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
In
Mechanical Engineering
By
Jeffrey William Schultz
2010
iii
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.
iv
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
15%.
v
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
vi
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
vii
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
viii
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
ix
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
x
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
xi
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]
xii
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]
xiii
∆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
xiv
Superscripts
𝑥 modified term
+ dimensionless
∗ modified term
1
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
𝜍
𝑔(𝜌𝑙 − 𝜌𝑣)
1 2
Correlation balances buoyancy force with surface tension force
(Staniszewski, 1959) 𝑑𝑑𝑒𝑝𝑡 = 0.0071𝛽
𝜍
𝑔(𝜌𝑙 − 𝜌𝑣)
1/2
1 + 0.435𝑑𝐷
𝑑𝑡
Correlation includes affect of bubble growth
rate
(Zuber, 1959) 𝑑𝑑𝑒𝑝𝑡 =
𝜍
𝑔(𝜌𝑙 − 𝜌𝑣)
1 3
6𝑘𝑙 𝑇𝑤𝑎𝑙𝑙 − 𝑇𝑠𝑎𝑡 𝑝∞
𝑞"
1/3
2
Source Departure Diameter Model Comments
(Ruckenstein, 1961) and (Zuber, 1964)
𝑑𝑑𝑒𝑝𝑡
= 3𝜋2𝜌𝑙𝛼𝑙
1 2 𝑔1 2 𝜌𝑙−𝜌𝑣 1 2
𝜍3 2
1 3
𝐽𝑎4 3 𝜍
𝑔 𝜌𝑙 − 𝜌𝑣
1 2
(Borishanskiy & Fokin, Heat transfer and hydrodynamics in steam generators,
1963)
𝑑𝑑𝑒𝑝𝑡 = −𝐶
2+
𝐶
2
2
+ 𝑅𝐹2
1 2
𝑅𝐹 = 𝑟𝑎𝑑𝑖𝑢𝑠 𝑓𝑟𝑜𝑚 𝐹𝑟𝑖𝑡𝑧 𝐶𝑜𝑟𝑟𝑒𝑙𝑎𝑡𝑖𝑜𝑛
𝐶 = 6
𝑔
𝜌𝑙𝜌𝑙−𝜌𝑣
𝜌𝑣𝜌𝑙
0.4
𝑞"
𝜌𝑣𝑓𝑔
(Cole & Shulman, 1966a) 𝑑𝑑𝑒𝑝𝑡 =
1000
𝑝
𝜍
𝑔(𝜌𝑙 − 𝜌𝑣)
1 2
𝑤𝑒𝑟𝑒 𝑝 𝑖𝑠 𝑖𝑛 𝑚𝑚𝐻𝑔
(Cole, 1967) 𝑑𝑑𝑒𝑝𝑡 = 0.04𝐽𝑎
𝜍
𝑔(𝜌𝑙 − 𝜌𝑣)
1 2
(Cole & Rohsenow, 1969) 𝑑𝑑𝑒𝑝𝑡 = 𝐶 𝐽𝑎𝑥 5 4
𝜍
𝑔(𝜌𝑙 − 𝜌𝑣)
1 2
𝐽𝑎𝑥 =𝑇𝑐𝑐𝑝 ,𝑙𝜌𝑙𝜌𝑣𝑓𝑔
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
1 3
𝛽𝑑𝑘𝑙 𝑇𝑤𝑎𝑙𝑙 − 𝑇𝑠𝑎𝑡
𝜌𝑣𝑓𝑔
2 3
𝑑∗ = 6.0𝑥10−3𝑚𝑚
𝛽𝑑 = 6.0
Correlation includes dynamic relationship
(Kutateladze & Gogonin, 1980) 𝑑𝑑𝑒𝑝𝑡 =. 25 1 + 105𝐾1
1 2 𝜍
𝑔 𝜌𝑙 − 𝜌𝑣
1 2
𝐾1 = 𝐽𝑎
𝑃𝑟𝑙
𝑔𝜌𝑙 𝜌𝑙 − 𝜌𝑣
𝜇𝑙2
𝜍
𝑔 𝜌𝑙 − 𝜌𝑣
3 2
−1
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
𝜍
𝑔(𝜌𝑙 − 𝜌𝑣)
1 2
𝜌𝑙 − 𝜌𝑣
𝜌𝑣
0.9
Expansion of Fritz
correlation to include high pressure systems
(Jensen & Memmel, 1986) 𝑑𝑑𝑒𝑝𝑡 = 0.19 1.8 + 105𝐾1
2 3 𝜍
𝑔(𝜌𝑙 − 𝜌𝑣)
1 2
Correlation is a
proposed improvement to (Kutateladze &
3
Source Departure Diameter Model Comments
Gogonin, 1980)
(Gorenflo, Knabe, & Bieling, 1986) 𝑑𝑑𝑒𝑝𝑡 = 𝐶1
𝐽𝑎4𝑘𝑙2
𝑔
1 3
1 + 1 +2𝜋
3𝐽𝑎
1 2
4 3
Correlation for high heat fluxes
(Stephan, 1992)
𝑑𝑑𝑒𝑝𝑡 = 0.25 𝜍
𝑔(𝜌𝑙 − 𝜌𝑣)
1 2
1 + 𝐽𝑎
𝑃𝑟𝑙
2 1
𝐴𝑟
1 2
Correlation valid for
5𝑥10−7 ≤ 𝐽𝑎
𝑃𝑟𝑙
2 1
𝐴𝑟≤ 0.1
(Kim & Kim, 2006) 𝑑𝑑𝑒𝑝𝑡 = 0.1649𝐽𝑎0.7
𝜍
𝑔(𝜌𝑙 − 𝜌𝑣)
1 2
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
4
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
𝑔
(
A)
𝑅
(
B)
𝑃∞
𝑭𝒑
𝑃𝑣
𝑅𝑏
(
D)
𝛽
𝑭𝝈 𝑭𝝈
𝐹𝑆𝑢𝑟𝑓 𝑇𝑒𝑛
(
C)
𝜌∞
𝑭𝒊
𝑅
𝑑𝑅
𝜇∞
𝑭𝑫
(
E)
𝑅
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)
5
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 =
𝑑
𝑑𝑡 𝑚
𝑑𝑅
𝑑𝑡 ≅
𝑑
𝑑𝑡
11
16𝜌
4𝜋𝑅3
3
𝑑𝑅
𝑑𝑡
Surface Tension Force 𝐹𝜍 = 2𝜋𝑅𝑏𝜍 sin𝛽
Drag Force 𝐹𝐷 =
𝜋
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.
6
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 −
𝑑
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
7
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
𝐸
𝑄
𝑊
d
dt 𝑚
𝑖𝑛
8
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
− d
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.
9
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.
10
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
𝑉1
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
11
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
𝑅
𝑜.
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∞ −
1
4
𝑑𝑝𝑣𝑑𝑡
𝑅 +1
20
𝑑2𝑝𝑣𝑑𝑡2
𝑅2 −1
120
𝑑3𝑝𝑣𝑑𝑡3
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
4
𝑑2𝑝𝑣𝑑𝑡2
𝑅2 −1
20
𝑑3𝑝𝑣𝑑𝑡3
𝑅3 +1
120
𝑑4𝑝𝑣𝑑𝑡4
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
12
𝑝𝑣 = 𝑝∞ +2𝜍𝑙𝑅
+ 𝜌𝑙 3
2 𝑑𝑅
𝑑𝑡
2
+ 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
0
= 4π 𝜌𝑣𝑒𝑣
R
0
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.
13
𝐸 =4𝜋𝑅3
3 𝜌𝑣𝑒𝑣 −
1
4 𝜌𝑣
𝑑𝑒𝑣𝑑𝑡
+ 𝑒𝑣𝑑𝜌𝑣𝑑𝑡
𝑅 +1
20 𝜌𝑣
𝑑2𝑒𝑣𝑑𝑡2
+ 2𝑑𝑒𝑣𝑑𝑡
𝑑𝜌𝑣𝑑𝑡
+ 𝑒𝑣𝑑2𝜌𝑣𝑑𝑡2
𝑅2
−1
120 𝜌𝑣
𝑑3𝑒𝑣𝑑𝑡3
+ 3𝑑𝜌𝑣𝑑𝑡
𝑑2𝑒𝑣𝑑𝑡2
+ 3𝑑𝑒𝑣𝑑𝑡
𝑑2𝜌𝑣𝑑𝑡2
+ 𝑒𝑣𝑑3𝜌𝑣𝑑𝑡3
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
+ 𝑒𝑣𝑑𝜌𝑣𝑑𝑡
𝑅
+1
4 𝜌𝑣
𝑑2𝑒𝑣𝑑𝑡2
+ 2𝑑𝑒𝑣𝑑𝑡
𝑑𝜌𝑣𝑑𝑡
+ 𝑒𝑣𝑑2𝜌𝑣𝑑𝑡2
𝑅2
−1
20 𝜌𝑣
𝑑3𝑒𝑣𝑑𝑡3
+ 3𝑑𝜌𝑣𝑑𝑡
𝑑2𝑒𝑣𝑑𝑡2
+ 3𝑑𝑒𝑣𝑑𝑡
𝑑2𝜌𝑣𝑑𝑡2
+ 𝑒𝑣𝑑3𝜌𝑣𝑑𝑡3
𝑅3
+1
120 𝜌𝑣
𝑑4𝑒𝑣𝑑𝑡4
+ 4𝑑𝜌𝑣𝑑𝑡
𝑑3𝑒𝑣𝑑𝑡3
+ 6𝑑2𝜌𝑣𝑑𝑡2
𝑑2𝑒𝑣𝑑𝑡2
+ 4𝑑𝑒𝑣𝑑𝑡
𝑑3𝜌𝑣𝑑𝑡3
+ 𝑒𝑣𝑑4𝜌𝑣𝑑𝑡4
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
14
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.
𝑑
𝑑𝑡 𝑚 𝑖𝑛
R
0
By performing successive integration by parts and taking the derivative of
the resulting series, the following equation for energy transfer rate is derived.
𝑑
𝑑𝑡 𝑚𝑖𝑛 =
4𝜋𝑅2
3 3𝜌𝑣𝑣𝑅
+ 𝑅 − 1 − 𝜌𝑣𝑑𝑣𝑑𝑡
+ 𝑣𝑑𝜌𝑣𝑑𝑡
𝑅
+1
4 𝜌𝑣
𝑑2𝑣𝑑𝑡2
+ 2𝑑𝑣𝑑𝑡
𝑑𝜌𝑣𝑑𝑡
+ 𝑣𝑑2𝜌𝑣𝑑𝑡2
𝑅2
−1
20 𝜌𝑣
𝑑3𝑣𝑑𝑡3
+ 3𝑑𝜌𝑣𝑑𝑡
𝑑2𝑣𝑑𝑡2
+ 3𝑑𝑣𝑑𝑡
𝑑2𝜌𝑣𝑑𝑡2
+ 𝑣𝑑3𝜌𝑣𝑑𝑡3
𝑅3
+1
120 𝜌𝑣
𝑑4𝑣𝑑𝑡4
+ 4𝑑𝜌𝑣𝑑𝑡
𝑑3𝑣𝑑𝑡3
+ 6𝑑2𝜌𝑣𝑑𝑡2
𝑑2𝑣𝑑𝑡2
+ 4𝑑𝑣𝑑𝑡
𝑑3𝜌𝑣𝑑𝑡3
+ 𝑣𝑑4𝜌𝑣𝑑𝑡4
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.
𝑑
𝑑𝑡 𝑚𝑖𝑛 = 4𝜋𝑣𝜌𝑣𝑅
2𝑅
Heat Transfer
15
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 −𝑑𝑝𝑣𝑑𝑡
𝑅 +1
4
𝑑2𝑝𝑣𝑑𝑡2
𝑅2 −1
20
𝑑3𝑝𝑣𝑑𝑡3
𝑅3 +1
120
𝑑4𝑝𝑣𝑑𝑡4
𝑅4
+ −𝑑𝜌𝑣𝑑𝑡
𝑅 +1
4
𝑑2𝜌𝑣𝑑𝑡2
𝑅2 −1
20
𝑑3𝜌𝑣𝑑𝑡3
𝑅3 +1
120
𝑑4𝜌𝑣𝑑𝑡4
𝑅4 𝑒𝑣 − 𝑣
+ −𝜌𝑣𝑅 +1
2
𝑑𝜌𝑣𝑑𝑡
𝑅2 −3
20
𝑑2𝜌𝑣𝑑𝑡2
𝑅3 +1
30
𝑑3𝜌𝑣𝑑𝑡3
𝑅4 𝑑𝑒𝑣𝑑𝑡
−𝑑𝑣𝑑𝑡
+ 1
4𝜌𝑣𝑅
2 −3
20
𝑑𝜌𝑣𝑑𝑡
𝑅3 +1
20
𝑑2𝜌𝑣𝑑𝑡2
𝑅4 𝑑2𝑒𝑣𝑑𝑡2
−𝑑2𝑣𝑑𝑡2
+ −1
20𝜌𝑣𝑅
3 +1
30
𝑑𝜌𝑣𝑑𝑡
𝑅4 𝑑3𝑒𝑣𝑑𝑡3
−𝑑3𝑣𝑑𝑡3
+ 1
120𝜌𝑣𝑅
4 𝑑4𝑒𝑣𝑑𝑡4
−𝑑4𝑣𝑑𝑡4
+ ⋯
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
16
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 =
𝑑
𝑑𝑡 𝑠𝑣𝑚𝑣
R
0
=𝑑
𝑑𝑡 4π 𝜌𝑣𝑠𝑣𝑅
2𝑑𝑅R
0
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 −
1
4
𝑑 𝜌𝑣𝑠𝑣
𝑑𝑡𝑅 +
1
20
𝑑2 𝜌𝑣𝑠𝑣
𝑑𝑡2𝑅2 −
1
120
𝑑3 𝜌𝑣𝑠𝑣
𝑑𝑡3𝑅3 + ⋯
Taking the derivative with respect to time of the total accumulated entropy
leads to the following equation.
17
𝑆 =4𝜋
3𝑅2 3𝜌𝑣𝑠𝑣𝑅
+ 𝑅 − 1 −𝑑 𝜌𝑣𝑠𝑣
𝑑𝑡𝑅 +
𝑑2 𝜌𝑣𝑠𝑣
𝑑𝑡2𝑅2 −
1
20
𝑑3 𝜌𝑣𝑠𝑣
𝑑𝑡3𝑅3
−1
120
𝑑4 𝜌𝑣𝑠𝑣
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.
𝑄
𝑇𝑤𝑎𝑙𝑙= −
4𝜋
𝑇𝑤𝑎𝑙𝑙𝑝∞𝑅
2𝑅
Net Entropy Flow Rate
The net entropy flow rate is defined as follows.
𝑑
𝑑𝑡 𝑆𝑣
𝑖𝑛
=𝑑
𝑑𝑡 4π 𝜌𝑣𝑠𝑣
R
0
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.
18
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
𝑇𝑤𝑎𝑙𝑙𝑝∞𝑅
2𝑅
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
4𝜋
𝑇𝑤𝑝∞𝑅
2𝑅 =4𝜋
𝑇𝑤𝑝∞
𝑑
𝑑𝑡 𝑅2𝑅
𝑑𝑡
𝑑𝑅
This reduces to the following equation.
0 =4𝜋
𝑇𝑤𝑎𝑙𝑙𝑝∞𝑅 2𝑅 + 𝑅
𝑅
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
form.
19
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
20
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
21
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
form.
𝑅𝑑2𝑅
𝑑𝑡2+
3
2 𝑑𝑅
𝑑𝑡
2
=1
𝜌𝑙 𝑝𝑣 − 𝑝∞ −
2𝜍
𝑅
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+
3
2 𝑑𝑅
𝑑𝑡
2
=𝑓𝑔𝜌𝑣
𝜌𝑙 𝑇∞ − 𝑇𝑠𝑎𝑡
𝑇𝑠𝑎𝑡
Integration of the above equation leads to the Rayleigh equation for
bubble growth
𝑅 = 2
3
𝑓𝑔𝜌𝑣𝜌𝑙
𝑇∞ − 𝑇𝑠𝑎𝑡
𝑇𝑠𝑎𝑡
1 2
𝑡
This equation is commonly written as follows.
𝑅 = 𝐴𝑡
𝑤𝑒𝑟𝑒
𝐴 = 𝑏𝑓𝑔𝜌𝑣𝜌𝑙
𝑇∞ − 𝑇𝑠𝑎𝑡
𝑇𝑠𝑎𝑡
1 2
, 𝑏 =2
3
22
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
wall.
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
23
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
𝑊𝑒𝑟𝑒 𝑡𝑒 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
𝑡=𝑡𝑑𝑒𝑝𝑡
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
2
24
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
𝑛
𝑛
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.
25
𝐷 = 𝑅𝑒𝑥𝑝 ,𝑖
3 + 3𝑒−𝐶
𝐴𝑖 𝑅𝑒𝑥𝑝 ,𝑖
𝑛
𝑛
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
Experimental Departure Radius of (Van Stralen, Cole, Sluyter, &
Sohal, 1975)
𝑅𝑒𝑥𝑝 , m
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
1 3
At departure, this equation will be equivalent to the Rayleigh equation.
Setting the general solution equal to the Rayleigh equation produces a function
26
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.
0 = 𝑅𝑑𝑒𝑝𝑡3 +
1
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
Experimental Departure Radius of (Van Stralen, Cole, Sluyter, &
Sohal, 1975)
𝑅𝑒𝑥𝑝 , m
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
27
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.
28
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
29
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.
0 = 𝑅𝑑𝑒𝑝𝑡3 +
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.
30
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)
𝑅𝑒𝑥𝑝 , m
Predicted Departure
Radius
𝑅𝑑𝑒𝑝𝑡 , m
% 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.
31
Figure 6. Comparison of Predicted Departure Radii from Modified Rayleigh Based Equation with Experimental Departure Radii.
32
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
33
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
𝜋
1 2
𝑡1 2
The equation is commonly written as follows.
𝑅 = 𝐵𝑡1 2
𝑤𝑒𝑟𝑒
𝐵 = 𝐽𝑎 12𝛼𝑙
𝜋
1 2
𝐽𝑎 =𝜌𝑙𝑐𝑝 ,𝑙
𝜌𝑣𝑓𝑔 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
𝜋
1 2
𝐽𝑎∗ =𝜌𝑙𝑐𝑝 ,𝑙
𝜌𝑣𝑓𝑔 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=
1
2𝐵𝑡−1 2
34
𝑑2𝑅
𝑑𝑡2= −
1
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.
1
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).
35
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.
36
𝑅 = −3𝑒−𝐶𝑡 + 𝐷 1 3
𝑤𝑒𝑟𝑒 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
𝑡=𝑡𝑑𝑒𝑝𝑡
= − 𝑑𝑅
𝑑𝑡 𝐺𝑒𝑛 𝑆𝑜𝑙𝑢𝑡𝑖𝑜𝑛 −1
𝑡=𝑡𝑑𝑒𝑝𝑡
By substation of the appropriate equations into the boundary conditions
defined above, a system of equations is created. This system of equations is
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
3
𝐵2
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.
37
𝐶 =
−𝑙𝑛 2𝑅𝑑𝑒𝑝𝑡 ,𝑖
3
𝐵𝑖2
𝑛
𝑛
𝑖=1
Experimental data published by (Van Stralen, Cole, Sluyter, & Sohal,
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
Experimental Departure Radius of (Van Stralen, Cole, Sluyter, &
Sohal, 1975)
𝑅𝑒𝑥𝑝 , m
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
2
𝑛
𝑛
𝑖=1
38
Experimental data of (Van Stralen, Cole, Sluyter, & Sohal, 1975) is again
utilized to evaluate this equation. Results of this evaluation are shown in Table
8.
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
Experimental Departure Radius of (Van Stralen, Cole, Sluyter, &
Sohal, 1975)
𝑅𝑒𝑥𝑝 , m
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
Average 3.149594E-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.
0 = 𝑅𝑑𝑒𝑝𝑡3 +
3𝑒−𝐶
𝐵2 𝑅𝑑𝑒𝑝𝑡
2 − 𝐷
Substitution of the constants 𝐶 and 𝐷 results in the following equation.
39
0 = 𝑅𝑑𝑒𝑝𝑡3 +
1
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
Experimental Departure Radius of (Van Stralen, Cole, Sluyter, &
Sohal, 1975)
𝑅𝑒𝑥𝑝 , m
Predicted Departure Radius
𝑅𝑑𝑒𝑝𝑡 , m
% 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.
40
Figure 9. Comparison of Predicted Departure Radii from Plesset-Zwick Based Equation with Experimental Departure Radii.
41
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
42
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.
0 = 𝑅𝑑𝑒𝑝𝑡3 +
3𝑒 0.894132 𝑙𝑛 𝐵 −4.010944
𝐵2 𝑅𝑑𝑒𝑝𝑡
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
10.
43
Table 10. Error Analysis of Predicted Departure Radii based on Plesset-Zwick Based Modified General Solution.
Bubble Number Ja
Experimental Departure Radius of (Van Stralen, Cole, Sluyter, &
Sohal, 1975)
𝑅𝑒𝑥𝑝 , m
Predicted Departure Radius
𝑅𝑑𝑒𝑝𝑡 , m
% 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.
44
Figure 11. Comparison of Predicted Departure Radii from Modified Plesset-Zwick Based Equation with Experimental Departure Radii.
45
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.
46
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+
𝑡+ + 𝑡𝑤+
1 2
1 2
− 𝑡+ 1 2
𝑤𝑒𝑟𝑒
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
47
Changing the equation back to its dimensional form produces the following
equation.
𝑅 =2𝐵2
3𝐴
𝐴2
𝐵2𝑡 + 1
3 2
− 𝐴2
𝐵2𝑡
3 2
− 1
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
𝐴2
𝐵2𝑡 + 1
1 2
− 𝐴2
𝐵2𝑡
1 2
𝑑2𝑅
𝑑𝑡2=
𝐴3
2𝐵2
𝐴2
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
3𝐴
𝐴2
𝐵2𝑡𝑑𝑒𝑝𝑡 + 1
3 2
− 𝐴2
𝐵2𝑡𝑑𝑒𝑝𝑡
3 2
− 1 𝐴3
2𝐵2
𝐴2
𝐵2𝑡𝑑𝑒𝑝𝑡 + 1
−1 2
− 𝐴2
𝐵2𝑡𝑑𝑒𝑝𝑡
−1 2
+ 2 𝐴 𝐴2
𝐵2𝑡𝑑𝑒𝑝𝑡 + 1
1 2
− 𝐴2
𝐵2𝑡𝑑𝑒𝑝𝑡
1 2
2
= 0
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)
48
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
49
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
𝑑𝑡 𝐺𝑒𝑛 𝑆𝑜𝑙𝑢𝑡𝑖𝑜𝑛 −1
𝑡=𝑡𝑑𝑒𝑝𝑡
By substitution of the appropriate equations into the boundary conditions
defined above, a system of equations is created. This system of equations is
utilized to solve for constant 𝐶.
𝐶 = −𝑙𝑛
4𝐵4
9𝐴3
𝐴2
𝐵2 𝑡𝑑𝑒𝑝𝑡 + 1 3 2
− 𝐴2
𝐵2 𝑡𝑑𝑒𝑝𝑡 3 2
− 1
𝐴2
𝐵2 𝑡𝑑𝑒𝑝𝑡 + 1 1 2
− 𝐴2
𝐵2 𝑡𝑑𝑒𝑝𝑡 1 2
2
If the radial velocity is known at the point of departure, the constant 𝐶 may
be more easily solve using the following form.
50
𝐶 = −𝑙𝑛 𝑅𝑑𝑒𝑝𝑡
2
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
4
9𝐴𝑖3
𝐴𝑖
2
𝐵𝑖2 𝑡𝑑𝑒𝑝𝑡 ,𝑖 + 1
3 2
− 𝐴𝑖
2
𝐵𝑖2 𝑡𝑑𝑒𝑝𝑡 ,𝑖
3 2
− 1
𝐴𝑖
2
𝐵𝑖2 𝑡𝑑𝑒𝑝𝑡 ,𝑖 + 1
1 2
− 𝐴𝑖
2
𝐵𝑖2 𝑡𝑑𝑒𝑝𝑡 ,𝑖
1 2
2
𝑛
𝑛
𝑖=1
Experimental data published by (Van Stralen, Cole, Sluyter, & Sohal,
1975) is utilized to solve for this constant. Results of this analysis are shown in
Table 11.
51
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,
Sluyter, & Sohal, 1975)
𝑅𝑒𝑥𝑝 , m
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
2
3𝐴𝑖
3
𝐴𝑖
2
𝐵𝑖2 𝑡𝑑𝑒𝑝𝑡 ,𝑖 + 1
3 2
− 𝐴𝑖
2
𝐵𝑖2 𝑡𝑑𝑒𝑝𝑡
3 2
− 1
3
+ 3𝑒−𝐶𝑡𝑑𝑒𝑝𝑡 ,𝑖
𝑛
𝑛
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.
52
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
Experimental Departure
Radius of (Van Stralen, Cole,
Sluyter, & Sohal, 1975)
𝑅𝑒𝑥𝑝 , m
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
Average 3.912008E-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 −
𝐴2
𝐵23𝑒−𝐶𝑅𝑑𝑒𝑝𝑡
3 +𝐴2𝐷
𝐵23𝑒−𝐶+ 1
3 2
− −𝐴2
𝐵23𝑒−𝐶𝑅𝑑𝑒𝑝𝑡
3 +𝐴2𝐷
𝐵23𝑒−𝐶
3 2
− 1 −𝑅𝑑𝑒𝑝𝑡
Substitution of the constants 𝐶 and 𝐷 produces the following equation.
53
0 =2𝐵2
3𝐴 −
𝐴2
𝐵23𝑒−6.666687𝑅𝑑𝑒𝑝𝑡
3 +1.000685E − 03𝐴2
𝐵23𝑒−6.666687+ 1
3 2
− −𝐴2
𝐵23𝑒−6.666687𝑅𝑑𝑒𝑝𝑡
3 +1.000685E − 03𝐴2
𝐵23𝑒−6.666687
3 2
− 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
Experimental Departure Radius of (Van Stralen, Cole, Sluyter, &
Sohal, 1975)
𝑅𝑒𝑥𝑝 , m
Predicted Departure Radius
𝑅𝑑𝑒𝑝𝑡 , m
% 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.
54
Figure 14. Comparison of Predicted Departure Radii from MRG Based Equation with Experimental Departure Radii.
55
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
56
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
𝐵2
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
𝐴2
+ 1.124843𝐸 −07 𝐵2
𝐴+ 7.128086𝐸 −04
1 3
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.
57
0
=2𝐵2
3𝐴
−𝐴2
𝐵23𝑒− 5.814845𝐸 −02 𝐴2+8.891619𝐸(−01)𝐴+3.399097 𝑅𝑑𝑒𝑝𝑡
3
+ −1.957951𝐸 −12
𝐵4
𝐴2 + 1.124843𝐸 −07 𝐵2
𝐴+ 7.128086𝐸 −04 𝐴2
𝐵23𝑒− 5.814845𝐸 −02 𝐴2+8.891619𝐸(−01)𝐴+3.399097 + 1
3 2
− −𝐴2
𝐵23𝑒− 5.814845𝐸 −02 𝐴2+8.891619𝐸(−01)𝐴+3.399097 𝑅𝑑𝑒𝑝𝑡
3
+ −1.957951𝐸 −12
𝐵4
𝐴2 + 1.124843𝐸 −07 𝐵2
𝐴+ 7.128086𝐸 −04 𝐴2
𝐵23𝑒− 5.814845𝐸 −02 𝐴2+8.891619𝐸(−01)𝐴+3.399097
3 2
− 1
−𝑅𝑑𝑒𝑝𝑡
The predicted departure radii determined from the equation above are
presented in Table 14.
58
Table 14. Error Analysis of Predicted Departure Radii based on MRG Based Modified General Solution.
Bubble Number
Ja 𝐶 𝐷
Experimental Departure
Radius of (Van Stralen, Cole,
Sluyter, & Sohal, 1975)
𝑅𝑒𝑥𝑝 , m
Predicted Departure
Radius
𝑅𝑑𝑒𝑝𝑡 , m
% 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.
59
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.
60
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.
0 = 𝑅𝑑𝑒𝑝𝑡3 +
3𝑒 0.894132 𝑙𝑛 𝐵 −4.010944
𝐵2 𝑅𝑑𝑒𝑝𝑡
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.
61
Table 15. Comparison of Derived Equation with Experimental Data of (Cole & Shulman, 1966b)
Bubble Jakob Number
Experimental Departure Radius
of (Cole & Shulman, 1966b)
𝑅𝑒𝑥𝑝 , m
Predicted Departure Radius
𝑅𝑑𝑒𝑝𝑡 , m
% 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.
62
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.
63
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.
64
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
65
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.
𝑆 𝑔𝑒𝑛 =4𝜋
𝑇𝑤𝑎𝑙𝑙𝑝∞𝑅
2𝑅
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.
0 = 𝑅𝑑𝑒𝑝𝑡3 +
3𝑒 0.894132 𝑙𝑛 𝐵 −4.010944
𝐵2 𝑅𝑑𝑒𝑝𝑡
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
66
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.
67
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
68
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
𝑜
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
2
𝑑𝐹𝑛𝑒𝑡𝑑𝑡
𝑅 +1
6
𝑑2𝐹𝑛𝑒𝑡𝑑𝑡2
𝑅2 −1
24
𝑑3𝐹𝑛𝑒𝑡𝑑𝑡3
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
2
𝑑2𝐹𝑛𝑒𝑡𝑑𝑡2
𝑅2 −1
6
𝑑3𝐹𝑛𝑒𝑡𝑑𝑡3
𝑅3 +1
24
𝑑4𝐹𝑛𝑒𝑡𝑑𝑡4
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.
69
𝐹𝑛𝑒𝑡 =4𝜋𝑅3
3 𝜌𝑙 − 𝜌𝑣 𝑔 + 𝜋𝑅𝑏𝜍 sin𝛽 −
𝑑
𝑑𝑡
11
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
𝑔 𝜌𝑙 − 𝜌𝑣
1 2
Figure 20. Bubble Dimensions.
𝑅
𝑅𝑏
𝛽
70
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.
4𝜋
3𝑅𝑑𝑒𝑝𝑡
3𝑔(𝜌𝑙 − 𝜌𝑣) = 2𝜋𝑅𝑑𝑒𝑝𝑡 𝜍𝑙 0.0208𝛽 2
6
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
6
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
6
(Cole, 1967) 𝑆𝜍 =
1
6 0.04𝐽𝑎 2
(Cole & Rohsenow, 1969) 𝑆𝜍 =
𝐶2 𝐽𝑎𝑥 5 2
6
𝐽𝑎x =𝑇𝑐𝑐𝑝 ,𝑙𝜌𝑙𝜌𝑣𝑙𝑣
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.
71
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
3 3𝜌𝑣𝑒𝑣𝑅
+ 𝑅 − 1 − 𝜌𝑣𝑑𝑒𝑣𝑑𝑡
+ 𝑒𝑣𝑑𝜌𝑣𝑑𝑡
𝑅
+1
4 𝜌𝑣
𝑑2𝑒𝑣𝑑𝑡2
+ 2𝑑𝑒𝑣𝑑𝑡
𝑑𝜌𝑣𝑑𝑡
+ 𝑒𝑣𝑑2𝜌𝑣𝑑𝑡2
𝑅2
−1
20 𝜌𝑣
𝑑3𝑒𝑣𝑑𝑡3
+ 3𝑑𝜌𝑣𝑑𝑡
𝑑2𝑒𝑣𝑑𝑡2
+ 3𝑑𝑒𝑣𝑑𝑡
𝑑2𝜌𝑣𝑑𝑡2
+ 𝑒𝑣𝑑3𝜌𝑣𝑑𝑡3
𝑅3
+1
120 𝜌𝑣
𝑑4𝑒𝑣𝑑𝑡4
+ 4𝑑𝜌𝑣𝑑𝑡
𝑑3𝑒𝑣𝑑𝑡3
+ 6𝑑2𝜌𝑣𝑑𝑡2
𝑑2𝑒𝑣𝑑𝑡2
+ 4𝑑𝑒𝑣𝑑𝑡
𝑑3𝜌𝑣𝑑𝑡3
+ 𝑒𝑣𝑑4𝜌𝑣𝑑𝑡4
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.
72
𝑑
𝑑𝑡 𝑚𝑖𝑛 =
4𝜋𝑅2
3 3𝜌𝑣𝑣𝑅
+ 𝑅 − 1 − 𝜌𝑣𝑑𝑣𝑑𝑡
+ 𝑣𝑑𝜌𝑣𝑑𝑡
𝑅
+1
4 𝜌𝑣
𝑑2𝑣𝑑𝑡2
+ 2𝑑𝑣𝑑𝑡
𝑑𝜌𝑣𝑑𝑡
+ 𝑣𝑑2𝜌𝑣𝑑𝑡2
𝑅2
−1
20 𝜌𝑣
𝑑3𝑣𝑑𝑡3
+ 3𝑑𝜌𝑣𝑑𝑡
𝑑2𝑣𝑑𝑡2
+ 3𝑑𝑣𝑑𝑡
𝑑2𝜌𝑣𝑑𝑡2
+ 𝑣𝑑3𝜌𝑣𝑑𝑡3
𝑅3
+1
120 𝜌𝑣
𝑑4𝑣𝑑𝑡4
+ 4𝑑𝜌𝑣𝑑𝑡
𝑑3𝑣𝑑𝑡3
+ 6𝑑2𝜌𝑣𝑑𝑡2
𝑑2𝑣𝑑𝑡2
+ 4𝑑𝑣𝑑𝑡
𝑑3𝜌𝑣𝑑𝑡3
+ 𝑣𝑑4𝜌𝑣𝑑𝑡4
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.
𝑑
𝑑𝑡 𝑚𝑖𝑛 = 4𝜋𝑣𝜌𝑣𝑅
2𝑅
Heat Transfer
By substitution of the equations derived above into the first law of
thermodynamics, the heat transfer rate for the bubble is defined.
73
𝑄 = 𝐹𝑛𝑒𝑡𝑅 + 𝑅 − 1 −𝑑𝐹𝑛𝑒𝑡𝑑𝑡
𝑅 +1
2
𝑑2𝐹𝑛𝑒𝑡𝑑𝑡2
𝑅2 −1
6
𝑑3𝐹𝑛𝑒𝑡𝑑𝑡3
𝑅3 +1
24
𝑑4𝐹𝑛𝑒𝑡𝑑𝑡4
𝑅4 + ⋯
+4𝜋𝑅2
3 3𝜌𝑣 𝑒𝑣 − 𝑣 𝑅
+ 𝑅
− 1 −𝑑𝜌𝑣𝑑𝑡
𝑅 +1
4
𝑑2𝜌𝑣𝑑𝑡2
𝑅2 −1
20
𝑑3𝜌𝑣𝑑𝑡3
𝑅3 +1
120
𝑑4𝜌𝑣𝑑𝑡4
𝑅4 𝑒𝑣 − 𝑣
+ −𝜌𝑣𝑅 +1
2
𝑑𝜌𝑣𝑑𝑡
𝑅2 −3
20
𝑑2𝜌𝑣𝑑𝑡2
𝑅3 +1
30
𝑑3𝜌𝑣𝑑𝑡3
𝑅4 𝑑𝑒𝑣𝑑𝑡
−𝑑𝑣𝑑𝑡
+ 1
4𝜌𝑣𝑅
2 −3
20
𝑑𝜌𝑣𝑑𝑡
𝑅3 +1
20
𝑑2𝜌𝑣𝑑𝑡2
𝑅4 𝑑2𝑒𝑣𝑑𝑡2
−𝑑2𝑣𝑑𝑡2
+ −1
20𝜌𝑣𝑅
3 +1
30
𝑑𝜌𝑣𝑑𝑡
𝑅4 𝑑3𝑒𝑣𝑑𝑡3
−𝑑3𝑣𝑑𝑡3
+ 1
120𝜌𝑣𝑅
4 𝑑4𝑒𝑣𝑑𝑡4
−𝑑4𝑣𝑑𝑡4
This equation is further reduced application of the definition of enthalpy.
𝑒𝑣 − 𝑣 = −𝑝𝑣𝑣𝑣 = −𝑝𝑣𝜌𝑣
Substitution of this relationship results in the following equation.
𝑄 = 𝐹𝑛𝑒𝑡𝑅 + 𝑅 − 1 −𝑑𝐹𝑛𝑒𝑡𝑑𝑡
𝑅 +1
2
𝑑2𝐹𝑛𝑒𝑡𝑑𝑡2
𝑅2 −1
6
𝑑3𝐹𝑛𝑒𝑡𝑑𝑡3
𝑅3 +1
24
𝑑4𝐹𝑛𝑒𝑡𝑑𝑡4
𝑅4 + ⋯
+4𝜋𝑅2
3 −3𝑝𝑣𝑅
+ 𝑅 − 1 𝑅𝑑𝑝𝑣𝑑𝑡
−1
4𝑅2
𝑑2𝑝𝑣𝑑𝑡2
+1
20𝑅3
𝑑3𝑝𝑣𝑑𝑡3
−1
120𝑅4
𝑑4𝑝𝑣𝑑𝑡4
+ ⋯
74
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
3𝑅2 3𝜌𝑣𝑠𝑣𝑅
+ 𝑅 − 1 −𝑑 𝜌𝑣𝑠𝑣
𝑑𝑡𝑅 +
𝑑2 𝜌𝑣𝑠𝑣
𝑑𝑡2𝑅2 −
1
20
𝑑3 𝜌𝑣𝑠𝑣
𝑑𝑡3𝑅3
−1
120
𝑑4 𝜌𝑣𝑠𝑣
𝑑𝑡4𝑅4 + ⋯
75
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.
𝑄
𝑇𝑤𝑎𝑙𝑙=
𝐹𝑛𝑒𝑡𝑇𝑤𝑎𝑙𝑙
𝑅
+ 𝑅 − 1
𝑇𝑤𝑎𝑙𝑙 −
𝑑𝐹𝑛𝑒𝑡𝑑𝑡
𝑅 +1
2
𝑑2𝐹𝑛𝑒𝑡𝑑𝑡2
𝑅2 −1
6
𝑑3𝐹𝑛𝑒𝑡𝑑𝑡3
𝑅3 +1
24
𝑑4𝐹𝑛𝑒𝑡𝑑𝑡4
𝑅4 + ⋯
+4𝜋𝑅2
3𝑇𝑤𝑎𝑙𝑙 −3𝑝𝑣𝑅
+ 𝑅 − 1 𝑅𝑑𝑝𝑣𝑑𝑡
−1
4𝑅2
𝑑2𝑝𝑣𝑑𝑡2
+1
20𝑅3
𝑑3𝑝𝑣𝑑𝑡3
−1
120𝑅4
𝑑4𝑝𝑣𝑑𝑡4
+ ⋯
If the vapor pressure within the bubble is assumed constant, the equation
can be reduced to the following.
𝑄
𝑇𝑤𝑎𝑙𝑙=
𝐹𝑛𝑒𝑡𝑇𝑤𝑎𝑙𝑙
𝑅
+ 𝑅 − 1
𝑇𝑤𝑎𝑙𝑙 −
𝑑𝐹𝑛𝑒𝑡𝑑𝑡
𝑅 +1
2
𝑑2𝐹𝑛𝑒𝑡𝑑𝑡2
𝑅2 −1
6
𝑑3𝐹𝑛𝑒𝑡𝑑𝑡3
𝑅3 +1
24
𝑑4𝐹𝑛𝑒𝑡𝑑𝑡4
𝑅4 + ⋯
−4𝜋
𝑇𝑤𝑎𝑙𝑙𝑝𝑠𝑎𝑡 𝑇∞ 𝑅
2𝑅
If the affects of the net force derivatives are neglected, the rate of heat
transfer is further simplified.
76
𝑄
𝑇𝑤𝑎𝑙𝑙=
𝐹𝑛𝑒𝑡𝑇𝑤𝑎𝑙𝑙
𝑅 −4𝜋
𝑇𝑤𝑎𝑙𝑙𝑝𝑠𝑎𝑡 𝑇∞ 𝑅
2𝑅
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
− 𝑅 − 1
𝑇𝑤𝑎𝑙𝑙 −
𝑑𝐹𝑛𝑒𝑡𝑑𝑡
𝑅 +1
2
𝑑2𝐹𝑛𝑒𝑡𝑑𝑡2
𝑅2 −1
6
𝑑3𝐹𝑛𝑒𝑡𝑑𝑡3
𝑅3 +1
24
𝑑4𝐹𝑛𝑒𝑡𝑑𝑡4
𝑅4 + ⋯
−4𝜋𝑅2
3𝑇𝑤𝑎𝑙𝑙 −3𝑝𝑣𝑅
+ 𝑅 − 1 𝑅𝑑𝑝𝑣𝑑𝑡
−1
4𝑅2
𝑑2𝑝𝑣𝑑𝑡2
+1
20𝑅3
𝑑3𝑝𝑣𝑑𝑡3
−1
120𝑅4
𝑑4𝑝𝑣𝑑𝑡4
+ ⋯
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.
77
𝑆 𝑔𝑒𝑛 = −𝐹𝑛𝑒𝑡𝑇𝑤𝑎𝑙𝑙
𝑅 +4𝜋
𝑇𝑤𝑎𝑙𝑙𝑝𝑠𝑎𝑡 𝑇∞ 𝑅
2𝑅
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
𝑑2𝐹𝑛𝑒𝑡𝑑𝑡2
+ 1
6𝑅3𝑅 +
1
2𝑅2𝑅 2 +
1
2𝑅2 − 𝑅2𝑅
𝑑3𝐹𝑛𝑒𝑡𝑑𝑡3
+ 1
3𝑅3𝑅 −
1
24𝑅4𝑅 −
1
6𝑅3𝑅 2 −
1
6𝑅3
𝑑4𝐹𝑛𝑒𝑡𝑑𝑡4
+ 1
24𝑅4 −
1
24𝑅4𝑅
𝑑5𝐹𝑛𝑒𝑡𝑑𝑡5
−4𝜋
3 −6𝑅𝑅 2 − 3𝑅2𝑅 𝑝𝑣 + −6𝑅2𝑅 + 3𝑅2𝑅 2 + 𝑅3𝑅
𝑑𝑝𝑣𝑑𝑡
+ −𝑅3 + 2𝑅3𝑅 − 𝑅3𝑅 2 −1
4𝑅4𝑅
𝑑2𝑝𝑣𝑑𝑡2
+ +1
4𝑅4 −
1
2𝑅4𝑅 +
2
5𝑅4𝑅 2 +
1
20𝑅5𝑅
𝑑3𝑝𝑣𝑑𝑡3
+ −1
20𝑅5 +
3
20𝑅5𝑅 −
1
20𝑅5𝑅 2 −
1
120𝑅6𝑅
𝑑4𝑝𝑣𝑑𝑡4
+ 1
120𝑅6 −
1
120𝑅6𝑅
𝑑5𝑝𝑣𝑑𝑡5
+ ⋯
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∞
78
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 =
4𝜋
3𝑔 𝜌𝑙 − 𝜌𝑣 𝑅
3 − 𝜋𝜍𝑙𝑆𝜍𝑅 −11𝜋
12𝜌𝑙 3𝑅2𝑅 2 + 𝑅3𝑅 −
𝜋
4𝑎𝜇𝑙𝑅𝑅
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
79
Order of Derivative
Equation
3 𝑑3𝐹𝑛𝑒𝑡𝑑𝑡3
=4𝜋
3𝑔 6𝑅 3 + 18𝑅𝑅 𝑅 + 3𝑅2𝑅 𝜌𝑙 − 𝜌𝑣 − 18𝑅𝑅 2 + 9𝑅2𝑅
𝑑𝜌𝑣𝑑𝑡
− 9𝑅2𝑅 𝑑2𝜌𝑣𝑑𝑡2
− 𝑅3 𝑑3𝜌𝑣𝑑𝑡3
− 𝜋𝜍𝑙𝑆𝜍 𝑅
−11𝜋
12𝜌𝑙 60𝑅 3𝑅 + 90𝑅𝑅 𝑅 2 + 60𝑅𝑅 2𝑅 + 30𝑅2𝑅 𝑅 + 15𝑅2𝑅
𝑑4𝑅
𝑑𝑡4
+ 𝑅3𝑑5𝑅
𝑑𝑡5 −
𝜋
4𝑎𝜇𝑙 3𝑅
2 + 4𝑅 𝑅 + 𝑅𝑑4𝑅
𝑑𝑡4
4 𝑑4𝐹𝑛𝑒𝑡𝑑𝑡4
=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
− 𝑅3 𝑑4𝜌𝑣𝑑𝑡4
− 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
5 𝑑5𝐹𝑛𝑒𝑡𝑑𝑡5
=4𝜋
3𝑔 90𝑅 𝑅 2 + 60𝑅 2𝑅 + 60𝑅𝑅 𝑅 + 30𝑅𝑅
𝑑4𝑅
𝑑𝑡4+ 3𝑅2
𝑑5𝑅
𝑑𝑡5 𝜌𝑙 − 𝜌𝑣
− 180𝑅 2𝑅 + 90𝑅𝑅 2 + 120𝑅𝑅 𝑅 + 15𝑅2𝑑4𝑅
𝑑𝑡4 𝑑𝜌𝑣𝑑𝑡
− 60𝑅 3 + 180𝑅𝑅 𝑅 + 30𝑅2𝑅 𝑑2𝜌𝑣𝑑𝑡2
− 60𝑅 2 + 30𝑅2𝑅 𝑑3𝜌𝑣𝑑𝑡3
− 15𝑅2𝑅 𝑑4𝜌𝑣𝑑𝑡4
− 𝑅3 𝑑5𝜌𝑣𝑑𝑡5
− 𝜋𝜍𝑙𝑆𝜍 𝑑5𝑅
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
𝑑𝑡4+ 63𝑅2𝑅
𝑑5𝑅
𝑑𝑡5 + 126𝑅𝑅 2
𝑑5𝑅
𝑑𝑡5
+ 21𝑅2𝑅 𝑑6𝑅
𝑑𝑡6+ 𝑅3
𝑑7𝑅
𝑑𝑡7 −
𝜋
4𝑎𝜇𝑙 10𝑅 3 + 15𝑅
𝑑4𝑅
𝑑𝑡4+ 6𝑅
𝑑5𝑅
𝑑𝑡5+ 𝑅
𝑑6𝑅
𝑑𝑡5
80
Table 19. Vapor Pressure Derivatives
Order of Derivative
Equation
0 𝑝𝑣 = 𝑝∞ +
2𝜍𝑙𝑅
+ 𝜌𝑙 3
2 𝑑𝑅
𝑑𝑡
2
+ 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
3 𝑑3𝑝𝑣𝑑𝑡3
= −12𝜍𝑙𝑅4
𝑅 3 +12𝜍𝑙𝑅3
𝑅 𝑅 −2𝜍𝑙𝑅2
𝑅 + 𝜌𝑙 13𝑅 𝑅 + 6𝑅 𝑑4𝑅
𝑑𝑡4+ 𝑅
𝑑5𝑅
𝑑𝑡5
4 𝑑4𝑝𝑣𝑑𝑡4
=48𝜍𝑙𝑅5
𝑅 4 −72𝜍𝑙𝑅4
𝑅 2𝑅 +12𝜍𝑙𝑅3
𝑅 2 +4𝜍𝑙𝑅3
𝑅 𝑅 −2𝜍𝑙𝑅2
𝑑4𝑅
𝑑𝑡4
+ 𝜌𝑙 13𝑅 2 + 19𝑅 𝑑4𝑅
𝑑𝑡4+ 7𝑅
𝑑5𝑅
𝑑𝑡5+ 𝑅
𝑑6𝑅
𝑑𝑡6
5 𝑑5𝑝𝑣𝑑𝑡5
= −240𝜍𝑙𝑅6
𝑅 5 +480𝜍𝑙𝑅5
𝑅 3𝑅 −252𝜍𝑙𝑅4
𝑅 𝑅 2 +28𝜍𝑙𝑅3
𝑅 𝑅 −12𝜍𝑙𝑅3
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
81
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
3𝐴
𝐴2
𝐵2𝑡 + 1
3 2
− 𝐴2
𝐵2𝑡
3 2
− 1
1 𝑑𝑅
𝑑𝑡= 𝐴
𝐴2
𝐵2𝑡 + 1
1 2
− 𝐴2
𝐵2𝑡
1 2
2 𝑑2𝑅
𝑑𝑡2=
𝐴3
2𝐵2
𝐴2
𝐵2𝑡 + 1
−1 2
− 𝐴2
𝐵2𝑡
−1 2
3 𝑑3𝑅
𝑑𝑡3= −
𝐴5
4𝐵4
𝐴2
𝐵2𝑡 + 1
−3 2
− 𝐴2
𝐵2𝑡
−3 2
4 𝑑4𝑅
𝑑𝑡4=
3𝐴7
8𝐵6
𝐴2
𝐵2𝑡 + 1
−5 2
− 𝐴2
𝐵2𝑡
−5 2
5 𝑑5𝑅
𝑑𝑡5= −
15𝐴9
16𝐵8
𝐴2
𝐵2𝑡 + 1
−7 2
− 𝐴2
𝐵2𝑡
−7 2
6 𝑑6𝑅
𝑑𝑡6=
105𝐴11
32𝐵10
𝐴2
𝐵2𝑡 + 1
−9 2
− 𝐴2
𝐵2𝑡
−9 2
7 𝑑7𝑅
𝑑𝑡7= −
945𝐴13
64𝐵12
𝐴2
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
82
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
−11
16
𝜌𝑙𝑔 𝜌𝑙 − 𝜌𝑣
3𝑅2𝑅 2 + 𝑅3𝑅 −3𝑎
16
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.
𝑆 𝑔𝑒𝑛 = −𝐹𝑛𝑒𝑡𝑇𝑤𝑎𝑙𝑙
𝑅 +4𝜋
𝑇𝑤𝑎𝑙𝑙𝑝𝑠𝑎𝑡 𝑇∞ 𝑅
2𝑅
By plotting the equation above, it is confirmed that the calculated entropy
generation rate does not reach a maximum value (Figure 21).
83
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.
84
0 = −𝑅
𝑅 𝐹𝑛𝑒𝑡 + −2 + 𝑅 + 𝑅
𝑅
𝑅 𝑑𝐹𝑛𝑒𝑡𝑑𝑡
+ 2𝑅 − 𝑅𝑅 −𝑅
𝑅 −
1
2𝑅2
𝑅
𝑅 𝑑2𝐹𝑛𝑒𝑡𝑑𝑡2
+ −𝑅2 +1
2𝑅2𝑅 +
1
2
𝑅2
𝑅 +
1
6𝑅3
𝑅
𝑅 𝑑3𝐹𝑛𝑒𝑡𝑑𝑡3
+ 1
3𝑅3 −
1
6𝑅3𝑅 −
1
6
𝑅3
𝑅 −
1
24𝑅4
𝑅
𝑅 𝑑4𝐹𝑛𝑒𝑡𝑑𝑡4
+ −1
24𝑅4 +
1
24
𝑅4
𝑅 𝑑5𝐹𝑛𝑒𝑡𝑑𝑡5
+4𝜋𝑅
3 6𝑅 + 3𝑅
𝑅
𝑅 𝑝𝑣 + 6𝑅 − 3𝑅𝑅 − 𝑅2
𝑅
𝑅 𝑑𝑝𝑣𝑑𝑡
+ −2𝑅2 + 𝑅2𝑅 +𝑅2
𝑅 +
1
4𝑅3
𝑅
𝑅 𝑑2𝑝𝑣𝑑𝑡2
+ 1
2𝑅3 −
1
4𝑅3𝑅 −
1
4
𝑅3
𝑅 −
1
20𝑅4
𝑅
𝑅 𝑑3𝑝𝑣𝑑𝑡3
+ −1
10𝑅4 +
1
20𝑅4𝑅 +
1
20
𝑅4
𝑅 +
1
120𝑅5
𝑅
𝑅 𝑑4𝑝𝑣𝑑𝑡4
+ 1
120𝑅5 −
1
120
𝑅5
𝑅 𝑑5𝑝𝑣𝑑𝑡5
+ ⋯ + ⋯
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.
85
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.
86
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.
𝑆 𝑔𝑒𝑛 =4𝜋
𝑇𝑤𝑎𝑙𝑙𝑝∞𝑅
2𝑅
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
87
equation for bubble growth, a third order equation is derived which allows for the
prediction of bubble departure radii.
0 = 𝑅𝑑𝑒𝑝𝑡3 +
3𝑒 0.894132 𝑙𝑛 𝐵 −4.010944
𝐵2 𝑅𝑑𝑒𝑝𝑡
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.
𝑆 𝑔𝑒𝑛 = −𝐹𝑛𝑒𝑡𝑇𝑤𝑎𝑙𝑙
𝑅
− 𝑅 − 1
𝑇𝑤𝑎𝑙𝑙 −
𝑑𝐹𝑛𝑒𝑡𝑑𝑡
𝑅 +1
2
𝑑2𝐹𝑛𝑒𝑡𝑑𝑡2
𝑅2 −1
6
𝑑3𝐹𝑛𝑒𝑡𝑑𝑡3
𝑅3 +1
24
𝑑4𝐹𝑛𝑒𝑡𝑑𝑡4
𝑅4 + ⋯
−4𝜋𝑅2
3𝑇𝑤𝑎𝑙𝑙 −3𝑝𝑣𝑅
+ 𝑅 − 1 𝑅𝑑𝑝𝑣𝑑𝑡
−1
4𝑅2
𝑑2𝑝𝑣𝑑𝑡2
+1
20𝑅3
𝑑3𝑝𝑣𝑑𝑡3
−1
120𝑅4
𝑑4𝑝𝑣𝑑𝑡4
+ ⋯
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
88
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
89
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.
90
BIBLIOGRAPHY
Borishanskiy, V. M., & Fokin, F. S. (1963). Heat transfer and hydrodynamics in
steam generators. Trudy TsKTI 62, 1 .
Borishanskiy, V. M., Danilova, G. N., Gotovskiy, M. A., Borishanskaya, A. V.,
Danilova, G. P., & Kupriyanova, A. V. (1981). Correlation of data on heat
transfer in, and elementary characteristics of the nucleate boiling
mechanism. Heat Transfer - Soviet Research, 13, 100-116.
Cole, R. (1967). Bubble frequencies and departure volumes at subatmospheric
pressures. AIChE Journal, 13 (4), 779-783.
Cole, R., & Rohsenow, W. (1969). Correlation of bubble departure diameters for
boiling of saturated liquids. Chemical Engineering Progress Symposium
Series , 65 (92), 211-213.
Cole, R., & Shulman, H. L. (1966a). Bubble departure diameters at
subatmospheric pressures. Chemical Engineering Progress Symposium
Series, 62 (64), 6-16.
Cole, R., & Shulman, H. L. (1966b). Bubble growth rates at high jakob numbers.
International Journal of Heat and Mass Transfer, 9 (12), 1377-1390.
Ellion, M. (1954). A Study of the Mechanism of Boiling Heat Transfer. Pasadena:
California Institute of Technology.
Fritz, W. (1935). Maximum volume of vapor bubbles. Physik Zeitschr, 36, 379-
384.
91
Golorin, V. S., Kol'chugin, B. A., & Zakharova, E. A. (1978). Investigation of the
mechanism of nucleate boiling of ethyl alcohol and benzene by means of
high-speed motion-picture photography. Heat Transfer - Soviet Research,
10, 79-98.
Gorenflo, D., Knabe, V., & Bieling, V. (1986). Bubble density on surfaces with
nucleate boiling-It's influence on heat transfer and burnout heat fluxes at
elevated saturation pressures. Proceedings of the 8th International Heat
and Mass Transfer Conference, 4, pp. 1995-2000. San Francisco.
Han, C. Y., & Griffith, P. (1962). The mechanism of heat transfer in nucleate pool
boiling. TR 16, Massachusetts Institute of Technology.
(1994). IAPWS release on surface tension of ordinary water substance.
International Association for the Properties of Water and Steam.
Jensen, M. K., & Memmel, G. J. (1986). Evaluation of bubble departure diameter
correlations. Proceedings of the 8th International Heat Transfer
Conference, 4, pp. 1907-1912.
Keshock, E. G., & Siegel, R. (1964). Forces acting on bubbles in nucleate boiling
under normal and reduced gravity conditions. NASA-TN-D-2299.
Kim, J., & Kim, M. H. (2006). On the departure behaviors of bubble at nucleate
pool boiling. International Journal of Multiphase Flow, 32 (10-11), 1269-
1286.
Kocamustafaogullari, G. (1983). Pressure dependence of bubble departure
diameter for water. International Communications in Heat and Mass
Transfer, 10 (6), 501-509.
92
Kutateladze, S. S., & Gogonin, I. I. (1980). Growth rate and detachment diameter
of a vapor bubble in free convection boiling of a saturated liquid. High
Temperature, 17 (4), 667-671.
Lien, Y. C. (1969). Bubble Growth Rates at Reduced Pressure. Massachusetts
Institute of Technology, Department of Mechanical Engineering.
Mikic, B. B., Rohsenow, W. M., & Griffith, P. (1970). On bubble growth rates.
International Journal of Heat and Mass Transfer, 13 (4), 657-666.
Plesset, M. S., & Zwick, S. A. (1954). The growth of vapor bubbles in
superheated liquids. Journal of Applied Physics, 25 (4), 493-500.
Rayleigh, L. (1917). On the pressure developed in a liquid during the collapse of
a spherical cavity. Philosophical Magazine, 34 (200), 94-98.
(2007). Revised release on the IAPWS Industrial Formulation of 1997 for the
thermodynamic properties of water and steam. Internaional Association for
the Properteis of Water and Steam.
Ruckenstein, E. (1961). Physical model for nucleate boiling heat transfer from a
horizontal surface. Buletinul Institutului Politehnic Bucuresti, 33 (3), 79-88.
Staniszewski, B. E. (1959). Nucleate boiling bubble growth and departure.
Massachusetts Institute of Technology Cambridge Division of Sponsored
Research.
Stephan, K. (1992). Heat transfer in condensation and boiling. Berlin: Springer-
Verlag.
93
Van Stralen, S. J., Cole, R., Sluyter, W. M., & Sohal, M. S. (1975). Bubble growth
rates in nucleate boiling of water at subatmospheric pressures.
International Journal of Heat and Mass Transfer, 18 (5), 655-669.
Zuber, N. (1959). Hydrodynamic aspects of boiling heat transfer. U.S. AEC
Report AECU 4439. United States Atomic Energy Commission.
Zuber, N. (1964). Recent trends in boiling heat transfer research. Part I: Nucleate
pool boiling. Applied Mechanics Reviews, 17 (9), 663-672.
94
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
𝑅
𝑜
Integration by successive parts produces the following.
𝑊 =4𝜋
3𝑅3 𝑝𝑣 − 𝑝∞ −
1
4
𝑑𝑝𝑣𝑑𝑡
𝑅 +1
20
𝑑2𝑝𝑣𝑑𝑡2
𝑅2 −1
120
𝑑3𝑝𝑣𝑑𝑡3
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=
4𝜋
3𝑅2𝑅 3 𝑝𝑣 − 𝑝∞ −
3
4
𝑑𝑝𝑣𝑑𝑡
𝑅 +3
20
𝑑2𝑝𝑣𝑑𝑡2
𝑅2 −1
40
𝑑3𝑝𝑣𝑑𝑡3
𝑅3 + ⋯
+4𝜋
3𝑅3
𝑑𝑝𝑣𝑑𝑡
−1
4
𝑑2𝑝𝑣𝑑𝑡2
𝑅 −1
4
𝑑𝑝𝑣𝑑𝑡
𝑅 +1
20
𝑑3𝑝𝑣𝑑𝑡3
𝑅2 +1
10
𝑑2𝑝𝑣𝑑𝑡2
𝑅𝑅
−1
120
𝑑4𝑝𝑣𝑑𝑡4
𝑅3 −1
40
𝑑3𝑝𝑣𝑑𝑡3
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.
95
𝑊 =4𝜋𝑅2
3 3 𝑝𝑣 − 𝑝∞ 𝑅
+ 𝑅 − 1 −𝑑𝑝𝑣𝑑𝑡
𝑅 +1
4
𝑑2𝑝𝑣𝑑𝑡2
𝑅2 −1
20
𝑑3𝑝𝑣𝑑𝑡3
𝑅3 +1
120
𝑑4𝑝𝑣𝑑𝑡4
𝑅4 + ⋯
The total internal energy of a spherical bubble is defined by the following
equation.
𝐸 = 𝑒𝑣𝑚𝑣 = 4π 𝜌𝑣𝑒𝑣
R
0
𝑅2𝑑𝑅
Successive integration by parts leads to the following equation.
𝐸 =4𝜋𝑅3
3 𝜌𝑣𝑒𝑣 −
1
4 𝜌𝑣
𝑑𝑒𝑣𝑑𝑡
+ 𝑒𝑣𝑑𝜌𝑣𝑑𝑡
𝑅 +1
20 𝜌𝑣
𝑑2𝑒𝑣𝑑𝑡2
+ 2𝑑𝑒𝑣𝑑𝑡
𝑑𝜌𝑣𝑑𝑡
+ 𝑒𝑣𝑑2𝜌𝑣𝑑𝑡2
𝑅2
−1
120 𝜌𝑣
𝑑3𝑒𝑣𝑑𝑡3
+ 3𝑑𝜌𝑣𝑑𝑡
𝑑2𝑒𝑣𝑑𝑡2
+ 3𝑑𝑒𝑣𝑑𝑡
𝑑2𝜌𝑣𝑑𝑡2
+ 𝑒𝑣𝑑3𝜌𝑣𝑑𝑡3
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.
96
𝑑𝐸
𝑑𝑡=
4𝜋
33𝑅2𝑅 𝜌𝑣𝑒𝑣 −
1
4 𝜌𝑣
𝑑𝑒𝑣𝑑𝑡
+ 𝑒𝑣𝑑𝜌𝑣𝑑𝑡
𝑅
+1
20 𝜌𝑣
𝑑2𝑒𝑣𝑑𝑡2
+ 2𝑑𝑒𝑣𝑑𝑡
𝑑𝜌𝑣𝑑𝑡
+ 𝑒𝑣𝑑2𝜌𝑣𝑑𝑡2
𝑅2
−1
120 𝜌𝑣
𝑑3𝑒𝑣𝑑𝑡3
+ 3𝑑𝜌𝑣𝑑𝑡
𝑑2𝑒𝑣𝑑𝑡2
+ 3𝑑𝑒𝑣𝑑𝑡
𝑑2𝜌𝑣𝑑𝑡2
+ 𝑒𝑣𝑑3𝜌𝑣𝑑𝑡3
𝑅3 + ⋯
+4𝜋𝑅3
3 𝜌𝑣
𝑑𝑒𝑣𝑑𝑡
+ 𝑒𝑣𝑑𝜌𝑣𝑑𝑡
−1
4 𝜌𝑣
𝑑2𝑒𝑣𝑑𝑡2
+ 2𝑑𝑒𝑣𝑑𝑡
𝑑𝜌𝑣𝑑𝑡
+ 𝑒𝑣𝑑2𝜌𝑣𝑑𝑡2
𝑅
−1
4 𝜌𝑣
𝑑𝑒𝑣𝑑𝑡
+ 𝑒𝑣𝑑𝜌𝑣𝑑𝑡
𝑅
+1
20 𝜌𝑣
𝑑3𝑒𝑣𝑑𝑡3
+ 3𝑑𝜌𝑣𝑑𝑡
𝑑2𝑒𝑣𝑑𝑡2
+ 3𝑑𝑒𝑣𝑑𝑡
𝑑2𝜌𝑣𝑑𝑡2
+ 𝑒𝑣𝑑3𝜌𝑣𝑑𝑡3
𝑅2
+1
10 𝜌𝑣
𝑑2𝑒𝑣𝑑𝑡2
+ 2𝑑𝑒𝑣𝑑𝑡
𝑑𝜌𝑣𝑑𝑡
+ 𝑒𝑣𝑑2𝜌𝑣𝑑𝑡2
𝑅𝑅
−1
120 𝜌𝑣
𝑑4𝑒𝑣𝑑𝑡4
+ 4𝑑𝜌𝑣𝑑𝑡
𝑑3𝑒𝑣𝑑𝑡3
+ 6𝑑2𝑒𝑣𝑑𝑡2
𝑑2𝜌𝑣𝑑𝑡2
+ 4𝑑𝑒𝑣𝑑𝑡
𝑑3𝜌𝑣𝑑𝑡3
+ 𝑒𝑣𝑑4𝜌𝑣𝑑𝑡4
𝑅3
−1
40 𝜌𝑣
𝑑3𝑒𝑣𝑑𝑡3
+ 3𝑑𝜌𝑣𝑑𝑡
𝑑2𝑒𝑣𝑑𝑡2
+ 3𝑑𝑒𝑣𝑑𝑡
𝑑2𝜌𝑣𝑑𝑡2
+ 𝑒𝑣𝑑3𝜌𝑣𝑑𝑡3
𝑅2𝑅 + ⋯
Mathematical manipulation of the above equation allows it to be reduced
to the following.
97
𝐸 =4𝜋
3𝑅2 3𝜌𝑣𝑒𝑣𝑅
+ 𝑅 − 1 − 𝜌𝑣𝑑𝑒𝑣𝑑𝑡
+ 𝑒𝑣𝑑𝜌𝑣𝑑𝑡
𝑅
+1
4 𝜌𝑣
𝑑2𝑒𝑣𝑑𝑡2
+ 2𝑑𝑒𝑣𝑑𝑡
𝑑𝜌𝑣𝑑𝑡
+ 𝑒𝑣𝑑2𝜌𝑣𝑑𝑡2
𝑅2
−1
20 𝜌𝑣
𝑑3𝑒𝑣𝑑𝑡3
+ 3𝑑𝜌𝑣𝑑𝑡
𝑑2𝑒𝑣𝑑𝑡2
+ 3𝑑𝑒𝑣𝑑𝑡
𝑑2𝜌𝑣𝑑𝑡2
+ 𝑒𝑣𝑑3𝜌𝑣𝑑𝑡3
𝑅3
+1
120 𝜌𝑣
𝑑4𝑒𝑣𝑑𝑡4
+ 4𝑑𝜌𝑣𝑑𝑡
𝑑3𝑒𝑣𝑑𝑡3
+ 6𝑑2𝑒𝑣𝑑𝑡2
𝑑2𝜌𝑣𝑑𝑡2
+ 4𝑑𝑒𝑣𝑑𝑡
𝑑3𝜌𝑣𝑑𝑡3
+ 𝑒𝑣𝑑4𝜌𝑣𝑑𝑡4
𝑅4 + ⋯
The rate of energy flow to a spherical bubble is defined as follows.
𝑑
𝑑𝑡 𝑚
𝑖𝑛
=𝑑
𝑑𝑡 𝑚 𝑖𝑛
R
0
=𝑑
𝑑𝑡 4π 𝜌𝑣𝑖𝑛
R
0
𝑅2𝑑𝑅
Successive integration by parts leads to the following equation.
𝑑
𝑑𝑡 𝑚
𝑖𝑛
=𝑑
𝑑𝑡
4𝜋𝑅3
3 𝜌𝑣𝑒𝑣 −
1
4 𝜌𝑣
𝑑𝑣𝑑𝑡
+ 𝑣𝑑𝜌𝑣𝑑𝑡
𝑅
+1
20 𝜌𝑣
𝑑2𝑣𝑑𝑡2
+ 2𝑑𝑣𝑑𝑡
𝑑𝜌𝑣𝑑𝑡
+ 𝑣𝑑2𝜌𝑣𝑑𝑡2
𝑅2
−1
120 𝜌𝑣
𝑑3𝑣𝑑𝑡3
+ 3𝑑𝜌𝑣𝑑𝑡
𝑑2𝑣𝑑𝑡2
+ 3𝑑𝑣𝑑𝑡
𝑑2𝜌𝑣𝑑𝑡2
+ 𝑣𝑑3𝜌𝑣𝑑𝑡3
𝑅3 + ⋯
The rate of energy flow is solved by calculating the derivative with respect
to time.
98
𝑑
𝑑𝑡 𝑚
𝑖𝑛
=4𝜋
33𝑅2𝑅 𝜌𝑣𝑣 −
1
4 𝜌𝑣
𝑑𝑣𝑑𝑡
+ 𝑣𝑑𝜌𝑣𝑑𝑡
𝑅
+1
20 𝜌𝑣
𝑑2𝑣𝑑𝑡2
+ 2𝑑𝑣𝑑𝑡
𝑑𝜌𝑣𝑑𝑡
+ 𝑣𝑑2𝜌𝑣𝑑𝑡2
𝑅2
−1
120 𝜌𝑣
𝑑3𝑣𝑑𝑡3
+ 3𝑑𝜌𝑣𝑑𝑡
𝑑2𝑣𝑑𝑡2
+ 3𝑑𝑣𝑑𝑡
𝑑2𝜌𝑣𝑑𝑡2
+ 𝑣𝑑3𝜌𝑣𝑑𝑡3
𝑅3 + ⋯
+4𝜋𝑅3
3 𝜌𝑣
𝑑𝑣𝑑𝑡
+ 𝑣𝑑𝜌𝑣𝑑𝑡
−1
4 𝜌𝑣
𝑑2𝑣𝑑𝑡2
+ 2𝑑𝑣𝑑𝑡
𝑑𝜌𝑣𝑑𝑡
+ 𝑣𝑑2𝜌𝑣𝑑𝑡2
𝑅
−1
4 𝜌𝑣
𝑑𝑣𝑑𝑡
+ 𝑣𝑑𝜌𝑣𝑑𝑡
𝑅
+1
20 𝜌𝑣
𝑑3𝑣𝑑𝑡3
+ 3𝑑𝜌𝑣𝑑𝑡
𝑑2𝑣𝑑𝑡2
+ 3𝑑𝑣𝑑𝑡
𝑑2𝜌𝑣𝑑𝑡2
+ 𝑣𝑑3𝜌𝑣𝑑𝑡3
𝑅2
+1
10 𝜌𝑣
𝑑2𝑣𝑑𝑡2
+ 2𝑑𝑣𝑑𝑡
𝑑𝜌𝑣𝑑𝑡
+ 𝑣𝑑2𝜌𝑣𝑑𝑡2
𝑅𝑅
−1
120 𝜌𝑣
𝑑4𝑣𝑑𝑡4
+ 4𝑑𝜌𝑣𝑑𝑡
𝑑3𝑣𝑑𝑡3
+ 6𝑑2𝑣𝑑𝑡2
𝑑2𝜌𝑣𝑑𝑡2
+ 4𝑑𝑣𝑑𝑡
𝑑3𝜌𝑣𝑑𝑡3
+ 𝑣𝑑4𝜌𝑣𝑑𝑡4
𝑅3
−1
40 𝜌𝑣
𝑑3𝑣𝑑𝑡3
+ 3𝑑𝜌𝑣𝑑𝑡
𝑑2𝑣𝑑𝑡2
+ 3𝑑𝑣𝑑𝑡
𝑑2𝜌𝑣𝑑𝑡2
+ 𝑣𝑑3𝜌𝑣𝑑𝑡3
𝑅2𝑅 + ⋯
Simple mathematical manipulation of the above equation allows it to be
reduced to the following.
99
𝑑
𝑑𝑡 𝑚
𝑖𝑛
=4𝜋
3𝑅2 3𝜌𝑣𝑣𝑅
+ 𝑅 − 1 − 𝜌𝑣𝑑𝑣𝑑𝑡
+ 𝑣𝑑𝜌𝑣𝑑𝑡
𝑅
+1
4 𝜌𝑣
𝑑2𝑣𝑑𝑡2
+ 2𝑑𝑣𝑑𝑡
𝑑𝜌𝑣𝑑𝑡
+ 𝑣𝑑2𝜌𝑣𝑑𝑡2
𝑅2
−1
20 𝜌𝑣
𝑑3𝑣𝑑𝑡3
+ 3𝑑𝜌𝑣𝑑𝑡
𝑑2𝑣𝑑𝑡2
+ 3𝑑𝑣𝑑𝑡
𝑑2𝜌𝑣𝑑𝑡2
+ 𝑣𝑑3𝜌𝑣𝑑𝑡3
𝑅3
+1
120 𝜌𝑣
𝑑4𝑣𝑑𝑡4
+ 4𝑑𝜌𝑣𝑑𝑡
𝑑3𝑣𝑑𝑡3
+ 6𝑑2𝑣𝑑𝑡2
𝑑2𝜌𝑣𝑑𝑡2
+ 4𝑑𝑣𝑑𝑡
𝑑3𝜌𝑣𝑑𝑡3
+ 𝑣𝑑4𝜌𝑣𝑑𝑡4
𝑅4 + ⋯
The heat transfer rate is solved for by substitution of the equations derived
above in the first law of thermodynamics.
100
𝑄 =4𝜋𝑅2
3 3 𝑝𝑣 − 𝑝∞ 𝑅
+ 𝑅 − 1 −𝑑𝑝𝑣𝑑𝑡
𝑅 +1
4
𝑑2𝑝𝑣𝑑𝑡2
𝑅2 −1
20
𝑑3𝑝𝑣𝑑𝑡3
𝑅3 +1
120
𝑑4𝑝𝑣𝑑𝑡4
𝑅4 + ⋯
+ +3𝜌𝑣𝑒𝑣𝑅
+ 𝑅 − 1 − 𝜌𝑣𝑑𝑒𝑣𝑑𝑡
+ 𝑒𝑣𝑑𝜌𝑣𝑑𝑡
𝑅
+1
4 𝜌𝑣
𝑑2𝑒𝑣𝑑𝑡2
+ 2𝑑𝑒𝑣𝑑𝑡
𝑑𝜌𝑣𝑑𝑡
+ 𝑒𝑣𝑑2𝜌𝑣𝑑𝑡2
𝑅2
−1
20 𝜌𝑣
𝑑3𝑒𝑣𝑑𝑡3
+ 3𝑑𝜌𝑣𝑑𝑡
𝑑2𝑒𝑣𝑑𝑡2
+ 3𝑑𝑒𝑣𝑑𝑡
𝑑2𝜌𝑣𝑑𝑡2
+ 𝑒𝑣𝑑3𝜌𝑣𝑑𝑡3
𝑅3
+1
120 𝜌𝑣
𝑑4𝑒𝑣𝑑𝑡4
+ 4𝑑𝜌𝑣𝑑𝑡
𝑑3𝑒𝑣𝑑𝑡3
+ 6𝑑2𝑒𝑣𝑑𝑡2
𝑑2𝜌𝑣𝑑𝑡2
+ 4𝑑𝑒𝑣𝑑𝑡
𝑑3𝜌𝑣𝑑𝑡3
+ 𝑒𝑣𝑑4𝜌𝑣𝑑𝑡4
𝑅4 + ⋯ − 3𝜌𝑣𝑣𝑅
− 𝑅 − 1 − 𝜌𝑣𝑑𝑣𝑑𝑡
+ 𝑣𝑑𝜌𝑣𝑑𝑡
𝑅
+1
4 𝜌𝑣
𝑑2𝑣𝑑𝑡2
+ 2𝑑𝑣𝑑𝑡
𝑑𝜌𝑣𝑑𝑡
+ 𝑣𝑑2𝜌𝑣𝑑𝑡2
𝑅2
−1
20 𝜌𝑣
𝑑3𝑣𝑑𝑡3
+ 3𝑑𝜌𝑣𝑑𝑡
𝑑2𝑣𝑑𝑡2
+ 3𝑑𝑣𝑑𝑡
𝑑2𝜌𝑣𝑑𝑡2
+ 𝑣𝑑3𝜌𝑣𝑑𝑡3
𝑅3
+1
120 𝜌𝑣
𝑑4𝑣𝑑𝑡4
+ 4𝑑𝜌𝑣𝑑𝑡
𝑑3𝑣𝑑𝑡3
+ 6𝑑2𝑣𝑑𝑡2
𝑑2𝜌𝑣𝑑𝑡2
+ 4𝑑𝑣𝑑𝑡
𝑑3𝜌𝑣𝑑𝑡3
+ 𝑣𝑑4𝜌𝑣𝑑𝑡4
𝑅4 + ⋯
Grouping of common terms allows the equation to be simplified.
101
𝑄 =4𝜋𝑅2
3 3 𝑝𝑣 − 𝑝∞ 𝑅 + 3𝜌𝑣 𝑒𝑣 − 𝑣 𝑅
+ 𝑅 − 1 −𝑑𝑝𝑣𝑑𝑡
𝑅 +1
4
𝑑2𝑝𝑣𝑑𝑡2
𝑅2 −1
20
𝑑3𝑝𝑣𝑑𝑡3
𝑅3 +1
120
𝑑4𝑝𝑣𝑑𝑡4
𝑅4
− 𝜌𝑣𝑑𝑒𝑣𝑑𝑡
+ 𝑒𝑣𝑑𝜌𝑣𝑑𝑡
− 𝜌𝑣𝑑𝑣𝑑𝑡
− 𝑣𝑑𝜌𝑣𝑑𝑡
𝑅
+1
4 𝜌𝑣
𝑑2𝑒𝑣𝑑𝑡2
+ 2𝑑𝑒𝑣𝑑𝑡
𝑑𝜌𝑣𝑑𝑡
+ 𝑒𝑣𝑑2𝜌𝑣𝑑𝑡2
− 𝜌𝑣𝑑2𝑣𝑑𝑡2
− 2𝑑𝑣𝑑𝑡
𝑑𝜌𝑣𝑑𝑡
− 𝑣𝑑2𝜌𝑣𝑑𝑡2
𝑅2
−1
20 𝜌𝑣
𝑑3𝑒𝑣𝑑𝑡3
+ 3𝑑𝜌𝑣𝑑𝑡
𝑑2𝑒𝑣𝑑𝑡2
+ 3𝑑𝑒𝑣𝑑𝑡
𝑑2𝜌𝑣𝑑𝑡2
+ 𝑒𝑣𝑑3𝜌𝑣𝑑𝑡3
− 𝜌𝑣𝑑3𝑣𝑑𝑡3
− 3𝑑𝜌𝑣𝑑𝑡
𝑑2𝑣𝑑𝑡2
− 3𝑑𝑣𝑑𝑡
𝑑2𝜌𝑣𝑑𝑡2
− 𝑣𝑑3𝜌𝑣𝑑𝑡3
𝑅3
+1
120 𝜌𝑣
𝑑4𝑒𝑣𝑑𝑡4
+ 4𝑑𝜌𝑣𝑑𝑡
𝑑3𝑒𝑣𝑑𝑡3
+ 6𝑑2𝑒𝑣𝑑𝑡2
𝑑2𝜌𝑣𝑑𝑡2
+ 4𝑑𝑒𝑣𝑑𝑡
𝑑3𝜌𝑣𝑑𝑡3
+ 𝑒𝑣𝑑4𝜌𝑣𝑑𝑡4
− 𝜌𝑣𝑑4𝑣𝑑𝑡4
− 4𝑑𝜌𝑣𝑑𝑡
𝑑3𝑣𝑑𝑡3
− 6𝑑2𝑣𝑑𝑡2
𝑑2𝜌𝑣𝑑𝑡2
− 4𝑑𝑣𝑑𝑡
𝑑3𝜌𝑣𝑑𝑡3
− 𝑣𝑑4𝜌𝑣𝑑𝑡4
𝑅4
+ ⋯
By further regrouping, the equation is reduced to the following form.
102
𝑄 =4𝜋𝑅2
3 3 𝑝𝑣 − 𝑝∞ 𝑅 + 3𝜌𝑣 𝑒𝑣 − 𝑣 𝑅
+ 𝑅 − 1 −𝑑𝑝𝑣𝑑𝑡
𝑅 +1
4
𝑑2𝑝𝑣𝑑𝑡2
𝑅2 −1
20
𝑑3𝑝𝑣𝑑𝑡3
𝑅3 +1
120
𝑑4𝑝𝑣𝑑𝑡4
𝑅4
+ −𝑑𝜌𝑣𝑑𝑡
𝑅 +1
4
𝑑2𝜌𝑣𝑑𝑡2
𝑅2 −1
20
𝑑3𝜌𝑣𝑑𝑡3
𝑅3 +1
120
𝑑4𝜌𝑣𝑑𝑡4
𝑅4 𝑒𝑣 − 𝑣
+ −𝜌𝑣𝑅 +1
2
𝑑𝜌𝑣𝑑𝑡
𝑅2 −3
20
𝑑2𝜌𝑣𝑑𝑡2
𝑅3 +1
30
𝑑3𝜌𝑣𝑑𝑡3
𝑅4 𝑑𝑒𝑣𝑑𝑡
−𝑑𝑣𝑑𝑡
+ 1
4𝜌𝑣𝑅
2 −3
20
𝑑𝜌𝑣𝑑𝑡
𝑅3 +1
20
𝑑2𝜌𝑣𝑑𝑡2
𝑅4 𝑑2𝑒𝑣𝑑𝑡2
−𝑑2𝑣𝑑𝑡2
+ −1
20𝜌𝑣𝑅
3 +1
30
𝑑𝜌𝑣𝑑𝑡
𝑅4 𝑑3𝑒𝑣𝑑𝑡3
−𝑑3𝑣𝑑𝑡3
+ 1
120𝜌𝑣𝑅
4 𝑑4𝑒𝑣𝑑𝑡4
−𝑑4𝑣𝑑𝑡4
+ ⋯
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.
103
𝑑𝑒𝑣𝑑𝑡
−𝑑𝑣𝑑𝑡
=𝑑
𝑑𝑡 𝑒𝑣 − 𝑣 =
1
𝜌𝑣2
𝑑𝜌𝑣𝑑𝑡
𝑝𝑣 −1
𝜌𝑣
𝑑𝑝𝑣𝑑𝑡
𝑑2𝑒𝑣𝑑𝑡2
−𝑑2𝑣𝑑𝑡2
=𝑑
𝑑𝑡2 𝑒𝑣 − 𝑣
= 1
𝜌𝑣2
𝑑2𝜌𝑣𝑑𝑡2
− 21
𝜌𝑣3 𝑑𝜌𝑣𝑑𝑡
2
𝑝𝑣 + 21
𝜌𝑣2
𝑑𝜌𝑣𝑑𝑡
𝑑𝑝𝑣𝑑𝑡
+ −1
𝜌𝑣 𝑑2𝑝𝑣𝑑𝑡2
𝑑3𝑒𝑣𝑑𝑡3
−𝑑3𝑣𝑑𝑡3
=𝑑
𝑑𝑡3 𝑒𝑣 − 𝑣
= 1
𝜌𝑣2
𝑑3𝜌𝑣𝑑𝑡3
+ 61
𝜌𝑣4 𝑑𝜌𝑣𝑑𝑡
3
− 61
𝜌𝑣3
𝑑𝜌𝑣𝑑𝑡
𝑑2𝜌𝑣𝑑𝑡2
𝑝𝑣
+ 31
𝜌𝑣2
𝑑2𝜌𝑣𝑑𝑡2
− 61
𝜌𝑣3 𝑑𝜌𝑣𝑑𝑡
2
𝑑𝑝𝑣𝑑𝑡
+ 31
𝜌𝑣2
𝑑𝜌𝑣𝑑𝑡
𝑑2𝑝𝑣𝑑𝑡2
+ −1
𝜌𝑣 𝑑3𝑝𝑣𝑑𝑡3
𝑑4𝑒𝑣𝑑𝑡4
−𝑑4𝑣𝑑𝑡4
=𝑑
𝑑𝑡4 𝑒𝑣 − 𝑣
= −81
𝜌𝑣3
𝑑𝜌𝑣𝑑𝑡
𝑑3𝜌𝑣𝑑𝑡3
+1
𝜌𝑣2
𝑑4𝜌𝑣𝑑𝑡4
− 241
𝜌𝑣5 𝑑𝜌𝑣𝑑𝑡
4
+ 361
𝜌𝑣4 𝑑𝜌𝑣𝑑𝑡
2 𝑑2𝜌𝑣𝑑𝑡2
− 61
𝜌𝑣3 𝑑2𝜌𝑣𝑑𝑡2
2
𝑝𝑣
+ 41
𝜌𝑣2
𝑑3𝜌𝑣𝑑𝑡3
+ 241
𝜌𝑣4 𝑑𝜌𝑣𝑑𝑡
3
− 241
𝜌𝑣3
𝑑𝜌𝑣𝑑𝑡
𝑑2𝜌𝑣𝑑𝑡2
𝑑𝑝𝑣𝑑𝑡
+ 61
𝜌𝑣2
𝑑2𝜌𝑣𝑑𝑡2
− 121
𝜌𝑣3 𝑑𝜌𝑣𝑑𝑡
2
𝑑2𝑝𝑣𝑑𝑡2
+ 41
𝜌𝑣2
𝑑𝜌𝑣𝑑𝑡
𝑑3𝑝𝑣𝑑𝑡3
+ −1
𝜌𝑣 𝑑4𝑝𝑣𝑑𝑡4
Substitution of the above relationships into the heat transfer rate results in
the following simplified equation.
𝑄 = −4𝜋𝑝∞𝑅2𝑅
104
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
R
0
𝑅2𝑑𝑅
By successive integration by parts, the following solution is determined.
𝑆 =4𝜋
3𝑅3 𝜌𝑣𝑠𝑣 −
1
4
𝑑 𝜌𝑣𝑠𝑣
𝑑𝑡𝑅 +
1
20
𝑑2 𝜌𝑣𝑠𝑣
𝑑𝑡2𝑅2 −
1
120
𝑑3 𝜌𝑣𝑠𝑣
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 −
3
4
𝑑 𝜌𝑣𝑠𝑣
𝑑𝑡𝑅𝑅 +
3
20
𝑑2 𝜌𝑣𝑠𝑣
𝑑𝑡2𝑅2𝑅 −
1
40
𝑑3 𝜌𝑣𝑠𝑣
𝑑𝑡3𝑅3𝑅 + ⋯
+4𝜋
3𝑅2
𝑑 𝜌𝑣𝑠𝑣
𝑑𝑡𝑅 −
1
4
𝑑2 𝜌𝑣𝑠𝑣
𝑑𝑡2𝑅2 −
1
4
𝑑 𝜌𝑣𝑠𝑣
𝑑𝑡𝑅𝑅 +
1
20
𝑑3 𝜌𝑣𝑠𝑣
𝑑𝑡3𝑅3
+1
10
𝑑2 𝜌𝑣𝑠𝑣
𝑑𝑡2𝑅2𝑅 −
1
120
𝑑4 𝜌𝑣𝑠𝑣
𝑑𝑡4𝑅4 −
1
40
𝑑3 𝜌𝑣𝑠𝑣
𝑑𝑡3𝑅3𝑅 + ⋯
This equation is simplified by grouping of common terms.
𝑆 =4𝜋
3𝑅2 3𝜌𝑣𝑠𝑣𝑅
+ 𝑅 − 1 −𝑑 𝜌𝑣𝑠𝑣
𝑑𝑡𝑅 +
𝑑2 𝜌𝑣𝑠𝑣
𝑑𝑡2𝑅2 −
1
20
𝑑3 𝜌𝑣𝑠𝑣
𝑑𝑡3𝑅3
−1
120
𝑑4 𝜌𝑣𝑠𝑣
𝑑𝑡4𝑅4 + ⋯
With the rate of entropy accumulation defined, it is necessary to define the
entropy transfer rate. This values defined by the following equation.
105
𝑄
𝑇𝑤= −
4𝜋
𝑇𝑤𝑝∞𝑅
2𝑅
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.
𝑑
𝑑𝑡 𝑠𝑣𝑚𝑣
𝑖𝑛
=𝑑
𝑑𝑡 4π 𝜌𝑣𝑠𝑣
R
0
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.
𝑑
𝑑𝑡 𝑠𝑣𝑚𝑣
𝑖𝑛
=4𝜋
3𝑅2 3𝜌𝑣𝑠𝑣𝑅
+ 𝑅 − 1 −𝑑 𝜌𝑣𝑠𝑣
𝑑𝑡𝑅 +
𝑑2 𝜌𝑣𝑠𝑣
𝑑𝑡2𝑅2 −
1
20
𝑑3 𝜌𝑣𝑠𝑣
𝑑𝑡3𝑅3
−1
120
𝑑4 𝜌𝑣𝑠𝑣
𝑑𝑡4𝑅4 + ⋯
Substitution of the equations defined above into the second law of
thermodynamics allows for the entropy generation rate to be defined.
106
𝑆 gen =4𝜋
3𝑅2 3𝜌𝑣𝑠𝑣𝑅
+ 𝑅 − 1 −𝑑 𝜌𝑣𝑠𝑣
𝑑𝑡𝑅 +
𝑑2 𝜌𝑣𝑠𝑣
𝑑𝑡2𝑅2 −
1
20
𝑑3 𝜌𝑣𝑠𝑣
𝑑𝑡3𝑅3
−1
120
𝑑4 𝜌𝑣𝑠𝑣
𝑑𝑡4𝑅4 + ⋯ +
4𝜋
𝑇𝑤𝑝∞𝑅
2𝑅
−4𝜋
3𝑅2 3𝜌𝑣𝑠𝑣𝑅
+ 𝑅 − 1 −𝑑 𝜌𝑣𝑠𝑣
𝑑𝑡𝑅 +
𝑑2 𝜌𝑣𝑠𝑣
𝑑𝑡2𝑅2 −
1
20
𝑑3 𝜌𝑣𝑠𝑣
𝑑𝑡3𝑅3
−1
120
𝑑4 𝜌𝑣𝑠𝑣
𝑑𝑡4𝑅4 + ⋯
Removal of common terms allows the equation to reduce to the following.
𝑆 gen =4𝜋
𝑇𝑤𝑝∞𝑅
2𝑅
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.
𝑑𝑆 𝑔𝑒𝑛𝑑𝑅
= 0 =4𝜋
𝑇𝑤𝑝∞
𝑑
𝑑𝑅 𝑅2𝑅 =
4𝜋
𝑇𝑤𝑝∞
𝑑
𝑑𝑡 𝑅2𝑅 ∗
𝑑𝑡
𝑑𝑅
The resulting equation is as follows.
0 =4𝜋𝑅
𝑇𝑤𝑝∞ 2𝑅 + 𝑅
𝑅
𝑅
By removal of common terms, the equation reduces to the following non-
linear, second order differential equation.
107
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=
𝑑
𝑑𝑡 𝑑𝑅
𝑑𝑡 =
𝑑
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
𝑅
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
1
3𝑅3 = 𝑒−2𝐶𝑡 + 𝐷
108
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
109
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
𝑡=𝑡𝑑𝑒𝑝𝑡
= − 𝑑𝑅
𝑑𝑡 𝐺𝑒𝑛 𝑆𝑜𝑙𝑢𝑡𝑖𝑜𝑛 −1
𝑡=𝑡𝑑𝑒𝑝𝑡
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.
110
𝐶 = −𝑙𝑛 𝐴𝑡𝑑𝑒𝑝𝑡2
The time of bubble departure is replaced by the Rayleigh equation.
𝑡𝑑𝑒𝑝𝑡 =𝑅𝑑𝑒𝑝𝑡
𝐴
Substitution into the equation for constant 𝐶 results in the following
equation.
𝐶 = −𝑙𝑛 1
𝐴𝑅𝑑𝑒𝑝𝑡
2
By averaging results for multiple experimental data sets, a value for
constant 𝐶 is defined.
𝐶 = −𝑙𝑛
1𝐴𝑖
𝑅𝑒𝑥𝑝 ,𝑖2
𝑛
𝑛
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
𝐴𝑅𝑑𝑒𝑝𝑡
A single constant 𝐷 is determined by taking the average of multiple
experimental data sets.
𝐷 = 𝑅𝑒𝑥𝑝 ,𝑖
3 + 3𝑒−𝐶
𝐴𝑖 𝑅𝑒𝑥𝑝 ,𝑖
𝑛
𝑛
𝑖=1
The same procedure is utilized for the Plesse-Zwick equation. Boundary
conditions are defined as follows.
111
𝑅𝑃𝑙𝑒𝑠𝑠𝑒𝑡 −𝑍𝑤𝑖𝑐𝑘 𝑡=𝑡𝑑𝑒𝑝𝑡 = 𝑅𝐺𝑒𝑛 𝑆𝑜𝑙𝑢𝑡𝑖𝑜𝑛 𝑡=𝑡𝑑𝑒𝑝𝑡
𝑑𝑅
𝑑𝑡 𝑃𝑙𝑒𝑠𝑠𝑒𝑡 −𝑍𝑤𝑖𝑐𝑘
𝑡=𝑡𝑑𝑒𝑝𝑡
= − 𝑑𝑅
𝑑𝑡 𝐺𝑒𝑛 𝑆𝑜𝑙𝑢𝑡𝑖𝑜𝑛 −1
𝑡=𝑡𝑑𝑒𝑝𝑡
By substation of the appropriate equations into the boundary conditions
defined above, the following system of equations is created.
𝐵𝑡𝑑𝑒𝑝𝑡1 2 = −3𝑒−𝐶𝑡𝑑𝑒𝑝𝑡 + 𝐷
1 3
1
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
3 2
𝑡𝑑𝑒𝑝𝑡−3 4 + 3𝑒−𝐶𝑡𝑑𝑒𝑝𝑡
The system of equations is combined.
𝐵3𝑡𝑑𝑒𝑝𝑡3 2 =
1
2𝐵𝑒−𝐶
3 2
𝑡𝑑𝑒𝑝𝑡−3 4
By rearranging the equation above, it is possible to solve for the constant
𝐶.
𝐶 = −𝑙𝑛 2𝐵𝑡𝑑𝑒𝑝𝑡3 2
The time at departure is replaced using the Plesset-Zwick equation.
𝑡𝑑𝑒𝑝𝑡 = 𝑅𝑑𝑒𝑝𝑡
𝐵
2
The resulting equation for the constant 𝐶 is a function of variable 𝐵 and
departure radius.
𝐶 = −𝑙𝑛 2𝑅𝑑𝑒𝑝𝑡
3
𝐵2
112
By averaging results for multiple experimental data sets, a value for
constant 𝐶 is defined.
𝐶 =
−𝑙𝑛 2𝑅𝑑𝑒𝑝𝑡 ,𝑖
3
𝐵𝑖2
𝑛
𝑛
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
2
Averaging results of experimental data results in the following equation.
𝐷 =
𝑅𝑑𝑒𝑝𝑡 ,𝑖3 +
3𝑒−𝐶
𝐵𝑖2 𝑅𝑑𝑒𝑝𝑡 ,𝑖
2
𝑛
𝑛
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
𝑡=𝑡𝑑𝑒𝑝𝑡
= − 𝑑𝑅
𝑑𝑡 𝐺𝑒𝑛 𝑆𝑜𝑙𝑢𝑡𝑖𝑜𝑛 −1
𝑡=𝑡𝑑𝑒𝑝𝑡
By substation of the appropriate equations into the boundary conditions
defined above, the following system of equations is created.
113
2𝐵2
3𝐴
𝐴2
𝐵2𝑡𝑑𝑒𝑝𝑡 + 1
3 2
− 𝐴2
𝐵2𝑡𝑑𝑒𝑝𝑡
3 2
− 1 = −3𝑒−𝐶𝑡𝑑𝑒𝑝𝑡 + 𝐷 1 3
𝐴 𝐴2
𝐵2𝑡𝑑𝑒𝑝𝑡 + 1
1 2
− 𝐴2
𝐵2𝑡𝑑𝑒𝑝𝑡
1 2
= 𝑒𝐶 −3𝑒−𝐶𝑡𝑑𝑒𝑝𝑡 + 𝐷 2 3
These equations is rewritten to solve for constant 𝐷.
𝐷 = 2𝐵2
3𝐴
3
𝐴2
𝐵2𝑡𝑑𝑒𝑝𝑡 + 1
3 2
− 𝐴2
𝐵2𝑡𝑑𝑒𝑝𝑡
3 2
− 1
3
+ 3𝑒−𝐶𝑡𝑑𝑒𝑝𝑡
𝐷 = 𝐴𝑒−𝐶 3 2 𝐴2
𝐵2𝑡𝑑𝑒𝑝𝑡 + 1
1 2
− 𝐴2
𝐵2𝑡𝑑𝑒𝑝𝑡
1 2
3 2
+ 3𝑒−𝐶𝑡𝑑𝑒𝑝𝑡
The system of equations is combined.
2𝐵2
3𝐴
3
𝐴2
𝐵2𝑡𝑑𝑒𝑝𝑡 + 1
3 2
− 𝐴2
𝐵2𝑡𝑑𝑒𝑝𝑡
3 2
− 1
3
= 𝐴𝑒−𝐶 3 2 𝐴2
𝐵2𝑡𝑑𝑒𝑝𝑡 + 1
1 2
− 𝐴2
𝐵2𝑡𝑑𝑒𝑝𝑡
1 2
3 2
This relationship is rewritten as follows.
𝑅𝑑𝑒𝑝𝑡3 = 𝑒−3𝐶 2
𝑑𝑅
𝑑𝑡 𝑑𝑒𝑝𝑡
3 2
By rearranging the equation above, it is possible to solve for the constant
𝐶.
𝐶 = −𝑙𝑛
4𝐵4
9𝐴3
𝐴2
𝐵2 𝑡𝑑𝑒𝑝𝑡 + 1 3 2
− 𝐴2
𝐵2 𝑡𝑑𝑒𝑝𝑡 3 2
− 1
𝐴2
𝐵2 𝑡𝑑𝑒𝑝𝑡 + 1 1 2
− 𝐴2
𝐵2 𝑡𝑑𝑒𝑝𝑡 1 2
2
114
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
2
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
4
9𝐴𝑖3
𝐴𝑖
2
𝐵𝑖2 𝑡𝑑𝑒𝑝𝑡 ,𝑖 + 1
3 2
− 𝐴𝑖
2
𝐵𝑖2 𝑡𝑑𝑒𝑝𝑡 ,𝑖
3 2
− 1
𝐴𝑖
2
𝐵𝑖2 𝑡𝑑𝑒𝑝𝑡 ,𝑖 + 1
1 2
− 𝐴𝑖
2
𝐵𝑖2 𝑡𝑑𝑒𝑝𝑡 ,𝑖
1 2
2
𝑛
𝑛
𝑖=1
The constant 𝐷 is determined by substitution of constant 𝐶 into the
equation below.
𝐷 = 2𝐵2
3𝐴
3
𝐴2
𝐵2𝑡𝑑𝑒𝑝𝑡 + 1
3 2
− 𝐴2
𝐵2𝑡𝑑𝑒𝑝𝑡
3 2
− 1
3
+ 3𝑒−𝐶𝑡𝑑𝑒𝑝𝑡
Averaging results for multiple experimental data sets results in the
following equation.
𝐷 =
2𝐵𝑖
2
3𝐴𝑖
3
𝐴𝑖
2
𝐵𝑖2 𝑡𝑑𝑒𝑝𝑡 ,𝑖 + 1
3 2
− 𝐴𝑖
2
𝐵𝑖2 𝑡𝑑𝑒𝑝𝑡
3 2
− 1
3
+ 3𝑒−𝐶𝑡𝑑𝑒𝑝𝑡 ,𝑖
𝑛
𝑛
𝑖=1
APPENDIX C: DERIVATION OF ENTROPY GENERATOIN RATE (NET
FORCE METHOD)
115
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
𝑜
Integration by successive parts leads to the following.
𝑊 = 𝑅 𝐹𝑛𝑒𝑡 −1
2
𝑑𝐹𝑛𝑒𝑡𝑑𝑡
𝑅 +1
6
𝑑2𝐹𝑛𝑒𝑡𝑑𝑡2
𝑅2 −1
24
𝑑3𝐹𝑛𝑒𝑡𝑑𝑡3
𝑅3 + ⋯
If the effect of changes in net force are ignored the equation is reduced to
the following.
𝑊 = 𝐹𝑛𝑒𝑡𝑅
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
2
𝑑2𝐹𝑛𝑒𝑡𝑑𝑡2
𝑅2 −1
6
𝑑3𝐹𝑛𝑒𝑡𝑑𝑡3
𝑅3 +1
24
𝑑4𝐹𝑛𝑒𝑡𝑑𝑡4
𝑅4 + ⋯
The assumption that changes in net force are negligible leads to the
following equation
𝑊 = 𝐹𝑛𝑒𝑡𝑅
The net force can be defined as the sum of the following forces as
described in Chapter 1.
𝐹𝑛𝑒𝑡 = 𝐹𝑏 + 𝐹𝑝 − 𝐹𝑖 − 𝐹𝜍 − 𝐹𝐷
116
Substitution of the appropriate equations results in the following.
𝐹𝑛𝑒𝑡 =4𝜋𝑅3
3 𝜌𝑙 − 𝜌𝑣 𝑔 + 𝜋𝑅𝑏𝜍 sin𝛽 −
𝑑
𝑑𝑡
11
16𝜌𝑙
4𝜋𝑅3
3 𝑅 − 2𝜋𝑅𝑏𝜍 sin𝛽
−𝜋
4𝑎𝜇𝑙𝑅𝑅
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
3𝑔 𝜌𝑙 − 𝜌𝑣 𝑅
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.
𝑑𝐹𝑛𝑒𝑡𝑑𝑡
=4𝜋
3𝑔 3 𝜌𝑙 − 𝜌𝑣 𝑅
2𝑅 −𝑑𝜌𝑣𝑑𝑡
𝑅3 − 𝜋𝜍𝑙𝑆𝜍 𝑅 −11𝜋
12𝜌𝑙 6𝑅𝑅
3 + 9𝑅2𝑅 𝑅 + 𝑅3𝑅
−𝜋
4𝑎𝜇𝑙 𝑅
2 + 𝑅𝑅
𝑑2𝐹𝑛𝑒𝑡𝑑𝑡2
=4𝜋
3𝑔 6𝑅𝑅 2 + 3𝑅2𝑅 𝜌𝑙 − 𝜌𝑣 − 6𝑅2𝑅
𝑑𝜌𝑣𝑑𝑡
− 𝑅3 𝑑2𝜌𝑣𝑑𝑡2
− 𝜋𝜍𝑙𝑆𝜍 𝑅
−11𝜋
12𝜌𝑙 6𝑅
4 + 36𝑅𝑅 2𝑅 + 9𝑅2𝑅 2 + 12𝑅2𝑅 𝑅 + 𝑅3𝑑4𝑅
𝑑𝑡4
−𝜋
4𝑎𝜇𝑙 3𝑅 𝑅 + 𝑅𝑅
117
𝑑3𝐹𝑛𝑒𝑡𝑑𝑡3
=4𝜋
3𝑔 6𝑅 3 + 18𝑅𝑅 𝑅 + 3𝑅2𝑅 𝜌𝑙 − 𝜌𝑣 − 18𝑅𝑅 2 + 9𝑅2𝑅
𝑑𝜌𝑣𝑑𝑡
− 9𝑅2𝑅 𝑑2𝜌𝑣𝑑𝑡2
− 𝑅3 𝑑3𝜌𝑣𝑑𝑡3
− 𝜋𝜍𝑙𝑆𝜍 𝑅
−11𝜋
12𝜌𝑙 60𝑅 3𝑅 + 90𝑅𝑅 𝑅 2 + 60𝑅𝑅 2𝑅 + 30𝑅2𝑅 𝑅 + 15𝑅2𝑅
𝑑4𝑅
𝑑𝑡4
+ 𝑅3𝑑5𝑅
𝑑𝑡5 −
𝜋
4𝑎𝜇𝑙 3𝑅
2 + 4𝑅 𝑅 + 𝑅𝑑4𝑅
𝑑𝑡4
𝑑4𝐹𝑛𝑒𝑡𝑑𝑡4
=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
− 𝑅3 𝑑4𝜌𝑣𝑑𝑡4
− 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
118
𝑑5𝐹𝑛𝑒𝑡𝑑𝑡5
=4𝜋
3𝑔 90𝑅 𝑅 2 + 60𝑅 2𝑅 + 60𝑅𝑅 𝑅 + 30𝑅𝑅
𝑑4𝑅
𝑑𝑡4+ 3𝑅2
𝑑5𝑅
𝑑𝑡5 𝜌𝑙 − 𝜌𝑣
− 180𝑅 2𝑅 + 90𝑅𝑅 2 + 120𝑅𝑅 𝑅 + 15𝑅2𝑑4𝑅
𝑑𝑡4 𝑑𝜌𝑣𝑑𝑡
− 60𝑅 3 + 180𝑅𝑅 𝑅 + 30𝑅2𝑅 𝑑2𝜌𝑣𝑑𝑡2
− 60𝑅 2 + 30𝑅2𝑅 𝑑3𝜌𝑣𝑑𝑡3
− 15𝑅2𝑅 𝑑4𝜌𝑣𝑑𝑡4
− 𝑅3 𝑑5𝜌𝑣𝑑𝑡5
− 𝜋𝜍𝑙𝑆𝜍 𝑑5𝑅
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
𝑑𝑡4+ 63𝑅2𝑅
𝑑5𝑅
𝑑𝑡5
+ 126𝑅𝑅 2𝑑5𝑅
𝑑𝑡5+ 21𝑅2𝑅
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.
119
𝐸 =4𝜋
3𝑅2 3𝜌𝑣𝑒𝑣𝑅
+ 𝑅 − 1 − 𝜌𝑣𝑑𝑒𝑣𝑑𝑡
+ 𝑒𝑣𝑑𝜌𝑣𝑑𝑡
𝑅
+1
4 𝜌𝑣
𝑑2𝑒𝑣𝑑𝑡2
+ 2𝑑𝑒𝑣𝑑𝑡
𝑑𝜌𝑣𝑑𝑡
+ 𝑒𝑣𝑑2𝜌𝑣𝑑𝑡2
𝑅2
−1
20 𝜌𝑣
𝑑3𝑒𝑣𝑑𝑡3
+ 3𝑑𝜌𝑣𝑑𝑡
𝑑2𝑒𝑣𝑑𝑡2
+ 3𝑑𝑒𝑣𝑑𝑡
𝑑2𝜌𝑣𝑑𝑡2
+ 𝑒𝑣𝑑3𝜌𝑣𝑑𝑡3
𝑅3
+1
120 𝜌𝑣
𝑑4𝑒𝑣𝑑𝑡4
+ 4𝑑𝜌𝑣𝑑𝑡
𝑑3𝑒𝑣𝑑𝑡3
+ 6𝑑2𝑒𝑣𝑑𝑡2
𝑑2𝜌𝑣𝑑𝑡2
+ 4𝑑𝑒𝑣𝑑𝑡
𝑑3𝜌𝑣𝑑𝑡3
+ 𝑒𝑣𝑑4𝜌𝑣𝑑𝑡4
𝑅4 + ⋯
The rate of energy flow to a spherical bubble is defined in Chapter 2 and is
defined by the following equation.
𝑑
𝑑𝑡 𝑚
𝑖𝑛
=4𝜋
3𝑅2 3𝜌𝑣𝑣𝑅
+ 𝑅 − 1 − 𝜌𝑣𝑑𝑣𝑑𝑡
+ 𝑣𝑑𝜌𝑣𝑑𝑡
𝑅
+1
4 𝜌𝑣
𝑑2𝑣𝑑𝑡2
+ 2𝑑𝑣𝑑𝑡
𝑑𝜌𝑣𝑑𝑡
+ 𝑣𝑑2𝜌𝑣𝑑𝑡2
𝑅2
−1
20 𝜌𝑣
𝑑3𝑣𝑑𝑡3
+ 3𝑑𝜌𝑣𝑑𝑡
𝑑2𝑣𝑑𝑡2
+ 3𝑑𝑣𝑑𝑡
𝑑2𝜌𝑣𝑑𝑡2
+ 𝑣𝑑3𝜌𝑣𝑑𝑡3
𝑅3
+1
120 𝜌𝑣
𝑑4𝑣𝑑𝑡4
+ 4𝑑𝜌𝑣𝑑𝑡
𝑑3𝑣𝑑𝑡3
+ 6𝑑2𝑣𝑑𝑡2
𝑑2𝜌𝑣𝑑𝑡2
+ 4𝑑𝑣𝑑𝑡
𝑑3𝜌𝑣𝑑𝑡3
+ 𝑣𝑑4𝜌𝑣𝑑𝑡4
𝑅4 + ⋯
120
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
2
𝑑2𝐹𝑛𝑒𝑡𝑑𝑡2
𝑅2 −1
6
𝑑3𝐹𝑛𝑒𝑡𝑑𝑡3
𝑅3 +1
24
𝑑4𝐹𝑛𝑒𝑡𝑑𝑡4
𝑅4 + ⋯
+4𝜋𝑅2
3 3𝜌𝑣𝑒𝑣𝑅
+ 𝑅 − 1 − 𝜌𝑣𝑑𝑒𝑣𝑑𝑡
+ 𝑒𝑣𝑑𝜌𝑣𝑑𝑡
𝑅
+1
4 𝜌𝑣
𝑑2𝑒𝑣𝑑𝑡2
+ 2𝑑𝑒𝑣𝑑𝑡
𝑑𝜌𝑣𝑑𝑡
+ 𝑒𝑣𝑑2𝜌𝑣𝑑𝑡2
𝑅2
−1
20 𝜌𝑣
𝑑3𝑒𝑣𝑑𝑡3
+ 3𝑑𝜌𝑣𝑑𝑡
𝑑2𝑒𝑣𝑑𝑡2
+ 3𝑑𝑒𝑣𝑑𝑡
𝑑2𝜌𝑣𝑑𝑡2
+ 𝑒𝑣𝑑3𝜌𝑣𝑑𝑡3
𝑅3
+1
120 𝜌𝑣
𝑑4𝑒𝑣𝑑𝑡4
+ 4𝑑𝜌𝑣𝑑𝑡
𝑑3𝑒𝑣𝑑𝑡3
+ 6𝑑2𝑒𝑣𝑑𝑡2
𝑑2𝜌𝑣𝑑𝑡2
+ 4𝑑𝑒𝑣𝑑𝑡
𝑑3𝜌𝑣𝑑𝑡3
+ 𝑒𝑣𝑑4𝜌𝑣𝑑𝑡4
𝑅4 + ⋯ − 3𝜌𝑣𝑣𝑅
− 𝑅 − 1 − 𝜌𝑣𝑑𝑣𝑑𝑡
+ 𝑣𝑑𝜌𝑣𝑑𝑡
𝑅
+1
4 𝜌𝑣
𝑑2𝑣𝑑𝑡2
+ 2𝑑𝑣𝑑𝑡
𝑑𝜌𝑣𝑑𝑡
+ 𝑣𝑑2𝜌𝑣𝑑𝑡2
𝑅2
−1
20 𝜌𝑣
𝑑3𝑣𝑑𝑡3
+ 3𝑑𝜌𝑣𝑑𝑡
𝑑2𝑣𝑑𝑡2
+ 3𝑑𝑣𝑑𝑡
𝑑2𝜌𝑣𝑑𝑡2
+ 𝑣𝑑3𝜌𝑣𝑑𝑡3
𝑅3
+1
120 𝜌𝑣
𝑑4𝑣𝑑𝑡4
+ 4𝑑𝜌𝑣𝑑𝑡
𝑑3𝑣𝑑𝑡3
+ 6𝑑2𝑣𝑑𝑡2
𝑑2𝜌𝑣𝑑𝑡2
+ 4𝑑𝑣𝑑𝑡
𝑑3𝜌𝑣𝑑𝑡3
+ 𝑣𝑑4𝜌𝑣𝑑𝑡4
𝑅4 + ⋯
121
Grouping of common terms allows the equation to be simplified.
𝑄 = 𝐹𝑛𝑒𝑡𝑅 + 𝑅 − 1 −𝑑𝐹𝑛𝑒𝑡𝑑𝑡
𝑅 +1
2
𝑑2𝐹𝑛𝑒𝑡𝑑𝑡2
𝑅2 −1
6
𝑑3𝐹𝑛𝑒𝑡𝑑𝑡3
𝑅3 +1
24
𝑑4𝐹𝑛𝑒𝑡𝑑𝑡4
𝑅4 + ⋯
+4𝜋𝑅2
3 3𝜌𝑣 𝑒𝑣 − 𝑣 𝑅
+ 𝑅 − 1 − 𝜌𝑣𝑑𝑒𝑣𝑑𝑡
+ 𝑒𝑣𝑑𝜌𝑣𝑑𝑡
− 𝜌𝑣𝑑𝑣𝑑𝑡
− 𝑣𝑑𝜌𝑣𝑑𝑡
𝑅
+1
4 𝜌𝑣
𝑑2𝑒𝑣𝑑𝑡2
+ 2𝑑𝑒𝑣𝑑𝑡
𝑑𝜌𝑣𝑑𝑡
+ 𝑒𝑣𝑑2𝜌𝑣𝑑𝑡2
− 𝜌𝑣𝑑2𝑣𝑑𝑡2
− 2𝑑𝑣𝑑𝑡
𝑑𝜌𝑣𝑑𝑡
− 𝑣𝑑2𝜌𝑣𝑑𝑡2
𝑅2
−1
20 𝜌𝑣
𝑑3𝑒𝑣𝑑𝑡3
+ 3𝑑𝜌𝑣𝑑𝑡
𝑑2𝑒𝑣𝑑𝑡2
+ 3𝑑𝑒𝑣𝑑𝑡
𝑑2𝜌𝑣𝑑𝑡2
+ 𝑒𝑣𝑑3𝜌𝑣𝑑𝑡3
− 𝜌𝑣𝑑3𝑣𝑑𝑡3
− 3𝑑𝜌𝑣𝑑𝑡
𝑑2𝑣𝑑𝑡2
− 3𝑑𝑣𝑑𝑡
𝑑2𝜌𝑣𝑑𝑡2
− 𝑣𝑑3𝜌𝑣𝑑𝑡3
𝑅3
+1
120 𝜌𝑣
𝑑4𝑒𝑣𝑑𝑡4
+ 4𝑑𝜌𝑣𝑑𝑡
𝑑3𝑒𝑣𝑑𝑡3
+ 6𝑑2𝑒𝑣𝑑𝑡2
𝑑2𝜌𝑣𝑑𝑡2
+ 4𝑑𝑒𝑣𝑑𝑡
𝑑3𝜌𝑣𝑑𝑡3
+ 𝑒𝑣𝑑4𝜌𝑣𝑑𝑡4
− 𝜌𝑣𝑑4𝑣𝑑𝑡4
− 4𝑑𝜌𝑣𝑑𝑡
𝑑3𝑣𝑑𝑡3
− 6𝑑2𝑣𝑑𝑡2
𝑑2𝜌𝑣𝑑𝑡2
− 4𝑑𝑣𝑑𝑡
𝑑3𝜌𝑣𝑑𝑡3
− 𝑣𝑑4𝜌𝑣𝑑𝑡4
𝑅4
+ ⋯
By further regrouping, the equation is reduced to the following form.
122
𝑄 = 𝐹𝑛𝑒𝑡𝑅 + 𝑅 − 1 −𝑑𝐹𝑛𝑒𝑡𝑑𝑡
𝑅 +1
2
𝑑2𝐹𝑛𝑒𝑡𝑑𝑡2
𝑅2 −1
6
𝑑3𝐹𝑛𝑒𝑡𝑑𝑡3
𝑅3 +1
24
𝑑4𝐹𝑛𝑒𝑡𝑑𝑡4
𝑅4 + ⋯
+4𝜋𝑅2
3 3𝜌𝑣 𝑒𝑣 − 𝑣 𝑅
+ 𝑅
− 1 −𝑑𝜌𝑣𝑑𝑡
𝑅 +1
4
𝑑2𝜌𝑣𝑑𝑡2
𝑅2 −1
20
𝑑3𝜌𝑣𝑑𝑡3
𝑅3 +1
120
𝑑4𝜌𝑣𝑑𝑡4
𝑅4 𝑒𝑣 − 𝑣
+ −𝜌𝑣𝑅 +1
2
𝑑𝜌𝑣𝑑𝑡
𝑅2 −3
20
𝑑2𝜌𝑣𝑑𝑡2
𝑅3 +1
30
𝑑3𝜌𝑣𝑑𝑡3
𝑅4 𝑑𝑒𝑣𝑑𝑡
−𝑑𝑣𝑑𝑡
+ 1
4𝜌𝑣𝑅
2 −3
20
𝑑𝜌𝑣𝑑𝑡
𝑅3 +1
20
𝑑2𝜌𝑣𝑑𝑡2
𝑅4 𝑑2𝑒𝑣𝑑𝑡2
−𝑑2𝑣𝑑𝑡2
+ −1
20𝜌𝑣𝑅
3 +1
30
𝑑𝜌𝑣𝑑𝑡
𝑅4 𝑑3𝑒𝑣𝑑𝑡3
−𝑑3𝑣𝑑𝑡3
+ 1
120𝜌𝑣𝑅
4 𝑑4𝑒𝑣𝑑𝑡4
−𝑑4𝑣𝑑𝑡4
+ ⋯
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.
123
𝑑𝑒𝑣𝑑𝑡
−𝑑𝑣𝑑𝑡
=𝑑
𝑑𝑡 𝑒𝑣 − 𝑣 =
1
𝜌𝑣2
𝑑𝜌𝑣𝑑𝑡
𝑝𝑣 −1
𝜌𝑣
𝑑𝑝𝑣𝑑𝑡
𝑑2𝑒𝑣𝑑𝑡2
−𝑑2𝑣𝑑𝑡2
=𝑑
𝑑𝑡2 𝑒𝑣 − 𝑣
= 1
𝜌𝑣2
𝑑2𝜌𝑣𝑑𝑡2
− 21
𝜌𝑣3 𝑑𝜌𝑣𝑑𝑡
2
𝑝𝑣 + 21
𝜌𝑣2
𝑑𝜌𝑣𝑑𝑡
𝑑𝑝𝑣𝑑𝑡
+ −1
𝜌𝑣 𝑑2𝑝𝑣𝑑𝑡2
𝑑3𝑒𝑣𝑑𝑡3
−𝑑3𝑣𝑑𝑡3
=𝑑
𝑑𝑡3 𝑒𝑣 − 𝑣
= 1
𝜌𝑣2
𝑑3𝜌𝑣𝑑𝑡3
+ 61
𝜌𝑣4 𝑑𝜌𝑣𝑑𝑡
3
− 61
𝜌𝑣3
𝑑𝜌𝑣𝑑𝑡
𝑑2𝜌𝑣𝑑𝑡2
𝑝𝑣
+ 31
𝜌𝑣2
𝑑2𝜌𝑣𝑑𝑡2
− 61
𝜌𝑣3 𝑑𝜌𝑣𝑑𝑡
2
𝑑𝑝𝑣𝑑𝑡
+ 31
𝜌𝑣2
𝑑𝜌𝑣𝑑𝑡
𝑑2𝑝𝑣𝑑𝑡2
+ −1
𝜌𝑣 𝑑3𝑝𝑣𝑑𝑡3
𝑑4𝑒𝑣𝑑𝑡4
−𝑑4𝑣𝑑𝑡4
=𝑑
𝑑𝑡4 𝑒𝑣 − 𝑣
= −81
𝜌𝑣3
𝑑𝜌𝑣𝑑𝑡
𝑑3𝜌𝑣𝑑𝑡3
+1
𝜌𝑣2
𝑑4𝜌𝑣𝑑𝑡4
− 241
𝜌𝑣5 𝑑𝜌𝑣𝑑𝑡
4
+ 361
𝜌𝑣4 𝑑𝜌𝑣𝑑𝑡
2 𝑑2𝜌𝑣𝑑𝑡2
− 61
𝜌𝑣3 𝑑2𝜌𝑣𝑑𝑡2
2
𝑝𝑣
+ 41
𝜌𝑣2
𝑑3𝜌𝑣𝑑𝑡3
+ 241
𝜌𝑣4 𝑑𝜌𝑣𝑑𝑡
3
− 241
𝜌𝑣3
𝑑𝜌𝑣𝑑𝑡
𝑑2𝜌𝑣𝑑𝑡2
𝑑𝑝𝑣𝑑𝑡
+ 61
𝜌𝑣2
𝑑2𝜌𝑣𝑑𝑡2
− 121
𝜌𝑣3 𝑑𝜌𝑣𝑑𝑡
2
𝑑2𝑝𝑣𝑑𝑡2
+ 41
𝜌𝑣2
𝑑𝜌𝑣𝑑𝑡
𝑑3𝑝𝑣𝑑𝑡3
+ −1
𝜌𝑣 𝑑4𝑝𝑣𝑑𝑡4
Substitution of the above relationships into the heat transfer rate results in
the following simplified equation.
124
𝑄 = 𝐹𝑛𝑒𝑡𝑅 + 𝑅 − 1 −𝑑𝐹𝑛𝑒𝑡𝑑𝑡
𝑅 +1
2
𝑑2𝐹𝑛𝑒𝑡𝑑𝑡2
𝑅2 −1
6
𝑑3𝐹𝑛𝑒𝑡𝑑𝑡3
𝑅3 +1
24
𝑑4𝐹𝑛𝑒𝑡𝑑𝑡4
𝑅4 + ⋯
+4𝜋𝑅2
3 −3𝑝𝑣𝑅
+ 𝑅 − 1 𝑅𝑑𝑝𝑣𝑑𝑡
−1
4𝑅2
𝑑2𝑝𝑣𝑑𝑡2
+1
20𝑅3
𝑑3𝑝𝑣𝑑𝑡3
−1
120𝑅4
𝑑4𝑝𝑣𝑑𝑡4
+ ⋯
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
3𝑅2 3𝜌𝑣𝑠𝑣𝑅
+ 𝑅 − 1 −𝑑 𝜌𝑣𝑠𝑣
𝑑𝑡𝑅 +
𝑑2 𝜌𝑣𝑠𝑣
𝑑𝑡2𝑅2 −
1
20
𝑑3 𝜌𝑣𝑠𝑣
𝑑𝑡3𝑅3
−1
120
𝑑4 𝜌𝑣𝑠𝑣
𝑑𝑡4𝑅4 + ⋯
The entropy transfer rate is defined using the following equation.
𝑄
𝑇𝑤=
𝐹𝑛𝑒𝑡𝑇𝑤
𝑅 + 𝑅 − 1
𝑇𝑤 −
𝑑𝐹𝑛𝑒𝑡𝑑𝑡
𝑅 +1
2
𝑑2𝐹𝑛𝑒𝑡𝑑𝑡2
𝑅2 −1
6
𝑑3𝐹𝑛𝑒𝑡𝑑𝑡3
𝑅3 +1
24
𝑑4𝐹𝑛𝑒𝑡𝑑𝑡4
𝑅4 + ⋯
+4𝜋𝑅2
3𝑇𝑤 −3𝑝𝑣𝑅
+ 𝑅 − 1 𝑅𝑑𝑝𝑣𝑑𝑡
−1
4𝑅2
𝑑2𝑝𝑣𝑑𝑡2
+1
20𝑅3
𝑑3𝑝𝑣𝑑𝑡3
−1
120𝑅4
𝑑4𝑝𝑣𝑑𝑡4
+ ⋯
125
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.
𝑑
𝑑𝑡 𝑠𝑣𝑚𝑣
𝑖𝑛
=4𝜋
3𝑅2 3𝜌𝑣𝑠𝑣𝑅
+ 𝑅 − 1 −𝑑 𝜌𝑣𝑠𝑣
𝑑𝑡𝑅 +
𝑑2 𝜌𝑣𝑠𝑣
𝑑𝑡2𝑅2 −
1
20
𝑑3 𝜌𝑣𝑠𝑣
𝑑𝑡3𝑅3
−1
120
𝑑4 𝜌𝑣𝑠𝑣
𝑑𝑡4𝑅4 + ⋯
Substitution of the equations defined above into the second law of
thermodynamics allows for the entropy generation rate to be defined.
126
𝑆 gen =4𝜋
3𝑅2 3𝜌𝑣𝑠𝑣𝑅
+ 𝑅 − 1 −𝑑 𝜌𝑣𝑠𝑣
𝑑𝑡𝑅 +
𝑑2 𝜌𝑣𝑠𝑣
𝑑𝑡2𝑅2 −
1
20
𝑑3 𝜌𝑣𝑠𝑣
𝑑𝑡3𝑅3
−1
120
𝑑4 𝜌𝑣𝑠𝑣
𝑑𝑡4𝑅4 + ⋯ −
𝐹𝑛𝑒𝑡𝑇𝑤
𝑅
− 𝑅 − 1
𝑇𝑤 −
𝑑𝐹𝑛𝑒𝑡𝑑𝑡
𝑅 +1
2
𝑑2𝐹𝑛𝑒𝑡𝑑𝑡2
𝑅2 −1
6
𝑑3𝐹𝑛𝑒𝑡𝑑𝑡3
𝑅3 +1
24
𝑑4𝐹𝑛𝑒𝑡𝑑𝑡4
𝑅4 + ⋯
−4𝜋𝑅2
3𝑇𝑤 −3𝑝𝑣𝑅
+ 𝑅 − 1 𝑅𝑑𝑝𝑣𝑑𝑡
−1
4𝑅2
𝑑2𝑝𝑣𝑑𝑡2
+1
20𝑅3
𝑑3𝑝𝑣𝑑𝑡3
−1
120𝑅4
𝑑4𝑝𝑣𝑑𝑡4
+ ⋯
−4𝜋
3𝑅2 3𝜌𝑣𝑠𝑣𝑅
+ 𝑅 − 1 −𝑑 𝜌𝑣𝑠𝑣
𝑑𝑡𝑅 +
𝑑2 𝜌𝑣𝑠𝑣
𝑑𝑡2𝑅2 −
1
20
𝑑3 𝜌𝑣𝑠𝑣
𝑑𝑡3𝑅3
−1
120
𝑑4 𝜌𝑣𝑠𝑣
𝑑𝑡4𝑅4 + ⋯
Removal of common terms reduces the equation to the following.
𝑆 gen = −𝐹𝑛𝑒𝑡𝑇𝑤
𝑅
− 𝑅 − 1
𝑇𝑤 −
𝑑𝐹𝑛𝑒𝑡𝑑𝑡
𝑅 +1
2
𝑑2𝐹𝑛𝑒𝑡𝑑𝑡2
𝑅2 −1
6
𝑑3𝐹𝑛𝑒𝑡𝑑𝑡3
𝑅3 +1
24
𝑑4𝐹𝑛𝑒𝑡𝑑𝑡4
𝑅4 + ⋯
−4𝜋𝑅2
3𝑇𝑤 −3𝑝𝑣𝑅
+ 𝑅 − 1 𝑅𝑑𝑝𝑣𝑑𝑡
−1
4𝑅2
𝑑2𝑝𝑣𝑑𝑡2
+1
20𝑅3
𝑑3𝑝𝑣𝑑𝑡3
−1
120𝑅4
𝑑4𝑝𝑣𝑑𝑡4
+ ⋯
127
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
2𝑅
The derivative of entropy generation rate with respect to bubble radius is
shown below.
𝑑𝑆 𝑔𝑒𝑛𝑑𝑅
= 0 = −𝑑
𝑑𝑡 𝐹𝑛𝑒𝑡𝑇𝑤
𝑅 𝑑𝑅
𝑑𝑡
−𝑑
𝑑𝑡 𝑅 − 1
𝑇𝑤 −
𝑑𝐹𝑛𝑒𝑡𝑑𝑡
𝑅 +1
2
𝑑2𝐹𝑛𝑒𝑡𝑑𝑡2
𝑅2 −1
6
𝑑3𝐹𝑛𝑒𝑡𝑑𝑡3
𝑅3 +1
24
𝑑4𝐹𝑛𝑒𝑡𝑑𝑡4
𝑅4
+ ⋯ 𝑑𝑅
𝑑𝑡
−𝑑
𝑑𝑡
4𝜋𝑅2
3𝑇𝑤 −3𝑝𝑣𝑅
+ 𝑅 − 1 𝑅𝑑𝑝𝑣𝑑𝑡
−1
4𝑅2
𝑑2𝑝𝑣𝑑𝑡2
+1
20𝑅3
𝑑3𝑝𝑣𝑑𝑡3
−1
120𝑅4
𝑑4𝑝𝑣𝑑𝑡4
+ ⋯ 𝑑𝑅
𝑑𝑡
Executing the derivatives and simplifying the resulting equation leads to
the following equation.
𝑑𝑆 𝑔𝑒𝑛
𝑑𝑅= 0 =
1
𝑇𝑤 −
𝑅
𝑅 𝐹𝑛𝑒𝑡 + −2 + 𝑅 + 𝑅
𝑅
𝑅 𝑑𝐹𝑛𝑒𝑡
𝑑𝑡+ 2𝑅 − 𝑅𝑅 −
𝑅
𝑅 −
1
2𝑅2 𝑅
𝑅 𝑑2𝐹𝑛𝑒𝑡
𝑑𝑡 2 + −𝑅2 +1
2𝑅2𝑅 +
1
2
𝑅2
𝑅 +
1
6𝑅3 𝑅
𝑅 𝑑3𝐹𝑛𝑒𝑡
𝑑𝑡 3 + 1
3𝑅3 −
1
6𝑅3𝑅 −
1
6
𝑅3
𝑅 −
1
24𝑅4 𝑅
𝑅 𝑑4𝐹𝑛𝑒𝑡
𝑑𝑡 4 + −1
24𝑅4 +
1
24
𝑅4
𝑅 𝑑5𝐹𝑛𝑒𝑡
𝑑𝑡 5 + ⋯ +4𝜋𝑅
3𝑇𝑤 6𝑅 + 3𝑅
𝑅
𝑅 𝑝𝑣 + 6𝑅 − 3𝑅𝑅 −
𝑅2 𝑅
𝑅 𝑑𝑝𝑣
𝑑𝑡+ −2𝑅2 + 𝑅2𝑅 +
𝑅2
𝑅 +
1
4𝑅3 𝑅
𝑅 𝑑2𝑝𝑣
𝑑𝑡 2 + 1
2𝑅3 −
1
4𝑅3𝑅 −
1
4
𝑅3
𝑅 −
1
20𝑅4 𝑅
𝑅 𝑑3𝑝𝑣
𝑑𝑡 3 +
−1
10𝑅4 +
1
20𝑅4𝑅 +
1
20
𝑅4
𝑅 +
1
120𝑅5 𝑅
𝑅 𝑑4𝑝𝑣
𝑑𝑡 4 + 1
120𝑅5 −
1
120
𝑅5
𝑅 𝑑5𝑝𝑣
𝑑𝑡 5 + ⋯
128
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]
129
%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]
%d5R 5th Derivative of Bubble Radius [m/s^5]
%d6R 6th Derivative of Bubble Radius [m/s^6]
%d7R 7th Derivative of Bubble Radius [m/s^2]
%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;
a=45;
%Define Experimental Data Sets
130
%**********************************************************************
****
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;
end
%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);
131
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;
132
[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;
end
%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;
end
133
%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,
i));
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;
end
end
end
%Define Method of Analysis (Pressure Method or Net Force Method)
%**********************************************************************
****
model=menu('Select a Model for Analysis','Pressure Method','Net Force
Method');
134
%**********************************************************************
****
%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);
end
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')
end
%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)
\n')
fprintf('**************************************************************
************\n')
elseif Data==3
fprintf('\n Data of Cole and Shulman
(1966) \n')
135
fprintf('**************************************************************
************\n')
end
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;
end
%Plot Results
%**************************************************************
figure
plot(t,Res(j,:))
axix auto
title('Residual vs. Time')
xlabel('Time, sec')
ylabel('Residual')
set(gcf,'color','w')
%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))
136
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')
end
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;
137
D=ones(1,max(size(A)))*3.912008E-04;
end
%******************************************************************
%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;
end
elseif submethod==2
for j=1:max(size(A));
C(j)=-0.894132*log(B(j))+4.010944;
D(j)=4.127304E-03*exp(-1.036544*A(j));
end
elseif submethod==3
for j=1:max(size(A));
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;
end
end
end
%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');
end
if Data==1
fprintf('\n Data of Van Stralen, Cole, Sluyter,
and Sohal (1975) \n')
fprintf('**************************************************************
************\n')
elseif Data==2
138
fprintf('\n Data of Ellion (1954)
\n')
fprintf('**************************************************************
************\n')
elseif Data==3
fprintf('\n Data of Cole and Shulman
(1966) \n')
fprintf('**************************************************************
************\n')
end
%Print System Properties
%******************************************************************
for j=1:max(size(A))
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));
%Define Polynomial Equations for Rayleigh and Plesset-Zwick
139
%**************************************************************
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)];
end
%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;
else
Rpre(j)=Rpre(j)+dr;
test=1;
end
step=step+1;
end
end
%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));
end
fprintf('\n');
end
%**********************************************************************
****
%Net Force Method
%**********************************************************************
****
140
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);
end
end
%**********************************************************************
%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 Data of Van Stralen, Cole, Sluyter,
and Sohal (1975) \n')
fprintf('**************************************************************
************\n')
elseif Data==2
fprintf('\n Data of Ellion (1954)
\n')
fprintf('**************************************************************
************\n')
elseif Data==3
fprintf('\n Data of Cole and Shulman
(1966) \n')
fprintf('**************************************************************
************\n')
141
end
%Initialize Vectors for Analysis
%******************************************************************
for j=1:m
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));
%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);
%Define Entropy Generation Rate
142
%**********************************************************
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;
end
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')
Rpre(j)=interp1(Rworking(j,:),R(j,:),0,'spline');
%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')
end
%**********************************************************************
%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 Data of Van Stralen, Cole, Sluyter,
and Sohal (1975) \n')
fprintf('**************************************************************
************\n')
143
elseif Data==2
fprintf('\n Data of Ellion (1954)
\n')
fprintf('**************************************************************
************\n')
elseif Data==3
fprintf('\n Data of Cole and Shulman
(1966) \n')
fprintf('**************************************************************
************\n')
end
for j=1:m
DenV(j,:)=DenVap(Tvap(j),Pbulk(j))*ones(1,n);
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:
Variable\n')
fprintf('Vapor Temperature: %f
[K]\n',Tvap(j))
fprintf('Vapor Density:
Variable\n')
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));
for i=1:n
144
%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);
end
%Define Vapor Density Behavior
%**************************************************************
for i=1:n
DenWork(j,i+3)=DenV(j,i);
end
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;
end
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)-
145
(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)-
9*R(j,i)^2*dR(j,i)*d2DenV(j,i)-R(j,i)^3*d3DenV(j,i))-
pi*St(j)*(S(j)*d3R(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)-
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+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,
i));
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)-
15*R(j,i)^2*dR(j,i)*d4DenV(j,i)-R(j,i)^3*d5DenV(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;
end
%Define Entropy Generation Rate
%**********************************************************
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-
146
(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))));
end
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')
for i=1:n
Rpre(j)=interp1(Rworking(j,:),R(j,:),0,'spline');
end
%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));
147
fprintf('**************************************************************
*************\n')
end
end
end
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')
set(gcf,'color','w')
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')
set(gcf,'color','w')
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.
148
function [Tsat]=SatTemp(P)
%Revised Release on the IAPWS Industrial Formulation 1997 for the
%Thermodynamic Properties of Water and Steam (The revision only relates
to
%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);
149
function [Psat]=SatPress(Tamb)
%Revised Release on the IAPWS Industrial Formulation 1997 for the
%Thermodynamic Properties of Water and Steam (The revision only relates
to
%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];
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;
150
function [Cp,Cv]=SpecHeatLiq(T,P)
%Revised Release on the IAPWS Industrial Formulation 1997 for the
%Thermodynamic Properties of Water and Steam (The revision only relates
to
%the extension of region 5 to 50 MPa)
%
%August 2007
%
%Section 8.3 - The Saturation-Temperature Equation (Backward Equation)
Pstar=16.53; %Reference Pressure, MPa
Tstar=1386; %Reference Temperature, K
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);
end
Cp=R*(-Tau^2)*GammaTauTau;
151
Cv=R*((-Tau^2)*GammaTauTau+(GammaPI-Tau*GammaPITau)^2/GammaPIPI);
152
function [Cp,Cv]=SpecHeatVap(T,P)
%Revised Release on the IAPWS Industrial Formulation 1997 for the
%Thermodynamic Properties of Water and Steam (The revision only relates
to
%the extension of region 5 to 50 MPa)
%
%August 2007
%
%Section 6 - Equations for Region 2
Tstar=540; %Reference Temperature, K
Pstar=1; %Reference Pressure, MPa
R=0.461526; %Gas Constant, kJ/kg-K
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);
153
GammaTauTau0=GammaTauTau0+n0(i)*J0(i)*(J0(i)-1)*Tau^(J0(i)-2);
end
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)-
1);
end
Cp=R*(-Tau^2*(GammaTauTau0+GammaTauTauR));
Cv=R*(-Tau^2*(GammaTauTau0+GammaTauTauR)-(1+PI*GammaPIR-
Tau*PI*GammaPITauR)^2/(1-PI^2*GammaPIPIR));
154
function [dl]=DenLiq(T,P)
%Revised Release on the IAPWS Industrial Formulation 1997 for the
%Thermodynamic Properties of Water and Steam (The revision only relates
to
%the extension of region 5 to 50 MPa)
%
%August 2007
%
%Section 8.3 - The Saturation-Temperature Equation (Backward Equation)
Pstar=16.53; %Reference Pressure, MPa
Tstar=1386; %Reference Temperature, K
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);
end
vl=R*T*PI*GammaPI/(1000*P);
155
dl=1/vl;
156
function [dv]=DenVap(T,P)
%Revised Release on the IAPWS Industrial Formulation 1997 for the
%Thermodynamic Properties of Water and Steam (The revision only relates
to
%the extension of region 5 to 50 MPa)
%
%August 2007
%
%Section 6 - Equations for Region 2
Tstar=540; %Reference Temperature, K
Pstar=1; %Reference Pressure, MPa
R=0.461526; %Gas Constant, kJ/kg-K
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);
157
GammaTauTau0=GammaTauTau0+n0(i)*J0(i)*(J0(i)-1)*Tau^(J0(i)-2);
end
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)-
1);
end
vv=R*T*(PI*(GammaPI0+GammaPIR))/(P*1000);
dv=1/vv;
158
function [hl]=EnthLiq(T,P)
%Revised Release on the IAPWS Industrial Formulation 1997 for the
%Thermodynamic Properties of Water and Steam (The revision only relates
to
%the extension of region 5 to 50 MPa)
%
%August 2007
%
%Section 8.3 - The Saturation-Temperature Equation (Backward Equation)
Pstar=16.53; %Reference Pressure, MPa
Tstar=1386; %Reference Temperature, K
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);
end
hl=R*T*Tau*GammaTau;
159
160
function [hv]=EnthVap(T,P)
%Revised Release on the IAPWS Industrial Formulation 1997 for the
%Thermodynamic Properties of Water and Steam (The revision only relates
to
%the extension of region 5 to 50 MPa)
%
%August 2007
%
%Section 6 - Equations for Region 2
Tstar=540; %Reference Temperature, K
Pstar=1; %Reference Pressure, MPa
R=0.461526; %Gas Constant, kJ/kg-K
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);
161
GammaTauTau0=GammaTauTau0+n0(i)*J0(i)*(J0(i)-1)*Tau^(J0(i)-2);
end
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)-
1);
end
hv=R*T*(Tau*(GammaTau0+GammaTauR));
function [hfg]=LatHeatVap(T,P)
%Revised Release on the IAPWS Industrial Formulation 1997 for the
%Thermodynamic Properties of Water and Steam (The revision only relates
to
%the extension of region 5 to 50 MPa)
%
%August 2007
%
%Section 6 - Equations for Region 2
Tstar=540; %Reference Temperature, K
Pstar=1; %Reference Pressure, MPa
R=0.461526; %Gas Constant, kJ/kg-K
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,-
162
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
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);
end
Gamma0=log(PI)+Gamma0;
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);
end
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-
163
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)-
1);
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);
end
hl=R*T*Tau*GammaTau2
%Latent Heat of Vaporization
hfg=hl-hv;
164
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;
165
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;
else
S=C(6)/dTbar^(3/5);
end
k0=0;
for i=1:4
k0=k0+Tbar^(1/2)*a(i)*Tbar^(i-1);
end
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);
166
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);
end
At=sum(A);
Vbar0=100*sqrt(Tbar)/At;
%Solve for Vbar1
B=0;
for i=1:6
for j=1:7
B=B+Dbar*(1/Tbar-1)^(i-1)*Table2(j,i)*(Dbar-1)^(j-1);
end
end
Vbar1=exp(B);
%Solve for Vbar2
Vbar2=1;
%Solve for Vbar
Vbar=Vbar0*Vbar1*Vbar2;
%Solve for Viscosity
V=Vbar*Vstar;