Entropy generatioin study for bubble separation in pool boiling

ENTROPY GENERATION STUDY FOR BUBBLE SEPARATION

IN POOL BOILING

A Project

Presented to the

Faculty of

California State Polytechnic University, Pomona

In Partial Fulfillment

Of the Requirements for the Degree

Master of Science

In

Mechanical Engineering

By

Jeffrey William Schultz

2010

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.

iii

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%.

iv

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

v

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

vi

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
Sluyter, & Sohal, 1975). ................................................................ 25

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

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

Plesset-Zwick Based General Solution. ........................................ 39

Plesset-Zwick Based Modified General Solution. .......................... 43

Table 11. Values of C for the General Solution Derived Using MRG
Sluyter, & Sohal, 1975). ................................................................ 51

Table 12. Values of D for the General Solution Derived Using MRG
Sluyter, & Sohal, 1975). ................................................................ 52

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

vii

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

viii

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

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

ix

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

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

x

NOMENCLATURE

General Symbols

�� parameter for Rayleigh Equation

�� Archimedes number

�� constant for Plesset-Zwick Equation

�� parameter for Plesset-Zwick Equation

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

�� constant of general solution

�� diameter [m]

�� diameter [m] or constant of general solution

�� internal energy per unit mass [J/kg]

�� internal energy [J]

�� energy change rage [W]

�� force [N]

�� buoyant force [N]

�� drag force [N]

�� inertia force [N]

�� net force[N]

�� excess pressure force [N]

�� surface tension force [N]

�� gravitational acceleration [m/s2]

�� enthalpy [J/kg]

xi

�� latent heat of vaporization [J/kg]

�� enthalpy [J]

�� Jakob number

�� thermal conductivity [W/m-K]

�� bubble mass [kg]

�� mass flow rate [kg/s]

�� pressure [Pa]

��∞ system pressure [Pa]

�� Prandtl number

��" heat transfer per area [W/m2]

�� heat transfer [J]

�� heat transfer rate [W]

�� bubble radius [m]

�� bubble growth rate [m/s]

�� radial acceleration of bubble [m/s2]

�� entropy [J/kg-K]

�� entropy change rate [W/K]

�� entropy generation rate [W/K]

�� dimensionless scaling factor for surface tension force

�� temperature [K]

��∞ uniform system temperature [K]

�� (��∞ ) saturation temperature at ��∞ [K]

xii

∆�� superheat [K]

�� time [s]

�� specific volume of liquid [m3/kg]

�� bubble volume [m3]

�� work [J]

�� rate of work [W]

Greek Symbols

�� thermal diffusivity of liquid

�� contact angle

�� viscosity

�� density of liquid [kg/ m3]

�� subcooling factor

�� surface tension [N/m]

Subscripts

�� base

�� departure

�� interface

�� liquid

�� vapor

�� wait

�� wall

xiii

Superscripts

�� modified term

+ dimensionless

∗ modified term

xiv

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) �� 1 2 Correlation balances
�� = 0.0208�� buoyancy force with
��(�� − �� )
surface tension force

(Staniszewski, 1959) �� 1/2 �� Correlation includes
�� = 0.0071�� 1 + 0.435 affect of bubble growth
��(�� − �� ) ��
rate

(Zuber, 1959) 1/3
�� 1 3 6�� − �� ∞
�� =
��(�� − �� ) ��"

1


(Ruckenstein, 1961) ��
and (Zuber, 1964) 1 2 1 3
3�� 2 �� 1 2 ��1 2
�� −�� 1 2
= ��4 3
�� 3 2 �� − ��

(Borishanskiy & Fokin, 2 1 2
��
Heat transfer and �� = − + + �� 2
hydrodynamics in 2 2
steam generators, �� = ��
1963)
6 �� 0.4 ��"
�� =
�� −��

(Cole & Shulman, 1000 �� 1 2
1966a) �� = ��
�� (�� − �� )

(Cole, 1967) �� 1 2
�� = 0.04��
��(�� − �� )

(Cole & Rohsenow, �� 1 2 Correlation for low
1969) �� = �� 5 4
pressure systems
��(�� − �� )
�� ,��
�� =
��
�� = 1.5��10−4 for water
�� = 4.65��10−4 for fluids other than water

�� Correlation includes
∗ 1 3 2 3 dynamic relationship
(Golorin, Kol'chugin, & 1.65�� 15.6�� − ��
= +
Zakharova, 1978) �� − �� − ��

�� ∗ = 6.0��10−3 ��
�� = 6.0

(Kutateladze & �� 1 2 ��1 ≤ 0.06
Gogonin, 1980) �� =. 25 1 + 105 ��1 1 2
�� − ��

�� − �� 3 2 −1
��1 =
�� 2 �� − ��

−0.46 −1 3
(Borishanskiy, ��
Danilova, Gotovskiy, �� = 5��105
��
Borishanskaya,
Danilova, &
Kupriyanova, 1981)

(Kocamustafaogullari, �� 1 2 �� − �� 0.9 Expansion of Fritz
1983) �� = 2.64��10−5 correlation to include
��(�� − �� ) ��
high pressure systems

(Jensen & Memmel, �� 1 2 Correlation is a
1986) �� = 0.19 1.8 + 105 ��1 2 3
proposed improvement
��(�� − �� )
to (Kutateladze &

2

Gogonin, 1980)

(Gorenflo, Knabe, & 1 3 1 2 4 3 Correlation for high
��4 �� 2 2��
Bieling, 1986) �� = ��1 1+ 1+ heat fluxes
�� 3��

(Stephan, 1992) 2 1 2 Correlation valid for
�� 1 2 �� 1
�� = 0.25 1+ 2
��(�� − �� ) �� 1
5��10−7 ≤
��
≤ 0.1

(Kim & Kim, 2006) �� 1 2 Correlation valid for
�� = 0.1649��0.7 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 Figure 1. Forces
Acting on a Bubble.
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

3

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.

��
��∞ ��∞

��
��

��
�� (A) ( (B) ( ��
��∞ ��∞
A) B)

��

��

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

4

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

Buoyancy Force 4��3
�� = �� − ��
3

Excess Pressure Force �� = �� sin ��

Inertia Force �� 11 4��3 ��
�� = �� ≅ ��
�� 16 3 ��

Surface Tension Force �� = 2�� sin ��

Drag Force ��
�� = �� , �� = 45
4 ��

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

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

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

5

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.

1 ��
�� = − �� + �� − ��
��

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

6

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
d
thermodynamics, it is possible to determine ��
dt
��
the rate of heat transfer. Heat transferred

to the bubble must result in changes to the
��
accumulated energy of the bubble, work
Figure 3. Balance of Energy for
First Law of Thermodynamics
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..

��
�� = �� + �� − ��
��
��

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.

7

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:

�� d
�� = �� − − ��
�� dt
��

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.

8

 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.

9

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.
��2
�� = ��
��1

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

10

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

relationship.

�� = 4��2 ��

Application of these relationships leads to the following equation for work

performed by the bubble on the surrounding fluid.
��
�� = 4��
�� − ��∞ ��2 �� .

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.

4�� 3 1 �� 1 �� 2 �� 2 1 �� 3 �� 3
�� = �� − ��∞ − �� + �� − �� + ⋯
3 4 �� 20 �� 2 120 �� 3

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.

4��2
�� = 3 �� − ��∞ ��
3

�� 1 ��2 �� 2 1 ��3 �� 3 1 ��4 �� 4
+ �� − 1 − �� + �� − �� + �� + ⋯
�� 4 �� 2 20 �� 3 120 �� 4

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.

2��
�� = �� +
��

11

2
2�� 3 �� 2 ��
�� = ��∞ + + �� + ��
�� 2 �� 2

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.

�� ≅ 4�� ∞ − ��∞ ��2

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.
R R
�� = �� = �� = 4π �� 2 ��
0 0

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.

12

4��3 1 �� 1 ��2 �� 2 �� 2
�� = �� − �� + �� + �� +2 + ��
3 4 �� 20 �� 2 �� 2

1 �� 3 �� 2 �� 2 �� 3 �� 3
− �� +3 +3 + �� + ⋯
120 �� 3 �� 2 �� 2 �� 3

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.

4��2
�� = 3��
3

��
+ �� − 1 − �� + ��
��

1 ��2 �� 2 �� 2
+ �� +2 + ��
4 �� 2 �� 2

1 ��3 �� 2 �� 2 �� 3 �� 3
− �� +3 +3 + ��
20 �� 3 �� 2 �� 2 �� 3

1 ��4 �� 3 �� 2 �� 2 �� 3 ��
+ �� +4 +6 2 +4
120 �� 4 �� 3 �� 2 �� 3

�� 4 �� 4
+ �� + ⋯
�� 4

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.

�� = 4�� 2 ��

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

13

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.
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.

�� 4��2
�� = 3��
�� 3

��
+ �� − 1 − �� + ��
��

1 ��2 �� 2 �� 2
+ �� +2 + ��
4 �� 2 �� 2

1 ��3 �� 2 �� 2 �� 3 �� 3
− �� +3 +3 + ��
20 �� 3 �� 2 �� 2 �� 3

1 ��4 �� 3 �� 2 �� 2 �� 3 ��
+ �� +4 +6 2 +4
120 �� 4 �� 3 �� 2 �� 3

�� 4 �� 4
+ �� + ⋯
�� 4

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.

��
�� = 4�� 2 ��
��

Heat Transfer

14

Substitution of the equations derived above into the first law of

thermodynamics produce the following equation for heat transfer rate.

4��2
�� = 3 �� − ��∞ �� + 3�� − ��
3

�� 1 �� 2 �� 2 1 ��3 �� 3 1 ��4 �� 4
+ �� − 1 − �� + �� − �� + ��
�� 4 �� 2 20 �� 3 120 �� 4

�� 1 ��2 �� 2 1 ��3 �� 3 1 ��4 �� 4
+ − �� + �� − �� + �� − ��
�� 4 �� 2 20 �� 3 120 �� 4

1 �� 2 3 ��2 �� 3 1 ��3 �� 4 ��
+ −�� + �� − �� + �� −
2 �� 20 �� 2 30 �� 3 ��

1 3 �� 3 1 ��2 �� 4 ��2 �� 2 ��
+ �� 2 − �� + �� −
4 20 �� 20 �� 2 �� 2 �� 2

1 1 �� 4 ��3 �� 3 ��
+ − �� 3 + �� −
20 30 �� 3 �� 3

1 ��4 �� 4 ��
+ �� 4 − +⋯
120 �� 4 �� 4

This equation can be further reduced application of the definition of

enthalpy.

��
�� − �� = −�� = −
��

Substitution of the above equation and its derivatives allows the heat

transfer rate equation for a spherical bubble to be reduced.

�� = −4��∞ ��2 ��

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

15

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.
R R
��
�� = �� = �� = 4π �� 2 ��
�� 0 �� 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.

4�� 3 1 �� 1 �� 2 �� 2 1 ��3 �� 3
�� = �� − �� + �� − �� + ⋯
3 4 �� 20 �� 2 120 �� 3

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

leads to the following equation.

16

4�� 2
�� = �� 3��
3

�� 2 �� 2 1 �� 3 �� 3
+ �� − 1 − �� + �� − ��
�� 2 20 �� 3

1 �� 4 �� 4
− �� + ⋯
120 �� 4

By applying the assumption of constant vapor properties at the saturation

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

�� = 4�� 2 ��

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.

�� 4��
=− �� 2 ��
�� ∞

Net Entropy Flow Rate

The net entropy flow rate is defined as follows.
R
��
�� = 4π �� 2 ��
�� 0
��

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.

17

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.

4��
�� = �� 2 ��
�� ∞

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.

�� 4�� 4��
=0= ��∞ ��2 �� = ��∞ ��2 ��
��

This reduces to the following equation.

4��
0= ��∞ �� 2�� + ��
��

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 = �� + 2��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.

18

�� = −3�� −2�� + 3�� 1 3

For this solution to be useful, variables �� and �� 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

19

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 ��∞

 Surface Tension of Liquid Saturated liquid at ��∞

 Specific Heat of Liquid Saturated liquid at ��∞

 Density of Liquid Saturated liquid at ��∞

 Latent Heat of Vaporization Saturated liquid at ��∞

 Density of Vapor Saturated liquid at ��∞

 Vapor Pressure Saturated liquid at ��∞

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

20

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.
2
�� 2 �� 3 �� 1 2��
�� 2 + = �� − ��∞ −
�� 2 ��

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

following form by utilization of a linearirzed Clausis-Clapeyron equation.
2
�� 2 �� 3 �� ∞ − ��
�� 2 + =
�� 2 ��

Integration of the above equation leads to the Rayleigh equation for

bubble growth

1 2
2 �� ∞ − ��
�� = ��
3 ��

This equation is commonly written as follows.

�� = ��

��

1 2
�� ∞ − �� 2
�� = �� , �� =
�� 3

21

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.

��
= ��
��

�� 2 ��
=0
�� 2

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.

�� + 2��2 = 0

Substitution of the Rayleigh equations into the equation above produces

the following relationship.

2��2 = 0

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

non-zero value of ��. 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

22

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

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

�� = −3�� −�� + �� 1 3

�� :

��
= −�� −�� −3�� −�� + �� −1 3
��

Utilization of the general solution requires that constants �� and �� 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.

�� = �� =��
��=��

−1
��
=−
�� =��
�� =��

By substation of the appropriate equations into the boundary conditions

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

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

�� = −�� 2

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

of the Rayleigh equation.

1
�� = −�� 2
��

23

Solving for constant �� requires experimental data including system

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

sets, a value for constant �� can be defined.

�� 1 2
−��
�� ,��
�� =
��
��=1

To define the constant ��, 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).
Experimental
Departure Radius
of (Van Stralen,
Bubble Number Cole, Sluyter, & Ja ��
Sohal, 1975)
�� , m

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 �� by substitution of constant �� into the

general solution and rearranging.

24

3�� −��
�� ,�� 3 + �� ,��
��
�� =
��
��=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 �� derived above. Results are shown in Table 4.

Table 4. Values of D for the General Solution Derived from Rayleigh Equation with
Experimental
Departure Radius
of (Van Stralen,
Sohal, 1975)
�� , m

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.

1 3
1
�� = − �� + 1.683431E − 05
8354.181454

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

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

25

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

departure time is replaced by utilizing the Rayleigh equation.

3�� −��
0 = �� 3 + �� − ��
��

Substitution of the constants �� and �� results in the following equation.

1
0 = �� 3 + �� − 1.683431E − 05
8354.181454 ∗ ��

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.
Experimental
Departure Radius
Predicted
of (Van Stralen,
Bubble Number Ja Departure Radius % Error
Cole, Sluyter, &
Sohal, 1975) �� , m
�� , m

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

26

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.

27

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 �� and ��. Results are improved

by modifying constants �� and �� to be functions of system values �� and/or ��.

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

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

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

approximated by the following equation.

�� = 7.459635��(��) + 2.607226

This equation fits the values of �� presented in Table 3 with a ��2 value of

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

28

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

calculated constant �� are now used to calculate modified values for constant ��.

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

�� and ��, a relationship is determined. Constant �� is best estimated with a ��2

value of 0.9971 by the following equation.

�� = 2.278040 −11 ��2 + 6.485067�� −09 �� − 3.367751��(−07)

Comparison of constant �� with constant �� fails to create an equally good

curve fit.

The derived equations for constants �� and �� are substituted into the

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

the following form.

�� = −3�� − 7.459635 �� (��)+2.607226 �� + 2.278040 −11 ��2 + 6.485067�� −09 ��

1 3
− 3.367751��(−07)

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

relationship is derived.

3 3�� − 7.459635 �� (��)+2.607226
0 = �� + �� − 2.278040 −11 ��2
��

− 6.485067�� −09 �� + 3.367751��(−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

�� and �� 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.

29

Table 6. Error Analysis of Predicted Departure Radii based on Rayleigh Based Modified
General Solution.
Experimenta
l Departure
Radius of
(Van Predicted
Bubble Stralen, Departure
Ja �� Cole, Radius % Error
Number
Sluyter, &
Sohal, �� , m
1975)
�� , m

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.

30

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

31

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

32

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.

1 2
12��
�� = �� 1 2
��

The equation is commonly written as follows.

�� = �� 1 2

��

1 2
12��
�� = ��
��
�� ,��
�� = �� − ��
�� ∞

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

number are rewritten as follows.

1 2
∗
12��
�� = ��
��
�� ,��
��∗ = �� − ��
��

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

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

�� 1 −1 2
= ��
�� 2

33

��2 �� 1
= − �� −3 2
�� 2 4

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 2 −1
�� = 0
4

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

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

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).

34

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.

35

�� = −3�� −�� + �� 1 3

��

��
= −�� −�� −3�� −�� + �� −1 3
��

Utilization of the general solution requires that constants �� and �� 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.

�� −�� =�� = �� =��

−1
��
=−
�� −�� =�� =��

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 ��.

�� = −�� 2�� 3 2

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

2�� 3
�� = −��
��2

Solving for constant �� requires experimental data including environmental

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

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

36

2�� ,�� 3
�� −��
�� 2
�� =
��
��=1

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

1975) is used to determine a value for ��. 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
Departure Radius
of (Van Stralen,
Sohal, 1975)
�� , m

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 �� defined, constant �� is solved for. By substitution of the

constant �� into the general solution, a solution for constant �� is determined.

3�� −��
�� ,�� 3 + �� ,�� 2
�� 2
�� =
��
��=1

37

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
Departure Radius
of (Van Stralen,
Sohal, 1975)
�� , m

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.

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

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.

3�� −��
0 = �� 3 + �� 2 − ��
��2

Substitution of the constants �� and �� results in the following equation.

38

Entropy generatioin study for bubble separation in pool boiling

Entropy generatioin study for bubble separation in pool boiling

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