL(j)=kL(j)/(DenL(j)*CpL(j));

VisL(j)=VisLiq(Tbulk(j),DenL(j));

%Define Vapor Properties

[DenV(j)]=DenVap(Tvap(j),Pbulk(j));

hfg(j)=1E3*(EnthVap(Tvap(j),Pbulk(j))-EnthLiq(Tvap(j),Pbulk(j)));

%Define System Conditions

Tsupw(j)=Twall(j)-Tvap(j);

Ja(j)=DenL(j)*CpL(j)*Tsupw(j)/(DenV(j)*hfg(j));

A(j)=(b*Tsupw(j)*hfg(j)*DenV(j)/(DenL(j)*Tvap(j)))^.5;

B(j)=Ja(j)*(12*aL(j)/pi)^.5;

%Define Bubble Behavior

%**********************************************************************

for j=1:m

%Define Critical Radius

%**********************************************************************

Rc(j)=2*St(j)/((Pvap(j)-Pbulk(j)*1E6));

for i=1:n

%Bubble Growth Behavior (MRG Equation)

%******************************************************************

R(j,i)=(2*B(j)^2/(3*A(j)))*((A(j)^2*t(i)/B(j)^2+1)^(3/2)-

(A(j)^2*t(i)/B(j)^2)^(3/2)-1);

dR(j,i)=(A(j))*((A(j)^2*t(i)/B(j)^2+1)^(1/2)-

(A(j)^2*t(i)/B(j)^2)^(1/2));

d2R(j,i)=(A(j)^3/(2*B(j)^2))*((A(j)^2*t(i)/B(j)^2+1)^(-1/2)-

(A(j)^2*t(i)/B(j)^2)^(-1/2));

d3R(j,i)=-(A(j)^5/(4*B(j)^4))*((A(j)^2*t(i)/B(j)^2+1)^(-3/2)-

(A(j)^2*t(i)/B(j)^2)^(-3/2));

d4R(j,i)=(3*A(j)^7/(8*B(j)^6))*((A(j)^2*t(i)/B(j)^2+1)^(-5/2)-

(A(j)^2*t(i)/B(j)^2)^(-5/2));

d5R(j,i)=-(15*A(j)^9/(16*B(j)^8))*((A(j)^2*t(i)/B(j)^2+1)^(-

7/2)-(A(j)^2*t(i)/B(j)^2)^(-7/2));

d6R(j,i)=(105*A(j)^11/(32*B(j)^10))*((A(j)^2*t(i)/B(j)^2+1)^(-

9/2)-(A(j)^2*t(i)/B(j)^2)^(-9/2));

d7R(j,i)=-(945*A(j)^13/(64*B(j)^12))*((A(j)^2*t(i)/B(j)^2+1)^(-

11/2)-(A(j)^2*t(i)/B(j)^2)^(-11/2));

if R(j,i)<=Rc(j)

R(j,i)=Rc(j);

dR(j,i)=0;

d2R(j,i)=0;

d3R(j,i)=0;

d4R(j,i)=0;

d5R(j,i)=0;

d6R(j,i)=0;

d7R(j,i)=0;

%Define Variable Vapor Pressure Behavior (Equation of Motion)

%******************************************************************

P(j,i)=(Pbulk(j)*1E6)+(2*St(j)/R(j,i))+DenL(j)*(3*dR(j,i)^2/2+R(j,i)*d2

R(j,i));

dP(j,i)=(-

2*St(j)*dR(j,i)/R(j,i)^2+DenL(j)*(4*dR(j,i)*d2R(j,i)+R(j,i)*d3R(j,i)));

d2P(j,i)=(4*St(j)*dR(j,i)^2/R(j,i)^3-

2*St(j)*d2R(j,i)/R(j,i)^2+DenL(j)*(4*d2R(j,i)^2+5*dR(j,i)*d3R(j,i)+R(j,

i)*d4R(j,i)));

d3P(j,i)=(-

12*St(j)*dR(j,i)^3/R(j,i)^4+12*St(j)*dR(j,i)*d2R(j,i)/R(j,i)^3-

2*St(j)*d3R(j,i)/dR(j,i)^2+DenL(j)*(13*d2R(j,i)*d3R(j,i)+6*dR(j,i)*d4R(

j,i)+R(j,i)*d5R(j,i)));

d4P(j,i)=(4*pi*g/3)*((36*dR(j,i)^2*d2R(j,i)+18*R(j,i)*d2R(j,i)^2+24*R(j

,i)*dR(j,i)*d3R(j,i)+3*R(j,i)^2*d4R(j,i))*(DenL(j)-DenV(j,i))-

(24*dR(j,i)^3+72*R(j,i)*dR(j,i)*d2R(j,i)+12*R(j,i)^2*d3R(j,i))*dDenV(j,

i)-(36*R(j,i)*dR(j,i)^2+18*R(j,i)^2*d2R(j,i))*d2DenV(j,i)-

12*R(j,i)^2*dR(j,i)*d3DenV(j,i)-R(j,i)^3*d4DenV(j,i))-

pi*St(j)*(S(j)*d4R(j,i))-

(11*pi/12)*DenL(j)*(270*dR(j,i)^2*d2R(j,i)^2+120*dR(j,i)^3*d3R(j,i)+90*

R(j,i)*d2R(j,i)^3+360*R(j,i)*dR(j,i)*d2R(j,i)*d3R(j,i)+30*R(j,i)^2+d3R(

j,i)^2+90*R(j,i)*dR(j,i)^2*d4R(j,i)+45*R(j,i)^2*d2R(j,i)*d4R(j,i)+18*R(

j,i)^2*dR(j,i)*d5R(j,i)+R(j,i)^3*d6R(j,i))-

(pi/4)*a*VisL(j)*(10*d2R(j,i)*d3R(j,i)+5*dR(j,i)*d4R(j,i)+R(j,i)*d5R(j,

d5P(j,i)=(4*pi*g/3)*((90*dR(j,i)*d2R(j,i)^2+60*dR(j,i)^2*d3R(j,i)+60*R(

j,i)*d2R(j,i)*d3R(j,i)+30*R(j,i)*dR(j,i)*d4R(j,i)+3*R(j,i)^2*d5R(j,i))*

(DenL(j)-DenV(j,i)))-pi*St(j)*S(j)*d5R(j,i)-

(11*pi/12)*DenL(j)*(630*dR(j,i)*d2R(j,i)^3+1260*dR(j,i)^2*d2R(j,i)*d3R(

j,i)+210*dR(j,i)^3*d4R(j,i)+630*R(j,i)*d2R(j,i)^2*d3R(j,i)+420*R(j,i)*d

R(j,i)*d3R(j,i)^2+120*R(j,i)^2*d3R(j,i)*d4R(j,i)+660*R(j,i)*dR(j,i)*d2R

(j,i)*d4R(j,i)+126*R(j,i)*dR(j,i)^2*d5R(j,i)+78*R(j,i)^2*d2R(j,i)*d5R(j

,i)+21*R(j,i)^2*dR(j,i)*d6R(j,i)+R(j,i)^3*d7R(j,i))-

(pi/4)*a*VisL(j)*(10*d3R(j,i)^3+15*d2R(j,i)*d4R(j,i)+6*dR(j,i)*d5R(j,i)

+R(j,i)*d6R(j,i));

if P(j,i)>=Pvap(j)*1E6

P(j,i)=Pvap(j)*1E6;

dP(j,i)=0;

d2P(j,i)=0;

d3P(j,i)=0;

d4P(j,i)=0;

d5P(j,i)=0;

%Define Method of Analysis (Pressure Method or Net Force Method)

%**********************************************************************

model=menu('Select a Model for Analysis','Pressure Method','Net Force

Method');

%**********************************************************************

%Pressure Method

%**********************************************************************

if model==1

%Define Entropy Generation Rate

%**********************************************************************

for j=1:m

for i=1:n

Sgen(j,i)=(4*pi/Twall(j))*Pbulk(j)*1E6*R(j,i)^2*dR(j,i);

figure

axis auto

plot(R(j,:),Sgen(j,:))

xlabel('Radius, m')

ylabel('Entropy Generation Rate, W')

title('Entropy Generation Rate vs. Bubble Radius')

set(gcf,'color','w')

%Define Pressure Method Equation (Direct Sub or Gen Solution)

%**********************************************************************

equation=menu('Select Equation','Direct Substitution','General

Solution');

%**********************************************************************

%Pressure Method-Direct Substitution

%**********************************************************************

if equation==1

fprintf(' THE FOLLOWING RESULTS ARE OBTAINED USING THE

PRESSURE METHOD\n')

fprintf(' WITH DIRECT SUBSTITUTION\n')

if Data==1

fprintf('\n Data of Van Stralen, Cole, Sluyter,

and Sohal (1975) \n')

fprintf('**************************************************************

************\n')

elseif Data==2

fprintf('\n Data of Ellion (1954)

fprintf('**************************************************************

************\n')

elseif Data==3

fprintf('\n Data of Cole and Shulman

(1966) \n')

fprintf('**************************************************************

************\n')

for j=1:m

%Solve Second Order, Non-linear Differential Equation

%**************************************************************

for i=1:n

Res(j,i)=R(j,i)*d2R(j,i)+2*dR(j,i)^2;

%Plot Results

%**************************************************************

figure

plot(t,Res(j,:))

axix auto

title('Residual vs. Time')

xlabel('Time, sec')

ylabel('Residual')

%Print Results

%**************************************************************

fprintf('Bubble Number:

%1.0f\n\n',j);

fprintf('Liquid Pressure: %f

[MPa]\n',Pbulk(j))

fprintf('Liquid Temperature: %f

[K]\n',Tbulk(j))

fprintf('Liquid Density: %f

[kg/m^3]\n',DenL(j))

fprintf('Liquid Surface Tension: %f

[N/m]\n',St(j))

fprintf('Liquid Viscosity: %E

[Ns/m^2]\n',VisL(j))

fprintf('Liquid Specific Heat (Const Pressure): %f

[kJ/kg]\n',CpL(j)*1E-3)

fprintf('Liquid Specific Heat (Const Volume): %f

[kJ/kg]\n',CvL(j)*1E-3)

fprintf('Liquid Thermal Conductivity: %f

[W/mK]\n',kL(j))

fprintf('Liquid Thermal Diffusivity: %E

[m^2/s]\n',aL(j))

fprintf('Specific Enthalpy of Vaporization: %f

[kJ/kg]\n\n',hfg(j)*1E-3)

fprintf('Vapor Pressure: %f

[MPa]\n',Pvap(j))

fprintf('Vapor Temperature: %f

[K]\n',Tvap(j))

fprintf('Vapor Density: %f

[kg/m^3]\n\n',DenV(j))

fprintf('Wall Temperature: %f

[K]\n',Twall(j))

fprintf('Wall Superheat: %f

[K]\n\n',Tsupw(j))

fprintf('Ja: %f

\n',Ja(j))

fprintf('A:

%f\n',A(j));

fprintf('B:

%f\n\n',B(j));

fprintf('**************************************************************

************\n')

fprintf('\n');

%**********************************************************************

%Pressure Method-General Solution

%**********************************************************************

elseif equation==2;

%Define Method of Analysis

%******************************************************************

method=menu('Select Method of Analysis','Uniform Constants C

and D','System Dependant Constants C and D');

submethod=menu('Select Equation','Rayleigh Equation','Plesset-

Zwick Equation','Mikic Equation');

%Initiate Matrices

%******************************************************************

C=zeros(1,m);

D=zeros(1,m);

Rworking=zeros(m,3);

ErrR=zeros(1,m);

%******************************************************************

%Uniform Constants C and D

%******************************************************************

if method==1

%Define Constants

%**************************************************************

if submethod==1

C=ones(1,max(size(A)))*10.129130;

D=ones(1,max(size(A)))*1.683431E-05;

elseif submethod==2

C=ones(1,max(size(A)))*6.784382;

D=ones(1,max(size(A)))*3.149594E-04;

elseif submethod==3

C=ones(1,max(size(A)))*6.666687;

D=ones(1,max(size(A)))*3.912008E-04;

%******************************************************************

%System Dependant Constants C and D

%******************************************************************

elseif method==2

%Define Constants

%**************************************************************

if submethod==1

for j=1:max(size(A));

C(j)=7.459635*log(A(j))+2.607226;

D(j)=2.278040E-11*Ja(j)^2+6.485067E-09*Ja(j)-

3.367751E-07;

elseif submethod==2

C(j)=-0.894132*log(B(j))+4.010944;

D(j)=4.127304E-03*exp(-1.036544*A(j));

elseif submethod==3

C(j)=5.814845E-2*A(j)^2+8.891619E-1*A(j)+3.399097;

D(j)=-1.957951E-12*B(j)^4/A(j)^2+1.124843E-

7*B(j)^2/A(j)+7.128086E-4;

%Print Details of Analysis

%******************************************************************

fprintf('**************************************************************

*************\n')

fprintf(' PREDICTED RADII USING GENERAL

SOLUTION\n')

if method==1

fprintf(' USINIG UNIFORM CONSTANTS C AND

D\n');

elseif method==2

fprintf(' USING SYSTEM DEPENDANT CONSTANTS C

AND D\n');

if Data==1

fprintf('**************************************************************

************\n')

elseif Data==2

fprintf('**************************************************************

************\n')

elseif Data==3

(1966) \n')

fprintf('**************************************************************

************\n')

%Print System Properties

%******************************************************************

for j=1:max(size(A))

%1.0f\n\n',j);

[MPa]\n',Pbulk(j))

[K]\n',Tbulk(j))

[N/m]\n',St(j))

[W/mK]\n',kL(j))

[m^2/s]\n',aL(j))

[MPa]\n',Pvap(j))

[K]\n',Tvap(j))

[K]\n',Twall(j))

[K]\n\n',Tsupw(j))

fprintf('Ja: %f

\n',Ja(j))

fprintf('A:

%f\n',A(j));

fprintf('B:

%f\n\n',B(j));

%Define Polynomial Equations for Rayleigh and Plesset-Zwick

%**************************************************************

if submethod==1

Rworking=[1,0,3*exp(-C(j))/A(j),-D(j)];

elseif submethod==2

Rworking=[1,3*exp(-C(j))/B(j)^2,0,-D(j)];

%Determine Solutions to Polynomial Equations

%**************************************************************

if submethod<=2

Rint(j,:)=roots(Rworking);

Rpre(j)=Rint(j,3);

elseif submethod==3

Rpre(j)=0.00000005;

dr=0.00000005;

test=1;

step=1;

while test==1

Res=(2*B(j)^2/(3*A(j)))*((-

A(j)^2*(Rpre(j))^3/(3*B(j)^2*exp(-C(j)))+A(j)^2*D(j)/(3*B(j)^2*exp(-

C(j)))+1)^1.5-(-A(j)^2*(Rpre(j))^3/(B(j)^2*3*exp(-

C(j)))+A(j)^2*D(j)/(B(j)^2*3*exp(-C(j))))^1.5-1)-(Rpre(j));

if Res<0

test=2;

Rpre(j)=Rpre(j)+dr;

test=1;

step=step+1;

%Calculated Error of Predicted Radii

%**************************************************************

ErrR(j)=100*(Rpre(j)-Rexp(j))/Rexp(j);

%Print Results

%**************************************************************

fprintf('Experimental Departure Radius:

%f\n\n',Rexp(j));

fprintf('Predicted Departure Radius:

%f\n',Rpre(j));

fprintf('Error:

%f\n\n',ErrR(j));

fprintf('\n');

%**********************************************************************

%Net Force Method

%**********************************************************************

elseif model==2

%Define Net Force Equation for Analysis

%**********************************************************************

equation=menu('Select Equation','Constant Vapor Pressure &

Negligible Net Force Derivatives','Complete Equation');

%Define Scaling Factor Method

%**********************************************************************

method=menu('Select Scaling Factor','Fritz Based Equation','Cole

Based Equation','Cole & Rohsenow Based Equation');

%Define Scaling Factor

%**********************************************************************

for j=1:m

if method==1

beta=30;

S(j)=(0.0208*beta)^2/6;

elseif method==2

S(j)=(1/6)*(0.04*Ja(j))^2;

elseif method==3

Jastar(j)=DenL(j)*CpL(j)*Tvap(j)/(DenV(j)*hfg(j));

S(j)=((1.5E-4)^2/6)*Jastar(j)^(5/2);

%**********************************************************************

%Constant Vapor Pressure and Negligible Net Force Derivatives

%**********************************************************************

if equation==1

fprintf(' THE FOLLOWING RESULTS ARE OBTAINED USING THE NET

FORCE METHOD\n')

fprintf(' WITH SIMPLIFIED SOLUTION\n')

if Data==1

fprintf('**************************************************************

************\n')

elseif Data==2

fprintf('**************************************************************

************\n')

elseif Data==3

(1966) \n')

fprintf('**************************************************************

************\n')

%Initialize Vectors for Analysis

%******************************************************************

for j=1:m

%1.0f\n\n',j);

[MPa]\n',Pbulk(j))

[K]\n',Tbulk(j))

[N/m]\n',St(j))

[W/mK]\n',kL(j))

[m^2/s]\n',aL(j))

[MPa]\n',Pvap(j))

[K]\n',Tvap(j))

[K]\n',Twall(j))

[K]\n\n',Tsupw(j))

fprintf('Ja: %f

\n',Ja(j))

fprintf('A:

%f\n',A(j));

fprintf('B:

%f\n\n',B(j));

%Define Bubble Growth Behavior

%**************************************************************

for i=1:n

F(j,i)=(4*pi/3)*g*(DenL(j)-DenV(j))*R(j,i)^3-

pi*St(j)*S(j)*R(j,i)-

(11*pi/12)*DenL(j)*(3*R(j,i)^2*dR(j,i)^2+R(j,i)^3*d2R(j,i))-

(pi/4)*a*VisL(j)*R(j,i)*dR(j);

%**********************************************************

Sgen(j,i)=-

F(j,i)*dR(j,i)/Twall(j)+(4*pi/Twall(j))*Pbulk(j)*1E6*R(j,i)^2*dR(j,i);

%Define Residual of Entropy Generation Rate Derivative

%**********************************************************

Rworking(j,i)=-

F(j,i)+4*pi*(2*R(j,i)*dR(j,i)^2/d2R(j,i)+R(j,i)^2)*Pbulk(j)*1E6;

figure

axis auto

xlabel('Radius, m')

Rpre(j)=interp1(Rworking(j,:),R(j,:),0,'spline');

%**************************************************************

%Print Results

%**************************************************************

%f\n\n',Rexp(j));

%f\n',Rpre(j));

fprintf('Error:

%f\n\n',ErrR(j));

fprintf('**************************************************************

*************\n')

%**********************************************************************

%Complete Equation

%**********************************************************************

elseif equation==2

fprintf(' THE FOLLOWING RESULTS ARE OBTAINED USING THE NET

FORCE METHOD\n')

fprintf(' WITH COMPLETE SOLUTION\n')

if Data==1

fprintf('**************************************************************

************\n')

elseif Data==2

fprintf('**************************************************************

************\n')

elseif Data==3

(1966) \n')

fprintf('**************************************************************

************\n')

for j=1:m

DenV(j,:)=DenVap(Tvap(j),Pbulk(j))*ones(1,n);

%1.0f\n\n',j);

[MPa]\n',Pbulk(j))

[K]\n',Tbulk(j))

[N/m]\n',St(j))

[W/mK]\n',kL(j))

[m^2/s]\n',aL(j))

fprintf('Vapor Pressure:

Variable\n')

[K]\n',Tvap(j))

fprintf('Vapor Density:

Variable\n')

[K]\n',Twall(j))

[K]\n\n',Tsupw(j))

fprintf('Ja: %f

\n',Ja(j))

fprintf('A:

%f\n',A(j));

fprintf('B:

%f\n\n',B(j));

for i=1:n

%Define Variable Vapor Temperature

%**********************************************************

Tvap(j,i)=SatTemp(P(j,i)/1E6);

%Define Variable Vapor Properties

%**********************************************************

DenV(j,i)=DenVap(Tvap(j,i),P(j,i)/1E6);

%Define Vapor Density Behavior

%**************************************************************

for i=1:n

DenWork(j,i+3)=DenV(j,i);

DenWork(j,1)=interp1(t,DenV(j,:),-3*dt,'spline','extrap');

DenWork(j,2)=interp1(t,DenV(j,:),-2*dt,'spline','extrap');

DenWork(j,3)=interp1(t,DenV(j,:),-dt,'spline','extrap');

DenWork(j,n+4)=interp1(t,DenV(j,:),tmax+dt,'spline','extrap');

DenWork(j,n+5)=interp1(t,DenV(j,:),tmax+2*dt,'spline','extrap');

DenWork(j,n+6)=interp1(t,DenV(j,:),tmax+3*dt,'spline','extrap');

for i=1:n

k=i+3;

dDenV(j,i)=(-DenWork(j,k+2)+8*DenWork(j,k+1)-

8*DenWork(j,k-1)+DenWork(j,k-1))/(12*dt);

d2DenV(j,i)=(-DenWork(j,k+2)+16*DenWork(j,k+1)-

30*DenWork(j,k)+16*DenWork(j,k-1)-DenWork(j,k-2))/(12*dt^2);

d3DenV(j,i)=(-DenWork(j,k+3)+8*DenWork(j,k+2)-

13*DenWork(j,k+1)+13*DenWork(j,k-1)-8*DenWork(j,k-

2)+DenWork(j,k+3))/(8*dt^3);

d4DenV(j,i)=(-

DenWork(j,k+3)+12*DenWork(j,k+2)+39*DenWork(j,k+1)+56*DenWork(j,k)-

39*DenWork(j,k-1)-8*DenWork(j,k-2)+DenWork(j,k+3))/(6*dt^4);

d5DenV(j,i)=0;

for i=1:n

%Define Net Force Behavior

%**********************************************************

F(j,i)=(4*pi*g/3)*R(j,i)^3*(DenL(j)-DenV(j,i))-

pi*St(j)*S(j)*R(j,i)-

(11*pi/12)*DenL(j)*(3*R(j,i)^2*dR(j,i)^2+R(j,i)^3*d2R(j,i))-

(pi/4)*a*VisL(j)*R(j,i)*dR(j,i);

dF(j,i)=(4*pi*g/3)*(3*R(j,i)^2*dR(j,i)*(DenL(j)-

DenV(j,i))-R(j,i)^3*dDenV(j,i))-pi*St(j)*S(j)*dR(j,i)-

(11*pi/12)*DenL(j)*(6*R(j,i)*dR(j,i)^3+9*R(j,i)^2*dR(j,i)*d2R(j,i)+R(j,

i)^3*d3R(j,i))-(pi/4)*a*VisL(j)*(R(j,i)*d2R(j,i)+dR(j,i)^2);

d2F(j,i)=(4*pi*g/3)*((6*(R(j,i)*dR(j,i)^2+3*R(j,i)^2*d2R(j,i)))*(DenL(j

)-DenV(j,i))-6*R(j,i)^2*dR(j,i)*dDenV(j,i)-R(j,i)^3*d2DenV(j,i))-

pi*St(j)*S(j)*d2R(j,i)-

(11*pi/12)*DenL(j)*(6*dR(j,i)^4+36*R(j,i)*dR(j,i)^2*d2R(j,i)+9*R(j,i)^2

*d2R(j,i)^2+12*R(j,i)^2*dR(j,i)*d3R(j,i)+R(j,i)^3*d4R(j,i))-

(pi/4)*a*VisL(j)*(3*dR(j,i)*d2R(j,i)+R(i)*d3R(j,i));

d3F(j,i)=(4*pi*g/3)*((6*dR(j,i)^3+18*R(j,i)*dR(j,i)*d2R(j,i)+3*R(j,i)^2

*d3R(j,i))*(DenL(j)-DenV(j,i))-

(18*R(j,i)*dR(j,i)^2+9*R(j,i)^2*d2R(j,i))*dDenV(j,i)-

(11*pi/12)*DenL(j)*(60*dR(j,i)^3*d2R(j,i)+90*R(j,i)*dR(j,i)*d2R(j,i)^2+

60*R(j,i)*dR(j,i)^2*d3R(j,i)+30*R(j,i)^2*d2R(j,i)*d3R(j,i)+15*R(j,i)^2*

dR(j,i)*d4R(j,i)+R(j,i)^3*d5R(j,i))-

(pi/4)*a*VisL(j)*(3*d2R(j,i)^2+4*dR(j,i)*d3R(j,i)+R(j,i)*d4R(j,i));

d4F(j,i)=(4*pi*g/3)*((36*dR(j,i)^2*d2R(j,i)+18*R(j,i)*d2R(j,i)^2+24*R(j

,i)*dR(j,i)*d3R(j,i)+3*R(j,i)^2*d4R(j,i))*(DenL(j)-DenV(j,i))-

(24*dR(j,i)^3+72*R(j,i)*dR(j,i)*d2R(j,i)+12*R(j,i)^2*d3R(j,i))*dDenV(j,

i)-(36*R(j,i)*dR(j,i)^2+18*R(j,i)^2*d2R(j,i))*d2DenV(j,i)-

(11*pi/12)*DenL(j)*(270*dR(j,i)^2*d2R(j,i)^2+120*dR(j,i)^3*d3R(j,i)+90*

R(j,i)*d2R(j,i)^3+360*R(j,i)*dR(j,i)*d2R(j,i)*d3R(j,i)+30*R(j,i)^2+d3R(

j,i)^2+45*R(j,i)*dR(j,i)^2*d4R(j,i)+60*R(j,i)^2*d2R(j,i)*d4R(j,i)+18*R(

j,i)^2*dR(j,i)*d5R(j,i)+R(j,i)^3*d6R(j,i))-

(pi/4)*a*VisL(j)*(10*d2R(j,i)*d3R(j,i)+5*dR(j,i)*d4R(j,i)+R(j,i)*d5R(j,

d5F(j,i)=(4*pi*g/3)*((90*dR(j,i)*d2R(j,i)^2+60*dR(j,i)^2*d3R(j,i)+60*R(

j,i)*d2R(j,i)*d3R(j,i)+30*R(j,i)*dR(j,i)*d4R(j,i)+3*R(j,i)^2*d5R(j,i))*

(DenL(j)-DenV(j,i))-

(180*dR(j,i)^2*d2R(j,i)+90*R(j,i)*d2R(j,i)^2+120*R(j,i)*dR(j,i)*d3R(j,i

)+15*R(j,i)^2*d4R(j,i)*dDenV(j,i)-

(60*dR(j,i)^3+180*R(j,i)*dR(j,i)*d2R(j,i)+30*R(j,i)^2*d3R(j,i))*d2DenV(

j,i)-(60*R(j,i)*dR(j,i)^2+30*R(j,i)^2*d2R(j,i))*d3DenV(j,i)-

pi*St(j)*S(j)*d5R(j,i)-(11*pi/12)*DenL(j)

*(630*dR(j,i)*d2R(j,i)^3+1260*dR(j,i)^2*d2R(j,i)*d3R(j,i)+120*dR(j,i)^3

*d4R(j,i)+630*R(j,i)*d2R(j,i)^2*d3R(j,i)+420*R(j,i)*dR(j,i)*d3R(j,i)^2+

630*R(j,i)*dR(j,i)*d2R(j,i)*d4R(j,i)+90*dR(j,i)^3*d4R(j,i)+105*R(j,i)^2

*d3R(j,i)*d4R(j,i)+63*R(j,i)^2*d2R(j,i)*d5R(j,i)+126*R(j,i)*dR(j,i)^2*d

5R(j,i)+21*R(j,i)^2*dR(j,i)*d6R(j,i)+R(j,i)^3*d7R(j,i))-

(pi/4)*a*VisL(j)*(10*d3R(j,i)^3+15*d2R(j,i)*d4R(j,i)+5*dR(j,i)*d5R(j,i)

+6*dR(j,i)*d5R(j,i)+dR(j,i)*d6R(j,i)));

if R(j,i)<=Rc(j)

F(j,i)=0;

dF(j,i)=0;

d2F(j,i)=0;

d3F(j,i)=0;

d4F(j,i)=0;

d5F(j,i)=0;

%**********************************************************

Sgen(j,i)=-F(j,i)*dR(j,i)/Twall(j)-((dR(j,i)-

1)/Twall(j))*(-dF(j,i)*R(j,i)+(1/2)*d2F(j,i)*R(j,i)^2-

(1/6)*d3F(j,i)^R(j,i)^3+(1/24)*d4F(j,i)*R(j,i)^4)+(4*pi*R(j,i)^2/(3*Twa

ll(j)))*(-3*P(j,i)*dR(j,i)+(dR(j,i)-1)*(R(j,i)*dP(j,i)-

(1/4)*R(j,i)^2*d2P(j,i)+(1/20)*R(j,i)^3*d3P(j,i)-

(1/120)*R(j,i)^4*d4P(j,i)));

%Define the Residual of Entropy Generation Rate

Derivative

%**********************************************************

Rworking(j,i)=(-1/Twall(j))*(-

d2R(j,i)*F(j,i)+(R(j,i)*d2R(j,i)+R(j,i)^2-

2*dR(j,i))*dF(j,i)+(2*R(j,i)*dR(j,i)-(1/2)*R(j,i)^2*d2R(j,i)-

R(j,i)*d2R(j,i)-

R(j,i))*d2F(j,i)+((1/6)*R(j,i)^3*d2R(j,i)+(1/2)*R(j,i)^2*dR(j,i)^2+(1/2

)*R(j,i)^2-R(j,i)^2*dR(j,i))*d3F(j,i)+((1/3)*R(j,i)^3*dR(j,i)-

(1/24)*R(j,i)^4*d2R(j,i)-(1/6)*R(j,i)^3*dR(j,i)^2-

(1/6)*R(j,i)^3)*d4F(j,i)+((1/24)*R(j,i)^4-

(1/24)*R(j,i)^4*dR(j,i))*d5F(j,i)-(4*pi/3)*((-6*R(j,i)*dR(j,i)^2-

3*R(j,i)^2*d2R(j,i))*P(j,i)+(-

6*R(j,i)^2*dR(j,i)+3*R(j,i)^2*dR(j,i)^2+R(j,i)^3*d2R(j,i))*dP(j,i)+(-

R(j,i)^3+2*R(j,i)^3*dR(j,i)-R(j,i)^3*dR(j,i)^2-

(1/4)*R(j,i)^4*d2R(j,i))*d2P(j,i)+((1/4)*R(j,i)-

(1/2)*R(j,i)^4*dR(j,i)+(2/5)*R(j,i)^4*dR(j,i)^2+(1/20)*R(j,i)^5*d2R(j,i

))*d3P(j,i)+(-(1/20)*R(j,i)^5+(3/20)*R(j,i)^5*dR(j,i)-

(1/20)*R(j,i)^5*dR(j,i)^2-

(1/120)*R(j,i)^6*d2R(j,i))*d4P(j,i)+((1/120)*R(j,i)^6-

(1/120)*R(j,i)^6*dR(j,i))));

figure

axis auto

xlabel('Radius, m')

for i=1:n

Rpre(j)=interp1(Rworking(j,:),R(j,:),0,'spline');

%**************************************************************

%Print Results

%**************************************************************

%f\n\n',Rexp(j));

%f\n',Rpre(j));

fprintf('Error:

%f\n\n',ErrR(j));

fprintf('**************************************************************

*************\n')

fprintf('\n');

%Plot Results

%**********************************************************************

figure

axis auto

plot(Ja,Rexp,'*r',Ja,Rpre,'ob')

xlabel('Jakob Number')

ylabel('Radius, m')

title('Departure Radius vs. Jakob Number')

legend('Experimental','Predicted','location','Best')

figure

axis square

plot(Rexp,Rexp,'-k',Rexp,0.85*Rexp,'--k',Rexp,0.7*Rexp,'--

k',0.85*Rexp,Rexp,'--k',0.7*Rexp,Rexp,'--k',Rexp,Rpre,'*b')

xlabel('Departure Radius, m')

ylabel('Predicted Departure Radius, m')

title('Error Analysis')

Thermal properties of the fluid and vapor have been solved for using the

following programs derived using the IAWPS standards for water and steam

properties.

function [Tsat]=SatTemp(P)

%Revised Release on the IAPWS Industrial Formulation 1997 for the

%Thermodynamic Properties of Water and Steam (The revision only relates

%the extension of region 5 to 50 MPa)

%August 2007

%Section 8.3 - The Saturation-Temperature Equation (Backward Equation)

Pstar=1; %Reference Pressure, MPa

Tstar=1; %Reference Temperature, K

n=[0.11670521452767E4,-0.72421316703206E6,-

0.17073846940092E2,0.12020824702470E5,-

0.32325550322333E7,0.14915108613530E2,-

0.48232657361591E4,0.40511340542057E6,-

0.23855557567849,0.65017534844798E3];

beta=(P/Pstar)^(1/4);

E=beta^2+n(3)*beta+n(6);

F=n(1)*beta^2+n(4)*beta+n(7);

G=n(2)*beta^2+n(5)*beta+n(8);

D=2*G/(-F-(F^2-4*E*G)^(1/2));

Tsat=Tstar*((n(10)+D-((n(10)+D)^2-4*(n(9)+n(10)*D))^(1/2))/2);

function [Psat]=SatPress(Tamb)

%August 2007

n=[0.11670521452767E4,-0.72421316703206E6,-

0.17073846940092E2,0.12020824702470E5,-

0.32325550322333E7,0.14915108613530E2,-

0.48232657361591E4,0.40511340542057E6,-

0.23855557567849,0.65017534844798E3];

Nu=(Tamb/Tstar)+n(9)/((Tamb/Tstar)-n(10));

A=Nu^2+n(1)*Nu+n(2);

B=n(3)*Nu^2+n(4)*Nu+n(5);

C=n(6)*Nu^2+n(7)*Nu+n(8);

Psat=Pstar*(2*C/(-B+(B^2-4*A*C)^(1/2)))^4;

function [Cp,Cv]=SpecHeatLiq(T,P)

%August 2007

Pstar=16.53; %Reference Pressure, MPa

R=0.461526; %Gas Constant, kJ/kg-K

PI=P/Pstar;

Tau=Tstar/T;

Table2=[0,-2,0.14632971213167;0,-1,-0.84548187169114;0,0,-

0.37563603672040E1;0,1,0.33855169168385E1;0,2,-

0.95791963387872;0,3,0.15772038513228;0,4,-0.16616417199501E-

1;0,5,0.81214629983568E-3;1,-9,0.28319080123804E-3;1,-7,-

0.60706301565874E-3;1,-1,-0.18990068218419E-1;1,0,-0.32529748770505E-

1;1,1,-0.21841717175414E-1;1,3,-0.52838357969930E-4;2,-3,-

0.47184321073267E-3;2,0,-0.30001780793026E-3;2,1,0.47661393906987E-

4;2,3,-0.44141845330846E-5;2,17,-0.72694996297594E-15;3,-4,-

0.31679644845054E-4;3,0,-0.28270797985312E-5;3,6,-0.85205128120103E-

9;4,-5,-0.22425281908000E-5;4,-2,-0.65171222895601E-6;4,10,-

0.14341729937924E-12;5,-8,-0.40516996860117E-6;8,-11,-

0.12734301741641E-8;8,-6,-0.17424871230634E-9;21,-29,-

0.68762131295531E-18;23,-31,0.14478307828521E-19;29,-

38,0.26335781662795E-22;30,-39,-0.11947622640071E-22;31,-40,-

.18228094581404E-23;32,-41,-0.93537087292458E-25];

I=Table2(:,1);

J=Table2(:,2);

n=Table2(:,3);

Gamma=0;

GammaPI=0;

GammaPIPI=0;

GammaTau=0;

GammaTauTau=0;

GammaPITau=0;

for i=1:34

Gamma=Gamma+n(i)*(7.1-PI)^I(i)*(Tau-1.222)^J(i);

GammaPI=GammaPI-n(i)*I(i)*(7.1-PI)^(I(i)-1)*(Tau-1.222)^J(i);

GammaPIPI=GammaPIPI+n(i)*I(i)*(I(i)-1)*(7.1-PI)^(I(i)-2)*(Tau-

1.222)^J(i);

GammaTau=GammaTau+n(i)*(7.1-PI)^I(i)*J(i)*(Tau-1.222)^(J(i)-1);

GammaTauTau=GammaTauTau+n(i)*(7.1-PI)^I(i)*J(i)*(J(i)-1)*(Tau-

1.222)^(J(i)-2);

GammaPITau=GammaPITau-n(i)*I(i)*(7.1-PI)^(I(i)-1)*J(i)*(Tau-

1.222)^(J(i)-1);

Cp=R*(-Tau^2)*GammaTauTau;

Cv=R*((-Tau^2)*GammaTauTau+(GammaPI-Tau*GammaPITau)^2/GammaPIPI);

function [Cp,Cv]=SpecHeatVap(T,P)

%August 2007

%Section 6 - Equations for Region 2

PI=P/Pstar;

Tau=Tstar/T;

Table10=[0,-.96927686500217E1;1,0.10086655968018E2;-5,-

0.56087911283020E-2;-4,0.71452738081455E-1;-3,-0.40710498223928;-

2,0.14240819171444E1;-1,-0.43839511319450E1;2,-

0.28408632460772;3,0.21268463753307E-1];

Table11=[1,0,-0.17731742473213E-2;1,1,-0.17834862292358E-1;1,2,-

0.45996013696365E-1;1,3,-0.57581259083432E-1;1,6,-0.50325278727930E-

1;2,1,-0.33032641670203E-4;2,2,-0.18948987516315E-3;2,4,-

0.39392777243355E-2;2,7,-0.43797295650573E-1;2,36,-0.26674547914087E-

4;3,0,0.20481737692309E-7;3,1,0.43870667284435E-6;3,3,-

0.32277677238570E-4;3,6,-0.15033924542148E-2;3,35,-0.40668253562649E-

1;4,1,-0.78847309559367E-9;4,2,0.12790717852285E-

7;4,3,0.48225372718507E-6;5,7,0.22922076337661E-5;6,3,-

0.16714766451061E-10;6,16,-0.21171472321355E-2;6,35,-

0.23895741934104E2;7,0,-0.59059564324270E-17;7,11,-0.12621808899101E-

5;7,25,-0.38946842435739E-1;8,8,.11256211360459E-10;8,36,-

0.82311340897998E1;9,13,0.19809712802088E-7;10,4,0.10406965210174E-

18;10,10,-0.10234747095929E-12;10,14,-0.10018179379511E-8;16,29,-

0.80882908646985E-10;16,50,0.10693031879409;18,57,-

0.33662250574171;20,20,0.89185845355421E-24;20,35,0.30629316876232E-

12;20,48,-0.42002467698208E-5;

21,21,-0.59056029685639E-25;22,53,0.37826947613457E-5;23,39,-

0.12768608934681E-14;24,26,0.73087610595061E-

28;24,40,0.55414715350778E-16;24,58,-0.94369707241210E-6];

J0=Table10(:,1);

n0=Table10(:,2);

I=Table11(:,1);

J=Table11(:,2);

n=Table11(:,3);

Gamma0=0;

GammaTau0=0;

GammaTauTau0=0;

GammaPI0=1/PI;

GammaPIPI0=-1/PI^2;

for i=1:9

Gamma0=Gamma0+n0(i)*Tau^J0(i);

GammaTau0=GammaTau0+n0(i)*J0(i)*Tau^(J0(i)-1);

GammaTauTau0=GammaTauTau0+n0(i)*J0(i)*(J0(i)-1)*Tau^(J0(i)-2);

Gamma0=log(PI)+Gamma0;

GammaR=0;

GammaPITau0=0;

GammaPIR=0;

GammaPIPIR=0;

GammaTauR=0;

GammaTauTauR=0;

GammaPITauR=0;

for i=1:43

GammaR=GammaR+n(i)*PI^I(i)*(Tau-0.5)^J(i);

GammaPIR=GammaPIR+n(i)*I(i)*PI^(I(i)-1)*(Tau-0.5)^J(i);

GammaPIPIR=GammaPIPIR+n(i)*I(i)*(I(i)-1)*PI^(I(i)-2)*(Tau-

0.5)^J(i);

GammaTauR=GammaTauR+n(i)*PI^I(i)*J(i)*(Tau-0.5)^(J(i)-1);

GammaTauTauR=GammaTauTauR+n(i)*PI^I(i)*J(i)*(J(i)-1)*(Tau-

0.5)^(J(i)-2);

GammaPITauR=GammaPITauR+n(i)*I(i)*PI^(I(i)-1)*J(i)*(Tau-0.5)^(J(i)-

Cp=R*(-Tau^2*(GammaTauTau0+GammaTauTauR));

Cv=R*(-Tau^2*(GammaTauTau0+GammaTauTauR)-(1+PI*GammaPIR-

Tau*PI*GammaPITauR)^2/(1-PI^2*GammaPIPIR));

function [dl]=DenLiq(T,P)

%August 2007

PI=P/Pstar;

Tau=Tstar/T;

Table2=[0,-2,0.14632971213167;0,-1,-0.84548187169114;0,0,-

0.37563603672040E1;0,1,0.33855169168385E1;0,2,-

0.95791963387872;0,3,0.15772038513228;0,4,-0.16616417199501E-

1;0,5,0.81214629983568E-3;1,-9,0.28319080123804E-3;1,-7,-

0.60706301565874E-3;1,-1,-0.18990068218419E-1;1,0,-0.32529748770505E-

1;1,1,-0.21841717175414E-1;1,3,-0.52838357969930E-4;2,-3,-

0.47184321073267E-3;2,0,-0.30001780793026E-3;2,1,0.47661393906987E-

4;2,3,-0.44141845330846E-5;2,17,-0.72694996297594E-15;3,-4,-

0.31679644845054E-4;3,0,-0.28270797985312E-5;3,6,-0.85205128120103E-

9;4,-5,-0.22425281908000E-5;4,-2,-0.65171222895601E-6;4,10,-

0.14341729937924E-12;5,-8,-0.40516996860117E-6;8,-11,-

0.12734301741641E-8;8,-6,-0.17424871230634E-9;21,-29,-

0.68762131295531E-18;23,-31,0.14478307828521E-19;29,-

38,0.26335781662795E-22;30,-39,-0.11947622640071E-22;31,-40,-

.18228094581404E-23;32,-41,-0.93537087292458E-25];

I=Table2(:,1);

J=Table2(:,2);

n=Table2(:,3);

Gamma=0;

GammaPI=0;

GammaPIPI=0;

GammaTau=0;

GammaTauTau=0;

GammaPITau=0;

for i=1:34

1.222)^J(i);

1.222)^(J(i)-2);

1.222)^(J(i)-1);

vl=R*T*PI*GammaPI/(1000*P);

dl=1/vl;

function [dv]=DenVap(T,P)

%August 2007

PI=P/Pstar;

Tau=Tstar/T;

Table10=[0,-.96927686500217E1;1,0.10086655968018E2;-5,-

0.56087911283020E-2;-4,0.71452738081455E-1;-3,-0.40710498223928;-

2,0.14240819171444E1;-1,-0.43839511319450E1;2,-

0.28408632460772;3,0.21268463753307E-1];

Table11=[1,0,-0.17731742473213E-2;1,1,-0.17834862292358E-1;1,2,-

0.45996013696365E-1;1,3,-0.57581259083432E-1;1,6,-0.50325278727930E-

1;2,1,-0.33032641670203E-4;2,2,-0.18948987516315E-3;2,4,-

0.39392777243355E-2;2,7,-0.43797295650573E-1;2,36,-0.26674547914087E-

4;3,0,0.20481737692309E-7;3,1,0.43870667284435E-6;3,3,-

0.32277677238570E-4;3,6,-0.15033924542148E-2;3,35,-0.40668253562649E-

1;4,1,-0.78847309559367E-9;4,2,0.12790717852285E-

7;4,3,0.48225372718507E-6;5,7,0.22922076337661E-5;6,3,-

0.16714766451061E-10;6,16,-0.21171472321355E-2;6,35,-

0.23895741934104E2;7,0,-0.59059564324270E-17;7,11,-0.12621808899101E-

5;7,25,-0.38946842435739E-1;8,8,.11256211360459E-10;8,36,-

0.82311340897998E1;9,13,0.19809712802088E-7;10,4,0.10406965210174E-

18;10,10,-0.10234747095929E-12;10,14,-0.10018179379511E-8;16,29,-

0.80882908646985E-10;16,50,0.10693031879409;18,57,-

0.33662250574171;20,20,0.89185845355421E-24;20,35,0.30629316876232E-

12;20,48,-0.42002467698208E-5;

21,21,-0.59056029685639E-25;22,53,0.37826947613457E-5;23,39,-

0.12768608934681E-14;24,26,0.73087610595061E-

28;24,40,0.55414715350778E-16;24,58,-0.94369707241210E-6];

J0=Table10(:,1);

n0=Table10(:,2);

I=Table11(:,1);

J=Table11(:,2);

n=Table11(:,3);

Gamma0=0;

GammaTau0=0;

GammaTauTau0=0;

GammaPI0=1/PI;

GammaPIPI0=-1/PI^2;

for i=1:9

GammaR=0;

GammaPITau0=0;

GammaPIR=0;

GammaPIPIR=0;

GammaTauR=0;

GammaTauTauR=0;

GammaPITauR=0;

for i=1:43

0.5)^J(i);

0.5)^(J(i)-2);

vv=R*T*(PI*(GammaPI0+GammaPIR))/(P*1000);

dv=1/vv;

function [hl]=EnthLiq(T,P)

%August 2007

PI=P/Pstar;

Tau=Tstar/T;

Table2=[0,-2,0.14632971213167;0,-1,-0.84548187169114;0,0,-

0.37563603672040E1;0,1,0.33855169168385E1;0,2,-

0.95791963387872;0,3,0.15772038513228;0,4,-0.16616417199501E-

1;0,5,0.81214629983568E-3;1,-9,0.28319080123804E-3;1,-7,-

0.60706301565874E-3;1,-1,-0.18990068218419E-1;1,0,-0.32529748770505E-

1;1,1,-0.21841717175414E-1;1,3,-0.52838357969930E-4;2,-3,-

0.47184321073267E-3;2,0,-0.30001780793026E-3;2,1,0.47661393906987E-

4;2,3,-0.44141845330846E-5;2,17,-0.72694996297594E-15;3,-4,-

0.31679644845054E-4;3,0,-0.28270797985312E-5;3,6,-0.85205128120103E-

9;4,-5,-0.22425281908000E-5;4,-2,-0.65171222895601E-6;4,10,-

0.14341729937924E-12;5,-8,-0.40516996860117E-6;8,-11,-

0.12734301741641E-8;8,-6,-0.17424871230634E-9;21,-29,-

0.68762131295531E-18;23,-31,0.14478307828521E-19;29,-

38,0.26335781662795E-22;30,-39,-0.11947622640071E-22;31,-40,-

.18228094581404E-23;32,-41,-0.93537087292458E-25];

I=Table2(:,1);

J=Table2(:,2);

n=Table2(:,3);

Gamma=0;

GammaPI=0;

GammaPIPI=0;

GammaTau=0;

GammaTauTau=0;

GammaPITau=0;

for i=1:34

1.222)^J(i);

1.222)^(J(i)-2);

1.222)^(J(i)-1);

hl=R*T*Tau*GammaTau;

function [hv]=EnthVap(T,P)

%August 2007

PI=P/Pstar;

Tau=Tstar/T;

Table10=[0,-.96927686500217E1;1,0.10086655968018E2;-5,-

0.56087911283020E-2;-4,0.71452738081455E-1;-3,-0.40710498223928;-

2,0.14240819171444E1;-1,-0.43839511319450E1;2,-

0.28408632460772;3,0.21268463753307E-1];

Table11=[1,0,-0.17731742473213E-2;1,1,-0.17834862292358E-1;1,2,-

0.45996013696365E-1;1,3,-0.57581259083432E-1;1,6,-0.50325278727930E-

1;2,1,-0.33032641670203E-4;2,2,-0.18948987516315E-3;2,4,-

0.39392777243355E-2;2,7,-0.43797295650573E-1;2,36,-0.26674547914087E-

4;3,0,0.20481737692309E-7;3,1,0.43870667284435E-6;3,3,-

0.32277677238570E-4;3,6,-0.15033924542148E-2;3,35,-0.40668253562649E-

1;4,1,-0.78847309559367E-9;4,2,0.12790717852285E-

7;4,3,0.48225372718507E-6;5,7,0.22922076337661E-5;6,3,-

0.16714766451061E-10;6,16,-0.21171472321355E-2;6,35,-

0.23895741934104E2;7,0,-0.59059564324270E-17;7,11,-0.12621808899101E-

5;7,25,-0.38946842435739E-1;8,8,.11256211360459E-10;8,36,-

0.82311340897998E1;9,13,0.19809712802088E-7;10,4,0.10406965210174E-

18;10,10,-0.10234747095929E-12;10,14,-0.10018179379511E-8;16,29,-

0.80882908646985E-10;16,50,0.10693031879409;18,57,-

0.33662250574171;20,20,0.89185845355421E-24;20,35,0.30629316876232E-

12;20,48,-0.42002467698208E-5;

21,21,-0.59056029685639E-25;22,53,0.37826947613457E-5;23,39,-

0.12768608934681E-14;24,26,0.73087610595061E-

28;24,40,0.55414715350778E-16;24,58,-0.94369707241210E-6];

J0=Table10(:,1);

n0=Table10(:,2);

I=Table11(:,1);

J=Table11(:,2);

n=Table11(:,3);

Gamma0=0;

GammaTau0=0;

GammaTauTau0=0;

GammaPI0=1/PI;

GammaPIPI0=-1/PI^2;

for i=1:9

GammaR=0;

GammaPITau0=0;

GammaPIR=0;

GammaPIPIR=0;

GammaTauR=0;

GammaTauTauR=0;

GammaPITauR=0;

for i=1:43

0.5)^J(i);

0.5)^(J(i)-2);

hv=R*T*(Tau*(GammaTau0+GammaTauR));

function [hfg]=LatHeatVap(T,P)

%August 2007

PI=P/Pstar;

Tau=Tstar/T;

%Enthalpy for Vapor

Table10=[0,-.96927686500217E1;1,0.10086655968018E2;-5,-

0.56087911283020E-2;-4,0.71452738081455E-1;-3,-0.40710498223928;-

2,0.14240819171444E1;-1,-0.43839511319450E1;2,-

0.28408632460772;3,0.21268463753307E-1];

Table11=[1,0,-0.17731742473213E-2;1,1,-0.17834862292358E-1;1,2,-

0.45996013696365E-1;1,3,-0.57581259083432E-1;1,6,-0.50325278727930E-

1;2,1,-0.33032641670203E-4;2,2,-0.18948987516315E-3;2,4,-

0.39392777243355E-2;2,7,-0.43797295650573E-1;2,36,-0.26674547914087E-

4;3,0,0.20481737692309E-7;3,1,0.43870667284435E-6;3,3,-

0.32277677238570E-4;3,6,-0.15033924542148E-2;3,35,-0.40668253562649E-

1;4,1,-0.78847309559367E-9;4,2,0.12790717852285E-

7;4,3,0.48225372718507E-6;5,7,0.22922076337661E-5;6,3,-

0.16714766451061E-10;6,16,-0.21171472321355E-2;6,35,-

0.23895741934104E2;7,0,-0.59059564324270E-17;7,11,-0.12621808899101E-

5;7,25,-0.38946842435739E-1;8,8,.11256211360459E-10;8,36,-

0.82311340897998E1;9,13,0.19809712802088E-7;10,4,0.10406965210174E-

18;10,10,-0.10234747095929E-12;10,14,-0.10018179379511E-8;16,29,-

0.80882908646985E-10;16,50,0.10693031879409;18,57,-

0.33662250574171;20,20,0.89185845355421E-24;20,35,0.30629316876232E-

12;20,48,-0.42002467698208E-5;

21,21,-0.59056029685639E-25;22,53,0.37826947613457E-5;23,39,-

0.12768608934681E-14;24,26,0.73087610595061E-

28;24,40,0.55414715350778E-16;24,58,-0.94369707241210E-6];

J0=Table10(:,1);

n0=Table10(:,2);

I1=Table11(:,1);

J1=Table11(:,2);

n1=Table11(:,3);

Gamma0=0;

GammaTau0=0;

GammaTauTau0=0;

GammaPI0=1/PI;

GammaPIPI0=-1/PI^2;

for i=1:9

GammaR1=0;

GammaPITau1=0;

GammaPIR1=0;

GammaPIPIR1=0;

GammaTauR1=0;

GammaTauTauR1=0;

GammaPITauR1=0;

for i=1:43

GammaR1=GammaR1+n1(i)*PI^I1(i)*(Tau-0.5)^J1(i);

GammaPIR1=GammaPIR1+n1(i)*I1(i)*PI^(I1(i)-1)*(Tau-0.5)^J1(i);

GammaPIPIR1=GammaPIPIR1+n1(i)*I1(i)*(I1(i)-1)*PI^(I1(i)-2)*(Tau-

0.5)^J1(i);

GammaTauR1=GammaTauR1+n1(i)*PI^I1(i)*J1(i)*(Tau-0.5)^(J1(i)-1);

GammaTauTauR1=GammaTauTauR1+n1(i)*PI^I1(i)*J1(i)*(J1(i)-1)*(Tau-

0.5)^(J1(i)-2);

GammaPITauR1=GammaPITauR1+n1(i)*I1(i)*PI^(I1(i)-1)*J1(i)*(Tau-

0.5)^(J1(i)-1);

hv=R*T*(Tau*(GammaTau0+GammaTauR1))

%Enthalpy of Liquid

Table2=[0,-2,0.14632971213167;0,-1,-0.84548187169114;0,0,-

0.37563603672040E1;0,1,0.33855169168385E1;0,2,-

0.95791963387872;0,3,0.15772038513228;0,4,-0.16616417199501E-

1;0,5,0.81214629983568E-3;1,-9,0.28319080123804E-3;1,-7,-

0.60706301565874E-3;1,-1,-0.18990068218419E-1;1,0,-0.32529748770505E-

1;1,1,-0.21841717175414E-1;1,3,-0.52838357969930E-4;2,-3,-

0.47184321073267E-3;2,0,-0.30001780793026E-3;2,1,0.47661393906987E-

4;2,3,-0.44141845330846E-5;2,17,-0.72694996297594E-15;3,-4,-

0.31679644845054E-4;3,0,-0.28270797985312E-5;3,6,-0.85205128120103E-

9;4,-5,-0.22425281908000E-5;4,-2,-0.65171222895601E-6;4,10,-

0.14341729937924E-12;5,-8,-0.40516996860117E-6;8,-11,-

0.12734301741641E-8;8,-6,-0.17424871230634E-9;21,-29,-

0.68762131295531E-18;23,-31,0.14478307828521E-19;29,-

38,0.26335781662795E-22;30,-39,-0.11947622640071E-22;31,-40,-

.18228094581404E-23;32,-41,-0.93537087292458E-25];

I2=Table2(:,1);

J2=Table2(:,2);

n2=Table2(:,3);

Gamma2=0;

GammaPI2=0;

GammaPIPI2=0;

GammaTau2=0;

GammaTauTau2=0;

GammaPITau2=0;

for i=1:34

Gamma2=Gamma2+n2(i)*(7.1-PI)^I2(i)*(Tau-1.222)^J2(i);

GammaPI2=GammaPI2-n2(i)*I2(i)*(7.1-PI)^(I2(i)-1)*(Tau-1.222)^J2(i);

GammaPIPI2=GammaPIPI2+n2(i)*I2(i)*(I2(i)-1)*(7.1-PI)^(I2(i)-

2)*(Tau-1.222)^J2(i);

GammaTau2=GammaTau2+n2(i)*(7.1-PI)^I2(i)*J2(i)*(Tau-1.222)^(J2(i)-

GammaTauTau2=GammaTauTau2+n2(i)*(7.1-PI)^I2(i)*J2(i)*(J2(i)-

1)*(Tau-1.222)^(J2(i)-2);

GammaPITau2=GammaPITau2-n2(i)*I2(i)*(7.1-PI)^(I2(i)-1)*J2(i)*(Tau-

1.222)^(J2(i)-1);

hl=R*T*Tau*GammaTau2

%Latent Heat of Vaporization

hfg=hl-hv;

function [ST]=SurfTen(T)

%Surface Temperature Tension of Ordinary Water Substance

%September 1994

Tc=647.096; %Reference Temperature

B=235.8;

b=-0.625;

u=1.256;

Tau=1-(T/Tc);

ST=B*(Tau*u)*(1+b*Tau)/1000;

function [k]=ThermCond(T,d)

%Revised Release on the IAPS Formulation 1985 for the Thermal

Conductivity

%of Ordinary Water Substance

%September 2008

Tstar=647.26; %Reference Temperature, K

dstar=317.7; %Reference Density, kg/m^3

kstar=1; %Reference Thermal Conductivity, W/m-K

Tbar=T/Tstar; %Dimensionless Temperature

dbar=d/dstar; %Dimensionless Density

a=[0.0102811,0.0299621,0.0156146,-0.00422464];

b=[-0.397070,0.400302,1.060000];

B=[-0.171587,2.392190];

d=[0.0701309,0.0118520,0.00169937,-1.0200];

C=[0.642857,-4.11717,-6.17937,0.00308976,0.0822994,10.0932];

dTbar=abs(Tbar-1)+C(4);

Q=2+C(5)/dTbar^(3/5);

if Tbar>=1

S=1/dTbar;

S=C(6)/dTbar^(3/5);

for i=1:4

k0=k0+Tbar^(1/2)*a(i)*Tbar^(i-1);

k1=b(1)+b(2)*dbar+b(3)*exp(B(1)*(dbar+B(2))^2);

k2=((d(1)/Tbar^10)+d(2))*dbar^(9/5)*exp(C(1)*(1-

dbar^(14/5)))+d(3)*S*dbar^Q*exp((Q/(1+Q))*(1-

dbar^(1+Q)))+d(4)*exp(C(2)*Tbar^(3/2)+C(3)/dbar^5);

k=1000*kstar*(k0+k1+k2);

function [V]=VisLiq(T,D)

%Release on the IAPWS Formulation 2008 for the Viscosity of Ordinary

%Water Substance

%September 2007

Tstar=647.096; %Reference Temperature, K

Dstar=322.0; %Reference Density, kg/m^3

Vstar=1E-6; %Reference Viscosity, Pa-s

Tbar=T/Tstar;

Dbar=D/Dstar;

Table1=[1.67752,2.20462,0.6366564,-0.241605];

Table2=[5.20094E-1,8.50895E-2,-1.08374,-2.89555E-1,0,0;2.22531E-

1,9.99115E-1,1.88797,1.26613,0,1.20573E-1;-2.81378E-1,-9.06851E-1,-

7.72479E-1,-4.89837E-1,-2.57040E-1,0;1.61913E-1,2.57399E-1,0,0,0,0;-

3.25372E-2,0,0,6.98452E-2,0,0;0,0,0,0,8.72102E-3,0;0,0,0,-4.35673E-

3,0,-5.93264E-4];

%Solve for Vbar0

A=zeros(1,4);

for i=1:4

A(i)=Table1(i)/Tbar^(i-1);

At=sum(A);

Vbar0=100*sqrt(Tbar)/At;

%Solve for Vbar1

for i=1:6

for j=1:7

B=B+Dbar*(1/Tbar-1)^(i-1)*Table2(j,i)*(Dbar-1)^(j-1);

Vbar1=exp(B);

%Solve for Vbar2

Vbar2=1;

%Solve for Vbar

Vbar=Vbar0*Vbar1*Vbar2;

%Solve for Viscosity

V=Vbar*Vstar;

entropy generatioin study for bubble separation in pool boiling

continued support

previous work

entropy generation study

rate of bubble grow

master of science degree

degree master of science

bubble separation

bubble departure radii

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