Entropy generatioin study for bubble separation in pool boiling

678 views

Published on

0 Comments
0 Likes
Statistics
Notes
  • Be the first to comment

  • Be the first to like this

No Downloads
Views
Total views
678
On SlideShare
0
From Embeds
0
Number of Embeds
15
Actions
Shares
0
Downloads
14
Comments
0
Likes
0
Embeds 0
No embeds

No notes for slide

Entropy generatioin study for bubble separation in pool boiling

  1. 1. 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
  2. 2. ACKNOWLEDGEMENTS I would like to start by thanking Dr. Hamed Khalkhali for his continuedsupport throughout the investigation. This work would not have been possiblewithout his suggestion of the problem statement. His advice and push to look atthe problem in a different light has been greatly appreciated and helped drive thisinvestigation to a successful conclusion. To my wife Melissa goes my greatest appreciation for her continuedsupport throughout my work towards a Master of Science degree and especiallyduring my work on this investigation. She has helped make an extremely busyschedule over the last two years manageable and enjoyable. Additionally I would like to thank my parents Nancy and Charlie, sisterKristen, mother and father in-law Peggy and Ed, sister in-law Margaret, andbrother in-law Mark for their continued support and motivation. I would also like to thank Dr Rajesh Pendekanti and Dr Keshava Datta forproviding me with the initial motivation to pursue a Master of Science degree inMechanical Engineering. Throughout my progress in the program at CaliforniaState Polytechnic University, Pomona, they have provided me with advice,support, and flexibility at work to allow me to pursue this degree. iii
  3. 3. ABSTRACT The current entropy generation rate study of spherical bubbles undergoinggrowth in nucleate pool boiling produces a novel correlation for predicting bubbledeparture radii. Two models for entropy generation rate in spherical bubbles aredeveloped by modeling the work performed by a bubble as that of athermodynamic system, and as a function of the net force acting on the bubbleand the rate of bubble grow. While the derived entropy generation rate equationsfail to support the hypothesis presented in this paper, one of the two modelsleads to a novel correlation which predicts published experimental data within15%. iv
  4. 4. TABLE OF CONTENTSSignature Page ...................................................................................................... iiAcknowledgements .............................................................................................. iiiAbstract ................................................................................................................ ivTable of Contents ................................................................................................. vList of Tables ....................................................................................................... viiList of Figures ....................................................................................................... ixNomenclature ....................................................................................................... xiIntroduction ........................................................................................................... 1 Previous Work ........................................................................................ 1 Problem Statement ................................................................................. 6 Methodology ........................................................................................... 6 General Assumptions ............................................................................. 8Entropy 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
  5. 5. 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 .............................................................................................. 85Conclusions ........................................................................................................ 86Bibliography ........................................................................................................ 90Appendix A: Derivation of Entropy Generation Rate (Pressure Method) ............ 94Appendix B: Defining the General Solution ...................................................... 109Appendix C: Derivation of Entropy Generatoin Rate (Net Force Method) ........ 114Appendix D: MatLab Programs......................................................................... 128 vi
  6. 6. LIST OF TABLESTable 1. Departure Diameter Correlations .................................................... 1Table 2. Forces acting on a bubble prior to separation. ................................ 5Table 3. Values of C for the General Solution Derived from Rayleigh Equation with Experimental Data of (Van Stralen, Cole, Sluyter, & Sohal, 1975). ................................................................ 24Table 4. Values of D for the General Solution Derived from Rayleigh Equation with Experimental Data of (Van Stralen, Cole, Sluyter, & Sohal, 1975). ................................................................ 25Table 5. Error Analysis of Predicted Departure Radii based on Rayleigh Based General Solution. ................................................ 26Table 6. Error Analysis of Predicted Departure Radii based on Rayleigh Based Modified General Solution. .................................. 30Table 7. Values of C for the General Solution Derived Using Plesset- Zwick Equation with Experimental Data of (Van Stralen, Cole, Sluyter, & Sohal, 1975). ....................................................... 37Table 8. Values of D for the General Solution Derived Using Plesset- Zwick Equation with Experimental Data of (Van Stralen, Cole, Sluyter, & Sohal, 1975). ....................................................... 38Table 9. Error Analysis of Predicted Departure Radii based on Plesset-Zwick Based General Solution. ........................................ 39Table 10. Error Analysis of Predicted Departure Radii based on Plesset-Zwick Based Modified General Solution. .......................... 43Table 11. Values of C for the General Solution Derived Using MRG Equation with Experimental Data of (Van Stralen, Cole, Sluyter, & Sohal, 1975). ................................................................ 51Table 12. Values of D for the General Solution Derived Using MRG Equation with Experimental Data of (Van Stralen, Cole, Sluyter, & Sohal, 1975). ................................................................ 52Table 13. Error Analysis of Predicted Departure Radii based on MRG Based General Solution. ............................................................... 53 vii
  7. 7. Table 14. Error Analysis of Predicted Departure Radii based on MRG Based Modified General Solution. ................................................. 58Table 15. Comparison of Derived Equation with Experimental Data of (Cole & Shulman, 1966b) .............................................................. 61Table 16. Comparison of Derived Equation with Experimental Data of (Ellion, 1954). ................................................................................ 63Table 17. Alternative dimensionless scaling factors calculated from bubble departure correlations. ...................................................... 70Table 18. Net Force Derivatives.................................................................... 78Table 19. Vapor Pressure Derivatives .......................................................... 80Table 20. MRG Equation Derivatives. ........................................................... 81 viii
  8. 8. LIST OF FIGURESFigure 1. Forces Acting on a Bubble. ............................................................. 3Figure 2. Forces Acting on Bubble. (A) Buoyancy Force, (B) Excess Pressure Force, (C) Inertia Force, (D) Surface Tension Force), (E) Drag Force. ................................................................... 4Figure 3. Balance of Energy for First Law of Thermodynamics ...................... 7Figure 5. Comparison of Predicted Departure Radii from Rayleigh Based Equation and Experimental Departure Radii. ..................... 27Figure 6. Error Plot of Predicted Departure Radii from Rayleigh Based Equation. ....................................................................................... 28Figure 7. Comparison of Predicted Departure Radii from Modified Rayleigh Based Equation with Experimental Departure Radii. ...... 31Figure 8. Error Plot of Predicted Departure Radii using Rayleigh Based Modified Equation. ............................................................. 32Figure 9. Entropy Generation Rate vs. Bubble Radius for Experimental Data Obtained from (Van Stralen, Cole, Sluyter, & Sohal, 1975) (A=1.924969, B=0.267915). ................................................ 35Figure 10. Comparison of Predicted Departure Radii from Plesset- Zwick Based Equation with Experimental Departure Radii. .......... 40Figure 11. Error Plot of Predicted Departure Radii using Plesset-Zwick Based Equation. ............................................................................ 41Figure 12. Comparison of Predicted Departure Radii from Modified Plesset-Zwick Based Equation with Experimental Departure Radii. ............................................................................................. 44Figure 13. Error Plot of Predicted Departure Radii using Plesset-Zwick Based Modified Equation. ............................................................. 45Figure 14. Residual Value vs. Time for Experimental Data Obtained from (Van Stralen, Cole, Sluyter, & Sohal, 1975) (A=1.924969, B=0.267915). .......................................................... 48Figure 15. Comparison of Predicted Departure Radii from MRG Based Equation with Experimental Departure Radii. ................................ 54 ix
  9. 9. Figure 16. Error Plot of Predicted Departure Radii using MRG Based Equation ........................................................................................ 55Figure 17. Comparison of Predicted Departure Radii from Modified MRG Based Equation with Experimental Departure Radii. ........... 59Figure 18. Error Plot of Predicted Departure Radii using MRG Based Modified Equation ......................................................................... 59Figure 19. Comparison of Predicted Departure Radii with Experimental Data of (Cole & Shulman, 1966b). ................................................ 62Figure 20. Comparison of Predicted Maximum Radii with Experimental Data of (Ellion, 1954). ................................................................... 64Figure 21. Bubble Dimensions. ...................................................................... 69Figure 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
  10. 10. NOMENCLATUREGeneral 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
  11. 11. 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
  12. 12. ∆ 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
  13. 13. Superscripts modified term+ dimensionless∗ modified term xiv
  14. 14. INTRODUCTIONPrevious Work Bubble departure diameters in nucleate pool boiling have been studiedextensively both analytically and experimentally. In 1935, Fritz developed acorrelation for bubble departure diameter in nucleate boiling by balancingbuoyancy and surface tension forces for a static bubble (Fritz, 1935). Thisequation has since been expanded by other investigators. Bubble growth ratewas included in a correlation by (Staniszewski, 1959) after observing that bubbledeparture diameter is dependent on the rate at which the bubble grows. Othershave expanded the range of the Fritz correlation to low pressure systems suchas (Cole Rohsenow, 1969), while (Kocamustafaogullari, 1983) have expandedit to fit high pressure systems. More recently, (Gorenflo, Knabe, Bieling, 1986)established an improved correlation for bubble departure at high heat fluxes. Asummary 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
  15. 15. Source Departure Diameter Model Comments (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 ( − ) , = = 1.510−4 for water = 4.6510−4 for fluids other than water Correlation includes ∗ 1 3 2 3 dynamic relationship(Golorin, Kolchugin, 1.65 15.6 − = + Zakharova, 1978) − − ∗ = 6.010−3 = 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, = 5105 Borishanskaya, Danilova, Kupriyanova, 1981)(Kocamustafaogullari, 1 2 − 0.9 Expansion of Fritz 1983) = 2.6410−5 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
  16. 16. Source Departure Diameter Model Comments 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 ( − ) 1 510−7 ≤ ≤ 0.1 (Kim Kim, 2006) 1 2 Correlation valid for = 0.16490.7 high and low Jakob ( − ) numbers An evaluation of forces acting on bubbles forming + in normal and reduced gravitational fields was performedby (Keshock Siegel, 1964). Five forces acting onbubbles during growth while attached to a wall wereidentified as buoyancy, excess pressure, inertia, surfacetension and drag forces; each of which acts to keep thebubble 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 liquidof the fluid pool promote bubble departure. Buoyancy is aided by the excesspressure force which accounts for the vapor pressure acting on the region of wallwithin the bubble base diameter. This force aids in pushing the liquid vaporinterface away from the wall. The resulting equation for this force takes the sameform 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
  17. 17. radial direction away from the bubble boundary due to bubble growth. As thefluid is displaced, its viscosity creates resistance to bubble growth. It can beseen in the equations in Table 2 that the inertia force is scaled by a factor of11/16. The scaling factor was proposed by (Han Griffith, 1962) to approximatemass of affected fluid around the outer surface of the bubble. The surfacetension force accounts for the force of the liquid vapor interface with the wall andthe drag force accounts for the motion of the growing bubble through thesurrounding liquid. These forces can be seen graphically in Figure 2 along withera 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
  18. 18. Table 2. Forces acting on a bubble prior to separation. Force Equation Buoyancy Force 43 = − 3 Excess Pressure Force = sin Inertia Force 11 43 = ≅ 16 3 Surface Tension Force = 2 sin Drag Force = , = 45 4 Bubble separation occurs when buoyancy and excess pressure forcesexceed the net affects of the inertia, surface tension, and drag forces. The workof (Keshock Siegel, 1964) demonstrated that varying system conditionsproduce varying levels of influence for each of the forces associated with bubbledeparture. While extensive research has led to the development a number ofcorrelations for bubble departure diameter, a universal correlation is lacking. Itcan be seen by analysis of the correlations provided in the Table 1 that bubbledeparture is a function of many variables including contact angle, bubble growthrate, Jakob number, thermal diffusivity, system temperatures, pressures, and anumber 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 systemand fluid properties which can be utilized to model a wide range of systemconditions. 5
  19. 19. Problem Statement Is it possible to develop a correlation for bubble departure radius ordiameter in nucleate pool boiling by analyzing entropy generation rate duringbubble growth? It is suspected that the rate entropy generation reaches a maximum valueat 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 sphericalbubble in nucleate pool boiling is defined by the equation below. 1 = − + − As the entropy generation rate reaches a maximum value, the sum of rateof work performed by the bubble on its surroundings and the rate of change ofinternal energy minus the rate of energy transfer to the bubble reaches aminimum. It is believed that at this point, the bubble reaches a state ofequilibrium which results in departure or collapse in the case of sub-cooledboiling. If this suspicion is correct, an entropy generation analysis of bubblegrowth using the second law of thermodynamics may lead to a novel correlationfor determination of bubble departure radius.Methodology The maximum rate of entropy generation can be determined by taking thederivative of entropy generation rate with respect to bubble radius and setting itequal to zero. This method requires that the net heat transfer rate for the bubblebe substituted into the entropy generation equation. The proposed method is 6
  20. 20. accomplished by evaluation of the bubble using the first and second laws ofthermodynamics.First Law of Thermodynamics The first law of thermodynamics states that energy must be conserved. Byanalyzing the bubble using the first law of dthermodynamics, it is possible to determine dt the rate of heat transfer. Heat transferredto the bubble must result in changes to the accumulated energy of the bubble, work Figure 3. Balance of Energy for First Law of Thermodynamicsperformed on the bubble boundary, andenergy flow at the bubble boundary. In the case of a bubble undergoing growthat a wall, the net energy flows into the bubble. Energy flow out of the bubble istherefore ignored. The resulting first law equation for a bubble reduces to thefollowing equation which can be seen graphically in Error! Reference sourcenot found.. = + − It is possible to determine the rate of heat transfer by determining the rateof work performed, the change rate for the accumulated energy, and the rate ofnet energy flow into the bubble. Given this value, it is then possible to solve forentropy generation rate using the second law of thermodynamics. 7
  21. 21. Second Law of Thermodynamics The second law of thermodynamics is a statement to the irreversibility of asystem. It states that entropy of a system not at equilibrium will increase withtime. For a system with open boundaries such as a bubble, entropy generationrate is a function of the rate of entropy accumulation inside a control volume, theentropy transfer rate, and net entropy flow rate at the boundaries of the controlvolume. The second law of thermodynamics can be written as follows: d = − − dt Given the heat transfer rate determined by the first law ofthermodynamics, it is possible to determine entropy generation rate using thesecond law of thermodynamics.General Assumptions The following chapters cover the derivation of two novel correlations forbubble departure radius in nucleate pool boiling. These derivations will be madebased 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
  22. 22.  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 thecorrelations for the purpose of simplifying equations.  Vapor pressure is constant and equal to the saturation pressure of the bulk liquid pool. 9
  23. 23. ENTROPY GENERATION RATE STUDY (PRESSURE-VOLUME METHOD) A novel correlation is derived for bubble departure radius using the secondlaw of thermodynamics. In this chapter, work performed by the bubble ismodeled as the integral of the system pressure multiplied by the rate of changein 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 understandingof the heat transfer rate for the system. This is accomplished by solving the firstlaw of thermodynamics. Equations will be derived for the rate of work performedby 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 usingthe equation for work done by a thermodynamic system. This equation is afunction of the driving pressure and the change in system volume. 2 = 1 For a bubble undergoing growth in a pool, the driving pressure isequivalent to the difference between vapor pressure within the bubble and theinterface pressure of the fluid surrounding the bubble. For the purposes of thisinvestigation, the interface pressure is assumed equivalent to the bulk fluidpressure. Furthermore, the bubble is assumed to maintain a spherical shape 10
  24. 24. which allows for the change in volume to be replaced by the followingrelationship. = 42 Application of these relationships leads to the following equation for workperformed 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 doneby 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 isdetermined by taking the derivative of the above equation with respect to time.Doing so results in the following relationship. 42 = 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 aspherical bubble is a function of the rate of bubble growth and the rate at whichvapor pressure changes. It is possible to reduce this equation to a function ofconstant fluid properties and bubble growth rate by utilization of the Young-Laplace equation or the equation of motion for a spherical bubble. 2 = + 11
  25. 25. 2 2 3 2 = ∞ + + + 2 2 For the purposes of this derivation, the rate of work performed by a bubblewill be maintained as a function of the rate of bubble growth and rate of vaporpressure change. If vapor pressure is assumed constant and equivalent to the saturationpressure of the bulk liquid pool through the life of the bubble, the equation can bereduced to the following. ≅ 4 ∞ − ∞ 2 This assumption will not accurately model the rate of work performed by abubble growth within the inertia controlled region as this region is characterizedby rapidly changing vapor pressures. However, it is believed to be an acceptablemodel for bubbles undergoing growth in the heat diffuse region in which the rateof vapor pressure change is minimal.Energy Change Rate The Internal energy of a system is a measure of its total kinetic andpotential energy. In the case of a bubble, internal energy can be determined bymultiplying 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 soleads to the following series for internal energy. 12
  26. 26. 43 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 bedetermined by taking the derivative of the internal energy with respect to time. 42 = 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 andequal to the saturation pressure of the bulk liquid pool, the above equation issimplified to the following form. = 4 2 Energy Transfer Rate The energy transfer across the bubble boundary is defined as derivativewith respect to time of the total vapor mass flowing across the boundary 13
  27. 27. multiplied by the enthalpy per unit mass of the transferred vapor. For thepurposes of this analysis, the state of the vapor entering the bubble is assumedto equivalent to that of the vapor within the bubble. This implies that enthalpy ofthe 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 ofthe resulting series, the following equation for energy transfer rate is derived. 42 = 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 saturationpressure of the bulk liquid pool, the energy transfer rate reduces to a function ofbubble growth rate. = 4 2 Heat Transfer 14
  28. 28. Substitution of the equations derived above into the first law ofthermodynamics produce the following equation for heat transfer rate. 42 = 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 ofenthalpy. − = − = − Substitution of the above equation and its derivatives allows the heattransfer rate equation for a spherical bubble to be reduced. = −4∞ 2 It is noted that this solution is identical to the solution derived by applyingthe assumption of constant vapor pressure. The rate of heat transfer for aspherical bubble is a function of bulk pressure and radial growth behavior of the 15
  29. 29. bubble. The assumption that vapor pressure is constant is acceptable fordetermination of heat transfer rate. However, the rate at which vapor pressurechanges may still have a significant influence on the rate of work, rate ofaccumulated energy, and rate of energy transfer for a spherical bubbleundergoing growth in the inertia controlled region.Derivation of Entropy Generation Rate With heat transfer rate defined, it is possible to determine the rate ofentropy generation. Like determination of heat transfer rate, this requiresrelationships for the rate of entropy accumulation, entropy transfer rate, and thenet entropy flow rate.Entropy Accumulation Rate Entropy accumulation rate within the bubble is determined by taking thederivative 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 integrationby 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 entropyleads to the following equation. 16
  30. 30. 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 saturationpoint 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 bydividing the heat transfer rate by the wall temperature. By substitution of thederived 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 beequivalent to the state of the vapor accumulated within the bubble, the equationfor net entropy flow rate takes the same form as that derived for the entropychange rate. 17
  31. 31. Entropy Generation Rate The rate of entropy generation is determined by substitution of the derivedequations into the second law of thermodynamics. As it was previously noted,the net entropy flow rate and the entropy transfer rate are equivalent andtherefore cancel. The resulting entropy generation rate equation is a function ofonly the heat transfer rate. 4 = 2 ∞ If entropy generation rate reaches a maximum value at the point of bubbledeparture as hypothesized, the bubble departure radius can be determined bytaking the derivative of entropy generation rate with respect to bubble radius andsetting 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 departureradius if the hypothesis is true. 0 = + 22 By utilization of substitution methods, it can be shown that the generalsolution to the second order, non-linear differential equation takes the followingform. 18
  32. 32. = −3 −2 + 3 1 3 For this solution to be useful, variables and must be defined. Thisrequires the application of two boundary conditions. The first boundary conditioncan be determined by evaluation of experimental data for bubble departure radii.Comparison of the rate of change for both the general solution and theexperimental bubble at departure can be used to satisfy the second boundarycondition.Analysis of Second Order, Non-Linear Differential Equation Analysis of the second order, non-linear differential equation requires anunderstanding of growth behavior of bubbles during pool boiling. Bubblebehavior has been described by a number of researchers including (Rayleigh,1917), (Plesset Zwick, 1954), and (Mikic, Rohsenow, Griffith, 1970). In thefollowing sections, the equations derived by these researches will be utilized tosolve 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-atmosphericpressures will be utilized for comparison and refinement of the second order,non-linear differential equation. Application of the equations for bubble growthrequires an understanding of both fluid and vapor properties. For the purposesof 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
  33. 33. the saturation point of the bulk liquid pressure. While the vapor pressure within abubble is highly dynamic, it approaches the bulk liquid pressure as growthtransitions from an inertia controlled region to heat diffuse region. As describedby (Lien, 1969), the following liquid properties will be utilized to solve for theJakob number of the system as well as additional system constants for use in thegrowth 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 byutilization of equations defined by the International Association for the Propertiesof Water and Steam (Revised release on the IAPWS Industrial Formulation of1997 for the thermodynamic properties of water and steam, 2007) (IAPWSrelease on surface tension of ordinary water substance, 1994)Analysis Using Rayleigh Equation Bubble growth is defined by two distinct regions. Initial bubble growth isdescribed as inertia controlled growth in which high internal pressures producerapid growth of the bubble. Growth in this region is limited by the amount of 20
  34. 34. momentum available to displace the surrounding fluid. As internal pressuresdrop and the effect of inertia becomes negligible, bubbles transition to heatdiffuse growth in which bubble growth is driven primarily by heat transfer.Correlations have been developed for each of these regions to describe thebubbles growth characteristics. In 1917, Rayleigh derived an equation of motion for the flow of andincompressible fluid around spherical bubble. The equation takes the followingform. 2 2 3 1 2 2 + = − ∞ − 2 It was shown by Rayleigh that this equation can be reduced to thefollowing form by utilization of a linearirzed Clausis-Clapeyron equation. 2 2 3 ∞ − 2 + = 2 Integration of the above equation leads to the Rayleigh equation forbubble growth 1 2 2 ∞ − = 3 This equation is commonly written as follows. = 1 2 ∞ − 2 = , = 3 21
  35. 35. From the relationship above, it is possible to determine the radial velocityand acceleration of a growing bubble by taking the first and second derivativeswith respect to time. = 2 =0 2 Utilization of the bubble growth equations defined above, the secondorder, non-linear differential equation derived in the section above may be solvedby direction substitution. If the hypothesis that entropy generation reaches amaximum value at the point of bubble departure, the solution to the equationbellow describes the departure radius for a bubble undergoing pool boiling on awall. + 22 = 0 Substitution of the Rayleigh equations into the equation above producesthe following relationship. 22 = 0 By observation, it can be seen that the above equation is invalid for anynon-zero value of . Furthermore, the equation is not a function of bubble radius.Substitution of the Rayleigh equation into the second order, non-linear differentialequation does not produce a departure radius for a spherical bubble. While direct substitution of the Rayleigh equation and it derivative into thesecond order, non-linear differential equation does not produce a departureradius, utilization of the general solution may provide improved results. Earlier in 22
  36. 36. 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 bedetermined. This is accomplished by applying boundary conditions. For thepurposes of this analysis the boundary conditions will be defined at the time ofbubble departure. At departure, the radius defined by the Rayleigh equation willbe set equal to the radius defined by the general solution. Additionally, the slopeof both equations will be assumed perpendicular at this time. = = = −1 =− = = By substation of the appropriate equations into the boundary conditionsdefined above, a system of equations is created. This system of equations isreduced 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 utilizationof the Rayleigh equation. 1 = − 2 23
  37. 37. Solving for constant requires experimental data including systemconditions and the departure radius. By averaging results for experimental datasets, 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 inTable 3.Table 3. Values of C for the General Solution Derived from Rayleigh Equation withExperimental 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 thegeneral solution and rearranging. 24
  38. 38. 3 − , 3 + , = =1 Evaluation of the equation above is again accomplished by utilizingexperimental data published by (Van Stralen, Cole, Sluyter, Sohal, 1975) andthe average constant derived above. Results are shown in Table 4.Table 4. Values of D for the General Solution Derived from Rayleigh Equation withExperimental 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 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 linearrelationship for bubble radius that satisfies the second order, non-lineardifferential 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
  39. 39. of the departure time. In order to solve for bubble departure radius, thedeparture 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, positiveand should be in the scale of expected results. Evaluation of experimental datafrom (Van Stralen, Cole, Sluyter, Sohal, 1975) with the equation aboveproduces the predicted departure radii presented in Table 5.Table 5. Error Analysis of Predicted Departure Radii based on Rayleigh Based GeneralSolution. 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
  40. 40. Analysis of the table indicates that predicted departure radii fail toconsistently fit with experimental data. This is seen graphically in Figure 4 andFigure 5.Figure 4. Comparison of Predicted Departure Radii from Rayleigh Based Equation andExperimental Departure Radii. 27
  41. 41. Figure 5. Error Plot of Predicted Departure Radii from Rayleigh Based Equation. The large error associated with predicted bubble departure radii isassociated with the average values of constants and . Results are improvedby modifying constants and to be functions of system values and/or .While values and are now variable from system to system, they are constantfor a given a given boiling condition. By comparison of the calculated values of presented in Table 3 with system constant , it is determined that isapproximated by the following equation. = 7.459635() + 2.607226 This equation fits the values of presented in Table 3 with a 2 value of0.9579. Comparison of the constant with Jakob numbers for the experimental 28
  42. 42. systems fails to produce a satisfactory curve fit. The modified values ofcalculated 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 2value 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 goodcurve fit. The derived equations for constants and are substituted into thegeneral solution to create a new correlation. The modified general solution takesthe 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 followingrelationship 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 theRayleigh 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 ispresented in Table 6. 29
  43. 43. Table 6. Error Analysis of Predicted Departure Radii based on Rayleigh Based ModifiedGeneral 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 theRayleigh equation demonstrate an improved fit with experimental data. This isseen graphically in Figure 6 and Figure 7. 30
  44. 44. Figure 6. Comparison of Predicted Departure Radii from Modified Rayleigh BasedEquation with Experimental Departure Radii. 31
  45. 45. Figure 7. Error Plot of Predicted Departure Radii using Rayleigh Based ModifiedEquation. The departure radii predicted using the Rayleigh based modified generalsolution demonstrates greatly improved fit with experimental data of (VanStralen, Cole, Sluyter, Sohal, 1975). For bubbles having a Jakob numbergreater than 100, experimental departure radii are predicted within 15% resultsobtained experimentally. Results improve as the Jakob number for the systemgrows.Analysis Using Plesset-Zwick Equation The previous section evaluated the use of the Rayleigh solution to providea departure radius for a bubble growing on a wall in pool boiling. It was noted 32
  46. 46. that the Rayleigh equation is only effective for modeling bubble growth occurringwithin the inertia controlled growth region. To better understand the growthbehavior of a bubble, another equation is required. In 1954, Plesset and Zwick developed an equation to describe bubblegrowth occurring in the heat diffuse region. The derived equation is a function ofthe Jakob number of the system and the thermal diffusivity of the surroundingliquid. 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 Jakobnumber are rewritten as follows. 1 2 ∗ 12 = , ∗ = − The Plesset-Zwick equation is utilized to determine the radial velocity andacceleration of a bubble by taking its first and second derivatives. 1 −1 2 = 2 33
  47. 47. 2 1 = − −3 2 2 4 The Plesset-Zwick equations defined above is used to solve the secondorder, non-linear differential equation derived in this chapter by directsubstitution. 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 theequation 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 abubble at departure. Furthermore, it indicates that the suspicion that entropygeneration reaches a maximum value at bubble departure may be invalid. Thisis confirmed by plotting the calculated entropy generation rate against the bubbleradius for on experimental data set from (Van Stralen, Cole, Sluyter, Sohal,1975). 34
  48. 48. Figure 8. Entropy Generation Rate vs. Bubble Radius for Experimental Data Obtainedfrom (Van Stralen, Cole, Sluyter, Sohal, 1975) (A=1.924969, B=0.267915). As seen in Figure 8, the calculated entropy generation rate does not reacha maximum value. This failure to reach a maximum entropy generation rate maybe associated with the inability to effectively model bubble radius and vaporproperties within the bubble. While direct substitution fails to produce a reasonable solution andidentifies a failure of the calculated entropy generation rate to reach a maximumvalue, utilization of the Plesset-Zwick equation to solve general solution to thesecond order, non-linear differential equation may result in a correlation whichpredicts departure radii of bubbles undergoing nucleate pool boiling. Aspreviously shown, the general solution takes the following form. 35
  49. 49. = −3 − + 1 3 = − − −3 − + −1 3 Utilization of the general solution requires that constants and besolved. This is accomplished by applying boundary conditions. For the purposesof this analysis the boundary conditions are defined at the time of bubbledeparture. At departure, the radius defined by the Plesset-Zwick equation is setequal to the radius defined by the general solution. Additionally, the slope ofboth equations is assumed perpendicular at this time. − = = = −1 =− − = = By substation of the appropriate equations into the boundary conditionsdefined above, a system of equations is created. This system of equations isarranged 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 environmentalconditions and the radius at bubble departure. By averaging results forexperimental data sets, a value for constant is defined. 36
  50. 50. 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 inTable 7Table 7. Values of C for the General Solution Derived Using Plesset-Zwick Equation withExperimental 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 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 theconstant into the general solution, a solution for constant is determined. 3 − , 3 + , 2 2 = =1 37

×