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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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∆ 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
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Superscripts modified term+ dimensionless∗ modified term xiv
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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= −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
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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
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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
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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
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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
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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.
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.
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.
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.
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.
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.
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.
Figure 6. Comparison of Predicted Departure Radii from Modified Rayleigh BasedEquation with Experimental Departure Radii. 31
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.
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.
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.
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.
= −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.
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
51.
Experimental data of (Van Stralen, Cole, Sluyter, & Sohal, 1975) is againutilized to evaluate this equation. Results of this evaluation are shown in Table8.Table 8. Values of D 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 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 differentialequation 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. Theequation can be re-written by replacing departure time using the Plesset-Zwickequation. 3 − 0 = 3 + 2 − 2 Substitution of the constants and results in the following equation. 38
52.
1 0 = 3 + 2 − 3.149594(−04) 294.644440 ∗ 2 The above equation has three possible solutions for the departure radius.The solution related to the departure radius of a bubble must be real, positiveand should be in the scale of expected results. Analysis of experimental datausing the equation above is shown in Table 9.Table 9. Error Analysis of Predicted Departure Radii based on Plesset-Zwick BasedGeneral 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 0.002976 223.4550 2 124.4618 0.0079 0.005379 -31.9130 3 200.1375 0.0119 0.008527 -28.3459 4 385.8247 0.0136 0.016043 17.9597 5 895.6793 0.0268 0.034263 27.8470 6 2038.6934 0.0415 0.055387 33.4620 The predicted radii from the general solution derived using the Plesset-Zwick equation is an improved fit with experimental data when compared topredicted values obtained using the averaged constants version of the Rayleighbased equation. This is seen graphically in Figure 9 and Figure 10. 39
53.
Figure 9. Comparison of Predicted Departure Radii from Plesset-Zwick Based Equationwith Experimental Departure Radii. 40
54.
Figure 10. Error Plot of Predicted Departure Radii using Plesset-Zwick Based Equation. While the predicted departure radii are a better fit with experimental data,an improved fit will be achieved solving for constants and and functions ofsystem properties. Like the solution derived using the Rayleigh equation, error isintroduced by determined constants and to be averages over a range ofexperimental data points. Results are improved by comparison of values forconstant with system properties , , and the Jakob number. Doing so leadsto the following relationship for constant . = −0.894132 + 4.010944 This equation is a poor fit with values of presented in Table 7 with a 2value of 0.2758. However new values of constant will be calculated and 41
55.
compensate for the error in this curve fit. By again comparing the modifiedvalues of constant with system values , , and Jakob number, a relationshipis determined. Constant is estimated with a 2 value of 0.9832 by use of thefollowing equation. = 4.127304 −03 1.036544 ∗ The derived equations for constants and are substituted into thegeneral solution to create a new relationship. The modified general solutiontakes the following form. 1 3 = −3 − −0.894132 +4.010944 + 4.127304 −03 1.036544 ∗ By setting this equation equivalent to the Plesset-Zwick equation, thefollowing equation is derived. 0.894132 −4.010944 3 0 = 3 + 2 − 4.127304 −03 1.036544 2 This equation takes the same form as that previously derived using thePlesset-Zwick equation. However, the equation is now a function of the systemvalue defined in the Rayleigh equation and the system value defined in thePlesset-Zwick equation. Analysis of experimental data from (Van Stralen, Cole,Sluyter, & Sohal, 1975) using the modified general solution is presented in Table10. 42
56.
Table 10. Error Analysis of Predicted Departure Radii based on Plesset-Zwick BasedModified 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 0.000935 1.6524 2 124.4618 0.0079 0.007791 -1.3751 3 200.1375 0.0119 0.010167 -14.5669 4 385.8247 0.0136 0.015478 13.8098 5 895.6793 0.0268 0.025268 -5.7164 6 2038.6934 0.0415 0.044919 8.2390 The results obtained using the modified constants are presentedgraphically in Figure 11 and Figure 12. 43
57.
Figure 11. Comparison of Predicted Departure Radii from Modified Plesset-Zwick BasedEquation with Experimental Departure Radii. 44
58.
Figure 12. Error Plot of Predicted Departure Radii using Plesset-Zwick Based ModifiedEquation. It is seen that the Plesset-Zwick based modified general solution showssignificantly improved fit with experimental data of (Van Stralen, Cole, Sluyter, &Sohal, 1975). The error associated with predicted values is within 15% ofexperimental values for the full range of Jakob numbers.Analysis Using MRG Equation The Rayleigh and Plesset-Zwick equations describe growth of a bubble inspecific regions. However, neither equation fully describes the growth of bubblethroughout all regions including the transition region from inertia controlledgrowth to heat-diffuse controlled growth. 45
59.
In 1970, Mikic, Rohsenow, and Griffith (MRG) developed an equationwhich spans all regions of growth. This was accomplished by writing both the −Rayleigh and Plesset-Zwick equations in terms of . They solved for this ∆term by rearranging the Plesset-Zwick equation and substituted into the Rayleighequation. The result was an equation which describes the growth through allregions for a bubble growing either on a wall or in an infinite body of liquid. Thisresulting dimensionless equation is a function of the variable introduced in theRayleigh equation, the variable introduced in the Plesset-Zwick equation,dimensionless waiting time, and a scaling factor which relates the wallsuperheat to the pool superheat. 1 2 1 2 + + = + + 1 + + − + 1 2 + + + + = 2 2 + = 2 − ∞ = − + = If the wait time is assumed to be very large, the equation reduces to thefollowing form. 2 + + = + 1 3 2 − + 3 2 −1 3 46
60.
Changing the equation back to its dimensional form produces the followingequation. 3 2 3 2 22 2 2 = + 1 − 2 −1 3 2 The radial velocity and acceleration of the bubble during its growth aredetermined by taking the first and second derivatives of the equation above. 1 2 1 2 2 2 = + 1 − 2 2 −1 2 −1 2 2 3 2 2 = + 1 − 2 2 22 2 Given the equations above for bubble growth behavior, the second order,non-linear differential equation derived in this chapter can be solved by directsubstitution. Substituting the MRG equations into the second order, non-lineardifferential equation results in the following. 3 2 3 2 −1 2 22 2 2 3 2 +1 − 2 −1 +1 3 2 22 2 −1 2 1 2 1 2 2 2 2 2 − 2 + 2 +1 − 2 =0 2 By observation, it is seen that a solution to the equation above is noteasily achieved. Plotting the left side of the equation above shows that thisrelationship only holds true at = ∞. This departure time is not feasiblesolution to the problem as it implies that the radius of departure is infinitely large.A plot of the value of the left side of the equation above (defined as residual) 47
61.
versus time is provided in Figure 13 for a set of experimental data obtained from(Van Stralen, Cole, Sluyter, & Sohal, 1975).Figure 13. Residual Value vs. Time for Experimental Data Obtained from (Van Stralen, Cole,Sluyter, & Sohal, 1975) (A=1.924969, B=0.267915). As shown above, direct substitution of the MRG equation and itsderivatives into the second order, non-linear differential equation fails to producea predicted departure radius. Additionally, it further supports that fact that thecalculated entropy generation rate fails to reach a maximum value. While the analysis above is further evidence that the derived entropygeneration rate equation fails to reach a maximum value, utilization of the MRGequation in the general solution may still produce a bubble departure radiuscorrelation. It was shown in earlier in this chapter that a general solution exists 48
62.
for the second order, non-linear differential equation. This general solution takesthe following form. = −3 − + 1 3 The derivative of the general solution takes the following form. = − − −3 − + −1 3 Solution of the general solution requires that constants and be solved.This is accomplished by applying boundary conditions. For the purposes of thisanalysis the boundary conditions are defined at the time of bubble departure. Atthe time, the radius defined by the MRG equation is set equal to the radiusdefined by the general solution. Additionally, the slope of both equations isassumed perpendicular at this time. = = = −1 =− = = By substitution of the appropriate equations into the boundary conditionsdefined above, a system of equations is created. This system of equations isutilized to solve for constant . 3 2 3 2 2 2 2 +1 − 2 −1 44 2 = − 93 2 1 2 2 1 2 +1 − 2 2 If the radial velocity is known at the point of departure, the constant maybe more easily solve using the following form. 49
63.
2 = − Unlike the analysis using the Rayleigh and Plesset-Zwick equation, theMRG equation cannot be rearranged to provide the time of bubble departure asfunction of departure radius. Determination of the departure time requiresnumerical analysis of the MRG equation. Solving for constant requires experimental data including environmentalconditions and the radius at bubble departure. By averaging results for multipleexperimental data sets, a value for constant is defined. 3 2 3 2 2 2 2 +1 − 2 , −1 4 4 2 , − 9 3 2 1 2 2 1 2 +1 − 2 , 2 , = =1 Experimental data published by (Van Stralen, Cole, Sluyter, & Sohal,1975) is utilized to solve for this constant. Results of this analysis are shown inTable 11. 50
64.
Table 11. Values of C for the General Solution Derived Using MRG Equation withExperimental Data of (Van Stralen, Cole, Sluyter, & Sohal, 1975). Experimental Departure Departure Radius of (Van Time from Bubble Stralen, Cole, Mikic Ja Number Sluyter, & Equation Sohal, 1975) , s , m 1 0.00092 9.070000E-03 64.4322 6.032567 0.009769 11.012151 2 0.0079 2.041900E-01 124.4618 2.572322 0.017661 5.746493 3 0.0119 1.866000E-01 200.1375 2.549390 0.028018 5.433665 4 0.0136 7.331000E-02 385.8247 2.411500 0.053010 6.269883 5 0.0268 6.485000E-02 895.6793 2.309636 0.120425 5.786335 6 0.0415 5.179000E-02 2038.6934 1.924969 0.267915 5.751597 Average 6.666687 With constant defined, constant is solved using the general solution.The resulting constant is defined by the following equation. 3 3 2 3 2 3 2 2 2 2 +1 − 2 −1 + 3 − , 3 2 , = =1 The experimental data of (Van Stralen, Cole, Sluyter, & Sohal, 1975) isagain utilized to solve for constant as shown in Table 12. 51
65.
Table 12. Values of D for the General Solution Derived Using MRG Equation withExperimental Data of (Van Stralen, Cole, Sluyter, & Sohal, 1975). Experimental Departure Departure Radius of (Van Time from Bubble Stralen, Cole, Mikic Ja Number Sluyter, & Equation Sohal, 1975) , s , m 1 0.00092 9.070000E-03 64.4322 6.032567 0.009769 3.462842E-05 2 0.0079 2.041900E-01 124.4618 2.572322 0.017661 7.800542E-04 3 0.0119 1.866000E-01 200.1375 2.549390 0.028018 7.140906E-04 4 0.0136 7.331000E-02 385.8247 2.411500 0.053010 2.823998E-04 5 0.0268 6.485000E-02 895.6793 2.309636 0.120425 2.668351E-04 2038.693 6 5.179000E-02 1.924969 0.267915 2.691968E-04 0.0415 4 Average 3.912008E-04 At departure, the fully defined general solution equation is equivalent tothe MRG equation. As previously stated, the resulting relationship cannot bechanged to a function of departure radius. However, the general solution may berearranged to define departure time as a function of departure radius.Substitution of this rearranged general solution in the MRG equation producesthe following function of bubble departure radius. 3 2 3 2 22 2 2 2 2 0= − 2 − 3 + 2 − + 1 − − 2 − 3 + 2 − 3 3 3 3 3 − 1 − Substitution of the constants and produces the following equation. 52
66.
3 2 22 2 3 1.000685E − 032 0= − + +1 3 2 3 −6.666687 2 3 −6.666687 3 2 2 3 1.000685E − 032 − − + − 1 − 2 3 −6.666687 2 3 −6.666687 Numerical evaluation of the equation above results in the predicteddeparture radii shown in Table 13.Table 13. Error Analysis of Predicted Departure Radii based on MRG 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 0.003116 238.7446 2 124.4618 0.0079 0.005572 -29.4677 3 200.1375 0.0119 0.008758 -26.4000 4 385.8247 0.0136 0.016121 18.5371 5 895.6793 0.0268 0.032843 22.5476 6 2038.6934 0.0415 0.050305 21.2171 It is seen in that table above that the general solution derived using theMRG equation predicts bubble departure radii within 30% for Jakob numbersgreater than 100. This is seen graphically in Figure 14. 53
67.
Figure 14. Comparison of Predicted Departure Radii from MRG Based Equation withExperimental Departure Radii. 54
68.
Figure 15. Error Plot of Predicted Departure Radii using MRG Based Equation While the predicted departure radii represent a reasonable prediction ofexperimental data, results can be further improved. Like with results obtainedusing the Rayleigh and Plesset-Zwick equations, solving for constants and as functions of constant system variables will improve results. Forcing constants and to be functions of system values such as , , or the Jakob number willfurther improve results. Analysis of the values for in Table 11 indicate a relationship with systemconstant . Constant is estimated with a 2 value of 0.9498 by the followingequation. = 5.814845 −02 2 + 8.891619(−01) + 3.399097 55
69.
The improved value for is used to generate new values for constant .The new values of D are again compared to the system constants , , and .Analysis indicates that the constant D is modeled with a 2 value of 0.9943 bythe following equation. 4 2 = −1.957951 −12 2 + 1.124843 −07 + 7.128086 −04 The modified general solution now takes the form shown below. −02 2 +8.891619(−01)+3.399097 4 = −3 − 5.814845 − 1.957951 −12 2 1 3 2 + 1.124843 −07 + 7.128086 −04 By rearranging the general solution above to solve for time, it may besubstituted into the MRG equation to generate of a function of departure radius.The solution to the following equation results in the radius of departure for abubble. 56
71.
Table 14. Error Analysis of Predicted Departure Radii based on MRG Based ModifiedGeneral Solution. Experimental Departure Predicted Radius of (Van Bubble Departure Ja Stralen, Cole, % Error Number Radius Sluyter, & Sohal, 1975) , m , m 1 64.4322 5.862968 7.452335E-05 0.00092 0.033729 3566.1467 2 124.4618 5.381753 1.507299E-03 0.0079 0.005585 -29.3044 3 200.1375 5.378633 1.057462E-03 0.0119 0.008660 -27.2248 4 385.8247 5.359889 8.078969E-04 0.0136 0.014679 7.9313 5 895.6793 5.346063 7.306733E-04 0.0268 0.028679 7.0112 6 2038.6934 5.294014 7.158238E-04 0.0415 0.037803 -8.9093 These results are seen graphically in Figure 16 and Figure 17. 58
72.
Figure 16. Comparison of Predicted Departure Radii from Modified MRG Based Equationwith Experimental Departure Radii.Figure 17. Error Plot of Predicted Departure Radii using MRG Based Modified Equation As shown above, the MRG based modified general solution has improvedpredicted departure radii for bubbles with higher Jakob numbers when comparedwith experimental data of (Van Stralen, Cole, Sluyter, & Sohal, 1975). Thepredicted departure radii fit experimental data within 30% for systems is a Jakobnumber larger than approximately 100. Additionally, it can be seen that the erroris further reduced for Jakob numbers of approximately 300 and larger. 59
73.
Confirmation of Correlation Comparison of the models generated in this chapter indicates that thegeneral solution to the second order, non-linear differential equation is capable ofpredicting departure radii of (Van Stralen, Cole, Sluyter, & Sohal, 1975) within15%. This is achieved by determining the real, non-negative solution to thefollowing system property dependant; third order equation derived using themodified Plesset-Zwick equation for bubble growth. 0.894132 −4.010944 3 0 = 3 + 2 − 4.127304 −03 1.036544 2 Confirmation of the equation is performed by analysis of additionalexperimental data sets. Data published by (Cole & Shulman, 1966b) for bubblesgrowing in sub atmospheric pressure water is utilized. This data set differs fromthat of (Van Stralen, Cole, Sluyter, & Sohal, 1975) in that the fluid temperature ismaintained at saturation temperature rather than superheated temperatures.Results of the analysis are presented in Table 16. 60
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Table 15. Comparison of Derived Equation with Experimental Data of (Cole & Shulman,1966b) Experimental Departure Radius Predicted Bubble Jakob Number of (Cole & Departure Radius % Error Shulman, 1966b) , m , m 1 89.2283 0.00900 0.003734 -58.5149 2 0.00775 -51.8237 3 0.00650 -42.559 Average 0.00775 -51.8237 4 191.9251 0.00925 0.007453 -19.4321 5 0.00800 -6.84342 Average 0.008625 -13.5933 6 296.9101 0.01900 0.015787 -16.913 7 0.01500 5.243592 8 0.01275 23.81599 9 0.01300 21.43491 10 0.00925 70.66528 11 0.01275 23.81599 12 0.01175 34.35352 13 0.01100 43.51399 14 0.01025 54.01501 15 0.00950 66.17409 Average 0.012425 27.08554 16 1993.5703 0.02075 0.02701 30.17266 17 0.02000 35.05414 18 0.01900 42.16225 Average 0.019917 35.61723 These results are presented graphically in Figure 18. 61
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Figure 18. Comparison of Predicted Departure Radii with Experimental Data of (Cole &Shulman, 1966b). Evaluation of the data indicates that the equation derived for departureradii is capable of estimated experimental departure radii within approximately50% for system undergoing boiling at saturated conditions. Further analysis of the equation is performed by comparison withexperimental data of (Ellion, 1954) for bubbles undergoing growth in sub cooledwater at atmospheric pressure. The experimental data utilized for this analysisrepresents the average maximum radii reached during sub cooled boiling atspecific system conditions. These predicted radii vary from departure radii in thatthe bubbles do not depart from the heating surface. Rather the bubbles reach amaximum radius at which point they begin to collapse. 62
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Results of the analysis are summarized in Table 16.Table 16. Comparison of Derived Equation with Experimental Data of (Ellion, 1954). Average Experimental Predicted Bubble Jakob Number Maximum Radius of Maximum Radius % Error (Ellion, 1954) , m , m 1 63.81317 0.000559 0.000956 71.0993 2 79.72192 0.000495 0.000763 54.0291 3 83.96245 0.004700 0.000718 52.8118 4 93.35631 0.000470 0.000630 34.1205 5 94.98782 0.000462 0.000616 33.1931 6 95.18446 0.000495 0.000616 24.2931 7 100.0765 0.000445 0.000575 29.2549 8 100.599 0.000437 0.000574 31.3915 9 106.478 0.000376 0.000533 41.7945 10 108.5569 0.000351 0.000518 47.9145 These results are shown graphically in Figure 19. 63
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Figure 19. Comparison of Predicted Maximum Radii with Experimental Data of (Ellion,1954). The analysis above indicates that the derived equation is capable ofpredicting maximum radii from the experimental data of (Ellion, 1954) withinapproximately 70% for systems with Jakob numbers ranging from approximately60 to 110.Summary A novel correlation for bubble departure radii is derived by performing anentropy generation study on a spherical bubble undergoing nucleate pool boiling.The entropy generation study results in a second order, non-linear differentialwhich is described by a general solution. This is achieved by modeling the rate 64
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of work performed by the bubble as that of a thermodynamic system. Rayleigh,Plesset-Zwick, and MRG equations for bubble growth are utilized to solve thesecond order, non-linear differential equation and its general solution. Direct substitution of the three equations into the differential equation isunsuccessful in producing predicted departure radii. Furthermore, results of thisanalysis, along with analysis of the derived entropy generation rate equationshown below, indicates that calculated entropy generation rates do not reach amaximum value. 4 = 2 ∞ Utilization of the Rayleigh, Plesset-Zwick, and MRG equations tosolve for the constants and of the general solution results in varying ability topredict departure radii. Most accurate predicted departure radii are achieved byutilization of the Plesset-Zwick equation. The resulting equation is presentedbelow. 0.894132 −4.010944 3 3 0 = + 2 − 4.127304 −03 1.036544 2 Further analysis of additional experimental data fails to reproduce thesame level of accuracy achieved for superheated boiling. The inability toaccurately predict departure radii for these data sets is likely associated with theinability of accurately consider all system properties in the derived equation.Evaluation of the equation above indicates that it is not a function of all variablescommonly associated with bubble departure correlations including gravity,contact, angle, water superheat and many others. Each of the data sets utilized 65
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for the analysis have been obtained at various levels of liquid superheat. Thisdistinction is not accounted for in the derived equation. Furthermore, the derivedequation relies on constant values which have been obtained by evaluation ofexperimental data of (Van Stralen, Cole, Sluyter, & Sohal, 1975). As a result, theequation is biased towards accurately predicting departure radii from this set ofexperimental data. While the derived correlation does not consider all variables commonlyassociated with bubble departure, the correlation and the method utilized toderive it are valuable. Utilizing the methods presented in this chapter, thecorrelation above can be adapted to more accurately predict departure radii for agiven heater system. This ability allows for the prediction of bubble departureradii within this given system at operating conditions other than those specificallytested. 66
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ENTROPY GENERATION RATE STUDY (NET FORCE METHOD) In Chapter 2, an entropy generation rate study was performed bymodeling the work performed by the bubble as that of a thermodynamic system.While the resulting equating fails to read a maximum value, it derivative led to acorrelation for bubble departure radii for bubbles undergoing growth on a wallduring nucleate pool boiling. It has been shown that the resulting correlation iscapable of predicting departure radii within 15% of experimental data of (VanStralen, Cole, Sluyter, & Sohal, 1975) but is less capable of predictingexperimental data sets for bubble growing in saturated and sub-cooling boiling.In this chapter, a novel correlation for bubble departure radius is determinedusing the second law of thermodynamics. The rate of work performed by thebubble on its surrounding will be calculated using the net force acting on abubble during growth on a wall. All steps of the following work are shown inAppendix C.Derivation of Heat Transfer Rate Determination of the heat transfer rate requires relationships for the rate ofwork performed by a bubble, the rate of energy accumulation within the bubble,and the rate at which energy is transferred across the bubble boundary.Rate or Work In this section rate of work performed by the bubble is modeled as afunction of the net forces acting on it. It is assumed that the net force acting on 67
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the bubble results in purely radial growth of the bubble. In this case, the totalwork done by a bubble is determined by integrating the product of the net forceacting on the bubble and the rate of radial growth of the bubble. = As the net force is a function of bubble radius, the total work performed issolved by performing successive integration by parts of the equation above.Doing so results in the following equation. 1 1 2 2 1 3 3 = − + − + ⋯ 2 6 2 24 3 The rate at which work is done by a bubble on its surrounding is found bytaking the derivative of the above equation with respect to time. 1 2 2 1 3 3 1 4 4 = + − 1 − + − + + ⋯ 2 2 6 3 24 4 If the influence of the net force derivatives is neglected, the equation forwork rate is reduced to the first term of the equation above. = Solution of the rate of work equation requires a definition for net force.The net force acting on a bubble is modeled by evaluation of the forcesdescribed by (Keshock & Siegel, 1964). = + − − − Substitution of the appropriate relationships for each force (defined inTable 2) into the equation leads to the following. 68
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43 11 43 = − + sin − − 2 sin 3 16 3 − 4 It is noted that the equations for excess pressure, surface tension, and theterm representing sum of the two forces are functions of the bubble base radius.Since this value may not be known or modeled, it is suggested that adimensionless scaling factor be introduced. = sin The proposed dimensionless scaling factor is a ratio of the bubble base radius, bubble radius, and contact angle. Each ofthese dimensions is seen graphically in Figure20. This dimensionless scaling factor allows for the excess pressure, surface tensions, andtheir sum to be calculated in terms of the Figure 20. Bubble Dimensions.bubble radius, allowing for consistency ofvariables throughout the model. While this scaling term requires the bubble base radius to calculate, it maybe approximated by evaluation of previous bubble departure diametercorrelations. For the sake of simplicity, the Fritz equation is utilized fordemonstration. 1 2 2 = 0.0208 − 69
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By simple mathematical analysis, the Fritz equation is rearranged in theform of a balance of the buoyant and surface tension forces at the point of bubbledeparture. 2 4 0.0208 3 ( − ) = 2 3 6 Comparison of the surface tension term of the Fritz correlation with theproposed form of the surface tension force equation leads to the followingdefinition for the dimensionless scaling factor . 2 0.0208 = 6 Analysis of other bubble departure diameter correlations leads toadditional definitions for the dimensionless scaling factor. Several scaling factorsare provided in Table 17.Table 17. Alternative dimensionless scaling factors calculated from bubble departurecorrelations. Correlation Derived Dimensionless Scaling Factor, 2 (Fritz, 1935) 0.0208 = 6 (Cole, 1967) 1 2 = 0.04 6 (Cole & Rohsenow, 1969) 2 5 2 = 6 , x = = 1.510−4 for water = 4.6510−4 for fluids other than water Rearranging the net force equation and applying the non-dimensionalscaling factor results in the following equation. 70
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4 11 = − 3 − − 32 2 + 3 − 3 12 4Energy Change Rate A relationship for rate of energy change was derived in Chapter 2. Thisfunction is utilized for the work presented in this section. 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 andequivalent to the saturation pressure of the bulk liquid, the above equation canbe simplified to the following form. = 4 2 Energy Transfer Rate As described in Chapter 2, the rate of energy transfer across the boundaryof a spherical bubble is defined by the following series. 71
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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, the energy transfer rate reduces to a function ofbubble growth rate. = 4 2 Heat Transfer By substitution of the equations derived above into the first law ofthermodynamics, the heat transfer rate for the bubble is defined. 72
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If vapor pressure is assumed constant and equivalent to the saturationpressure of the bulk liquid, and the affects of net force derivative are ignored, theequation reduces to the following. = − 4 ∞ 2 Unlike the equation derived in Chapter 2, the equations for heat transferderived using the net force method are functions of bubble radius, net force, andvapor pressure.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 The entropy accumulation rate for a spherical bubble is determined thesame way as described in Chapter 2. The resulting equation for the rate ofentropy accumulation is as follows. 4 2 = 3 3 2 2 1 3 3 + − 1 − + − 2 20 3 1 4 4 − + ⋯ 120 4 74
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By applying the assumption of constant vapor properties at the saturationpoint of the bulk liquid, 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. = − 1 1 2 2 1 3 3 1 4 4 + − + − + + ⋯ 2 2 6 3 24 4 42 + −3 3 1 2 2 1 3 1 4 4 + − 1 − + 3 − +⋯ 4 2 20 3 120 4 If the vapor pressure within the bubble is assumed constant, the equationcan be reduced to the following. = − 1 1 2 2 1 3 3 1 4 4 + − + − + + ⋯ 2 2 6 3 24 4 4 − 2 ∞ If the affects of the net force derivatives are neglected, the rate of heattransfer is further simplified. 75
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4 = − 2 ∞Net Entropy Flow Rate 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.Entropy Generation Rate The rate of entropy generation is determined by substitution of theequations derived above in the second law of thermodynamics. As noted inChapter 2, the influence of the rate of entropy accumulation is canceled by theinfluence of the rate of entropy transfer at the bubble boundary. = − − 1 1 2 2 1 3 3 1 4 4 − − + − + + ⋯ 2 2 6 3 24 4 42 − −3 3 1 2 2 1 3 1 4 4 + − 1 − + 3 − +⋯ 4 2 20 3 120 4 If the vapor pressure is assumed constant and equivalent to the saturationpressure of the bulk liquid, and the net force derivatives are neglected, theequation is reduced to the following form. 76
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4 = − + 2 ∞ If entropy generation rate reaches a maximum at the point of bubbledeparture as hypothesized, the bubble departure radius is determined by takingthe derivative of entropy generation rate with respect to radius and settingequivalent to zero. The resulting equations are shown below. 1 2 0 = − + + 2 − 2 + 2 − 2 − 2 − 2 2 1 3 1 1 3 + + 2 2 + 2 − 2 6 2 2 3 1 3 1 4 1 3 2 1 3 4 + − − − 3 24 6 6 4 1 4 1 4 5 + − 24 24 5 4 − −62 − 32 + −62 + 32 2 + 3 3 1 2 + −3 + 23 − 3 2 − 4 4 2 1 4 1 4 2 4 2 1 5 3 + + − + + 4 2 5 20 3 1 5 3 1 1 6 4 + − + 5 − 5 2 − 20 20 20 120 4 1 6 1 6 5 + − +⋯ 120 120 5 If the affects of the net force derivatives are neglected and the vaporpressure is assumed constant, the rate of heat transfer is simplified. 0 = − + 4 22 + 2 ∞ 77
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Rearranging the equation above allows it to be rewritten as a modifiedforce balance equation. 2 = 4 2 + 2 ∞ The equation above modifies the force balance analysis performed by(Keshock & Siegel, 1964) by implying that departure of a bubble undergoingnucleate pool boiling on a wall occurs at a value other than 0. Both the full equation and the simplified equations must be solvednumerically by substitution of relationships for net force and vapor pressure. Thenet force has already been defined in this chapter. A relationship for vaporpressure was introduced in Chapter 2. Both equations and their first fivederivatives are provided in Table 18 and Table 19.Table 18. Net Force Derivatives. Order of Equation Derivative 0 4 11 = − 3 − − 32 2 + 3 − 3 12 4 1 4 11 = 3 − 2 − 3 − − 63 + 92 + 3 3 12 − + 2 4 2 2 = 4 62 + 32 − − 62 − 3 − 2 11 4 − 64 + 362 + 92 2 + 122 + 3 4 12 − 3 + 4 78
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Table 19. Vapor Pressure Derivatives Order of Equation Derivative 2 0 2 3 2 = ∞ + + + 2 2 1 2 = − 2 + 3 + + 2 2 4 2 2 4 = 3 − 2 + 42 + 5 + 4 2 3 3 12 12 2 4 5 3 = − 4 3 + 3 − 2 + 13 + 6 4 + 5 4 4 48 4 72 2 12 4 2 4 4 = 5 − 4 + 3 2 + 3 − 2 4 4 5 6 + 132 + 19 4 + 7 5 + 6 5 5 240 5 480 3 252 28 12 10 4 2 5 5 =− 6 + 5 − 4 2 + 3 − 3 2 + 3 4 − 2 5 4 5 6 7 + 45 4 + 26 5 + 8 6 + 7 Analysis of Net Force Correlation Two equations have been derived for predicting departure radii bymodeling the work performed with the net force acting on the bubble. Theequations derived are based on the following sets of assumptions. Constant vapor pressure and negligible net force derivatives Vapor pressure and net force derivatives considered Solution of each equation requires a model for the growth behavior of abubble in nucleate pool boiling. For this analysis, the MRG equation derived by(Mikic, Rohsenow, & Griffith, 1970) is utilized as it accurately models bubbles 80
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growing in both the inertia and heat-diffuse regions. This equation and itsderivatives are listed in Table 20.Table 20. MRG Equation Derivatives. Order of Equation Derivative 0 3 2 3 2 22 2 2 = + 1 − −1 3 2 2 1 1 2 1 2 2 2 = + 1 − 2 2 2 −1 2 −1 2 2 3 2 2 2 = + 1 − 22 2 2 3 −3 2 −3 2 3 5 2 2 =− 4 + 1 − 3 4 2 2 4 −5 2 −5 2 4 37 2 2 = + 1 − 4 86 2 2 5 −7 2 −7 2 5 159 2 2 5 =− + 1 − 168 2 2 6 −9 2 −9 2 6 10511 2 2 = + 1 − 6 3210 2 2 7 −11 2 −11 2 7 94513 2 2 7 =− + 1 − 6412 2 2 Solution of the three equations is performed numerically using the MatLabcode provided in Appendix D.Constant Vapor Pressure and Negligible Net Force Derivatives A simplified equation has been derived by assuming that the influence ofthe rate of change of net force is negligible and that vapor pressure is assumed 81
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constant and equivalent to the saturation pressure of the bulk liquid. Theresulting equation is as follows. 2 = −4 2 + 2 ∞ Substitution of the net force equation allows it to be reduced to thefollowing. ∞ 2 3 0 = 3 + 3 2 + 2 − − 4 − 11 3 − 32 2 + 3 − 16 − 16 − Numerical analysis of this equation is performed using scaling factorsderived from the correlations of (Fritz, 1935), (Cole, 1967), and (Cole &Rohsenow, 1969). Results of the numerical analysis indicate that the derivedequation is unable to predict departure radii from experimental data. This is anindication that the calculated entropy generation rate does not reach a maximumvalue. Analysis of the derived entropy generation equation shown belowconfirms this. 4 = − + ∞ 2 By plotting the equation above, it is confirmed that the calculated entropygeneration rate does not reach a maximum value (Figure 21). 82
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Figure 21. Entropy Generation Rate vs. Bubble Radius for Experimental Data Obtainedfrom (Van Stralen, Cole, Sluyter, & Sohal, 1975) (A=1.924969, B=0.267915).Vapor Pressure and Net Force Derivatives Considered The derived equation is improved by including affects of net forcederivatives and changes in vapor pressures. By considering all variables, theequation expands to the following. 83
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1 2 2 0= − + −2 + + + 2 − − − 2 2 1 2 2 1 2 1 3 3 + − + + + 2 2 6 3 1 3 1 3 1 3 1 4 + − − − 4 3 6 6 24 4 1 4 1 4 5 + − + 24 24 5 4 + 6 + 3 + 6 − 3 − 2 3 2 2 1 3 2 2 + −2 + + + 4 2 1 3 1 3 1 3 1 4 3 + − − − 2 4 4 20 3 1 4 1 1 4 1 5 4 + − + 4 + + 10 20 20 120 4 1 5 1 5 5 + − +⋯ +⋯ 120 120 5 In the equation above, net force is now a function of the rate of change ofvapor properties and the vapor pressure now varies with bubble size. As with thesimplified equation, the equation above must be solved numerically. Numerical analysis of the equation above indicates that it is also unable topredict departure radii of experimental data. 84
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Summary An equation for entropy generation rate is derived by studying the rate ofentropy generation for a bubble undergoing nucleate pool boiling. In thederivation of this equation, work has been modeled as a function of the net forcesacting on the bubble and the rate at which the bubble grows. A derivative of thederived equation was taken in an attempt to solve for a departure radii. Attemptsto do so were unsuccessful and it has been confirmed that the derived entropygeneration rate equation does not reach a maximum value. 85
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CONCLUSIONS Entropy generation studies of spherical bubbles undergoing growth innucleate pool boiling have resulted in a novel correlation for bubble departureradii. Two equations for entropy generation rate have been derived for sphericalbubbles undergoing growth on a wall in nucleate pool boiling. These equationshave been derived by modeling the work performed by the bubble as that of athermodynamic system, and as a function of the net forces acting on the bubbleand the rate at which the bubble grows. When work performed by a spherical bubble is modeled using theequation for a thermodynamic system, the entropy generation rate takes thefollowing form. 4 = 2 ∞ The derivative of the above equation results in a separable second order,non-linear differential equation. Evaluation of this equation indicates that directsubstitution of the Rayleigh, Plesset-Zwick, and MRG equations fails to predictdeparture radii for bubbles undergoing growth in nucleate pool boiling. Furtherinvestigation indicates that this is caused by failure of the calculated entropygeneration rate to reach a maximum value. Analysis of the general solution to the second order, non-linear differentialequation produces a novel correlation for predicting departure radii. By settingthe general solution of the differential equation equivalent to the Plesset-Zwick 86
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equation for bubble growth, a third order equation is derived which allows for theprediction of bubble departure radii. 0.894132 −4.010944 3 3 0 = + 2 − 4.127304 −03 1.036544 2 The real, non-negative solution to the above equation estimates thedeparture radius of a spherical bubble undergoing growth in nucleate poolboiling. Predicted departure radii derived using this equation compare well withdata of (Van Stralen, Cole, Sluyter, & Sohal, 1975) but is less capable ofpredicting experimental data of (Cole & Shulman, 1966b) and (Ellion, 1954) forsystems undergoing saturated and sub-cooled boiling. The ability to accuratelypredict experimental departure data of (Van Stralen, Cole, Sluyter, & Sohal,1975) can be attributed that the method by which the equation is derived. Modeling the work performed by a bubble as the integral of the product ofthe net force acting on the bubble and the rate of growth of the bubble producesa complex equation for entropy generation rate. = − − 1 1 2 2 1 3 3 1 4 4 − − + − + + ⋯ 2 2 6 3 24 4 42 − −3 3 1 2 2 1 3 3 1 4 4 + − 1 − + − +⋯ 4 2 20 3 120 4 The derivative of the above equation and a reduced form of it fail to resultin predicted departure radii for experimental data. Analysis of the equation 87
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above and its simplified form indicate that both fail to reach a maximum value atbubble departure. The failure of both entropy generation rate equations to reach a maximumvalue may be associated with the equation(s) utilized to model the growth of thebubble. The Rayleigh, Plesset-Zwick, and MRG equations are intended toapproximate the growth a bubble through specific regions of growth, or in thecase of the MRG equation, through the life of the bubble. Analysis of allequations indicates that each will model the growth of the bubble to an infinitelylarge radius. This is not representative of real bubbles which reach a maximumradius at, or near, departure prior to shrinking. The utilization of these equationsin the development of an entropy generation rate model likely introduces somelevel of error near the point of departure. Furthermore, the equations derived for heat transfer rate indicates thatheat transfer rate is always positive and growing. This cannot be true for abubble departing from a heated surface. In this case, the heat is supplied to thebubble by means of a superheated surface and, potentially, a superheated liquid.Once the temperature within the bubble exceeds the temperature of the liquid,thermal energy is only transferred to the bubble by the wall. At the point ofbubble departure, this heat transfer rate disappears, or is greatly diminishes.This is not consistent with results of the derived equation. The findings of this entropy generation study do not disprove thehypothesis. However, they do indicate that the derived equations for entropygeneration rate do not accurately demonstrate the behavior of bubbles at, or 88
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near, the point of bubble departure. The development of improved entropygeneration rate models may lead to additional novel correlations for theprediction of bubble departure radii. 89
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BIBLIOGRAPHYBorishanskiy, V. M., & Fokin, F. S. (1963). Heat transfer and hydrodynamics in steam generators. Trudy TsKTI 62, 1 .Borishanskiy, V. M., Danilova, G. N., Gotovskiy, M. A., Borishanskaya, A. V., Danilova, G. P., & Kupriyanova, A. V. (1981). Correlation of data on heat transfer in, and elementary characteristics of the nucleate boiling mechanism. Heat Transfer - Soviet Research, 13, 100-116.Cole, R. (1967). Bubble frequencies and departure volumes at subatmospheric pressures. AIChE Journal, 13 (4), 779-783.Cole, R., & Rohsenow, W. (1969). Correlation of bubble departure diameters for boiling of saturated liquids. Chemical Engineering Progress Symposium Series , 65 (92), 211-213.Cole, R., & Shulman, H. L. (1966a). Bubble departure diameters at subatmospheric pressures. Chemical Engineering Progress Symposium Series, 62 (64), 6-16.Cole, R., & Shulman, H. L. (1966b). Bubble growth rates at high jakob numbers. International Journal of Heat and Mass Transfer, 9 (12), 1377-1390.Ellion, M. (1954). A Study of the Mechanism of Boiling Heat Transfer. Pasadena: California Institute of Technology.Fritz, W. (1935). Maximum volume of vapor bubbles. Physik Zeitschr, 36, 379- 384. 90
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Golorin, V. S., Kolchugin, B. A., & Zakharova, E. A. (1978). Investigation of the mechanism of nucleate boiling of ethyl alcohol and benzene by means of high-speed motion-picture photography. Heat Transfer - Soviet Research, 10, 79-98.Gorenflo, D., Knabe, V., & Bieling, V. (1986). Bubble density on surfaces with nucleate boiling-Its influence on heat transfer and burnout heat fluxes at elevated saturation pressures. Proceedings of the 8th International Heat and Mass Transfer Conference, 4, pp. 1995-2000. San Francisco.Han, C. Y., & Griffith, P. (1962). The mechanism of heat transfer in nucleate pool boiling. TR 16, Massachusetts Institute of Technology.(1994). IAPWS release on surface tension of ordinary water substance. International Association for the Properties of Water and Steam.Jensen, M. K., & Memmel, G. J. (1986). Evaluation of bubble departure diameter correlations. Proceedings of the 8th International Heat Transfer Conference, 4, pp. 1907-1912.Keshock, E. G., & Siegel, R. (1964). Forces acting on bubbles in nucleate boiling under normal and reduced gravity conditions. NASA-TN-D-2299.Kim, J., & Kim, M. H. (2006). On the departure behaviors of bubble at nucleate pool boiling. International Journal of Multiphase Flow, 32 (10-11), 1269- 1286.Kocamustafaogullari, G. (1983). Pressure dependence of bubble departure diameter for water. International Communications in Heat and Mass Transfer, 10 (6), 501-509. 91
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Kutateladze, S. S., & Gogonin, I. I. (1980). Growth rate and detachment diameter of a vapor bubble in free convection boiling of a saturated liquid. High Temperature, 17 (4), 667-671.Lien, Y. C. (1969). Bubble Growth Rates at Reduced Pressure. Massachusetts Institute of Technology, Department of Mechanical Engineering.Mikic, B. B., Rohsenow, W. M., & Griffith, P. (1970). On bubble growth rates. International Journal of Heat and Mass Transfer, 13 (4), 657-666.Plesset, M. S., & Zwick, S. A. (1954). The growth of vapor bubbles in superheated liquids. Journal of Applied Physics, 25 (4), 493-500.Rayleigh, L. (1917). On the pressure developed in a liquid during the collapse of a spherical cavity. Philosophical Magazine, 34 (200), 94-98.(2007). Revised release on the IAPWS Industrial Formulation of 1997 for the thermodynamic properties of water and steam. Internaional Association for the Properteis of Water and Steam.Ruckenstein, E. (1961). Physical model for nucleate boiling heat transfer from a horizontal surface. Buletinul Institutului Politehnic Bucuresti, 33 (3), 79-88.Staniszewski, B. E. (1959). Nucleate boiling bubble growth and departure. Massachusetts Institute of Technology Cambridge Division of Sponsored Research.Stephan, K. (1992). Heat transfer in condensation and boiling. Berlin: Springer- Verlag. 92
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Van Stralen, S. J., Cole, R., Sluyter, W. M., & Sohal, M. S. (1975). Bubble growth rates in nucleate boiling of water at subatmospheric pressures. International Journal of Heat and Mass Transfer, 18 (5), 655-669.Zuber, N. (1959). Hydrodynamic aspects of boiling heat transfer. U.S. AEC Report AECU 4439. United States Atomic Energy Commission.Zuber, N. (1964). Recent trends in boiling heat transfer research. Part I: Nucleate pool boiling. Applied Mechanics Reviews, 17 (9), 663-672. 93
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APPENDIX A: DERIVATION OF ENTROPY GENERATION RATE (PRESSURE METHOD) The first law of thermodynamics is utilized to derive a relationshipfor the rate of heat transfer to a spherical bubble undergoing growth in poolboiling. d = + − dt The rate of work performed by the bubble is defined at a given radius asfollows. = 4 − ∞ 2 Integration by successive parts produces the following. 4 3 1 1 2 2 1 3 3 = − ∞ − + − + ⋯ 3 4 20 2 120 3 The rate of work performed by the bubble at a radius is determined bytaking the derivative of the series above with respect to time. 4 2 3 3 2 2 1 3 3 = 3 − ∞ − + − + ⋯ 3 4 20 2 40 3 4 3 1 2 1 1 3 2 1 2 + − − + + 3 4 2 4 20 3 10 2 1 4 3 1 3 2 − − + ⋯ 120 4 40 3 Grouping common terms in the equation above results in the followingequation for rate of work performed by a bubble at a radius of . 94
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42 = 3 − ∞ 3 1 2 2 1 3 3 1 4 4 + − 1 − + − + + ⋯ 4 2 20 3 120 4 The total internal energy of a spherical bubble is defined by the followingequation. R = = 4π 2 0 Successive integration by parts leads to the following equation. 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 changes at any radius isdetermined by taking the derivative of the above equation with respect to time. 95
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To solve for entropy generation, a definition for rate of entropyaccumulation is required. This begins by defining the total entropy at a givenradius . R = = 4π 2 0 By successive integration by parts, the following solution is determined. 4 3 1 1 2 2 1 3 3 = − + − + ⋯ 3 4 20 2 120 3 The rate at which entropy accumulation occurs at a given radius isdetermined by taking the derivative of the equation above. 4 2 3 3 2 2 1 3 3 = 3 − + − + ⋯ 3 4 20 2 40 3 4 2 1 2 2 1 1 3 3 + − − + 3 4 2 4 20 3 1 2 2 1 4 4 1 3 3 + − − + ⋯ 10 2 120 4 40 3 This equation is simplified by grouping of common terms. 4 2 = 3 3 2 2 1 3 3 + − 1 − + − 2 20 3 1 4 4 − + ⋯ 120 4 With the rate of entropy accumulation defined, it is necessary to define theentropy transfer rate. This values defined by the following equation. 104
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4 =− 2 ∞ The final piece of the second law of thermodynamics required to solve forthe entropy generation rate is the net entropy flow rate at the boundaries of thebubble. R = 4π 2 0 By observation of the above equation, it is seen that it is takes the sameform as the equation defining the rate of entropy accumulation within the bubble.If the state of vapor entering the bubble is assumed to be at the same state asthe vapor accumulated within the bubble, the equation becomes identical to thatfor entropy accumulation rate. 4 2 = 3 3 2 2 1 3 3 + − 1 − + − 2 20 3 1 4 4 − + ⋯ 120 4 Substitution of the equations defined above into the second law ofthermodynamics allows for the entropy generation rate to be defined. 105
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4 2 gen = 3 3 2 2 1 3 3 + − 1 − + − 2 20 3 1 4 4 4 − + ⋯ + 2 120 4 ∞ 4 2 − 3 3 2 2 1 3 3 + − 1 − + − 2 20 3 1 4 4 − + ⋯ 120 4 Removal of common terms allows the equation to reduce to the following. 4 gen = 2 ∞ If the hypothesis is true, bubble departure occurs when the rate of entropygeneration reaches a maximum. The maximum occurs when the derivative ofentropy generation rate reaches zero. Therefore, a derivative with respect to istaken of the entropy generation rate equation above and set equal to zero. 4 4 =0= ∞ 2 = ∞ 2 ∗ The resulting equation is as follows. 4 0= ∞ 2 + By removal of common terms, the equation reduces to the following non-linear, second order differential equation. 106
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+ 22 = 0 The solution to the above non-linear second order differential equation isdetermined by substitution. The following variables are defined for thesubstitution = 2 2 = = ∗ = ′ By substitution of the above defined variables, the non-linear second orderdifferential equation is simplified. ′ + 22 = 0 This is further simplified by removal of common terms. ′ = −2 By applying the definition of ′ , the above equation is separable. = −2 Integration of the separated equation results in the following. = −2 − This equation is rewriten as follows by taking the exponent of both sides. = −2 −2 = −2 − By applying the definition of , the equation is rewritten again. 2 = − This equation is solved by integration 1 3 = −2 + 3 107
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Rearranging the equation results in the following equation for . = 3 − + 1 3The following solution also works. = −3 − + 1 3 108
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APPENDIX B: DEFINING THE GENERAL SOLUTION The general solution derived for the second order, non-linear differentialequation is fully defined by utilization of the Rayleigh, Plesset-Zwick, and MRGEquations. = −3 − + 1 3 The derivative of the general solution takes the following form. = − − −3 − + −1 3 For the Rayleigh solution, boundary conditions will be defined as follows. = = = −1 =− = = By substitution of the appropriate equations into the boundary conditionsdefined above, the following system of equation is created. 1 3 = −3 − + 2 3 = −3 − + Both equations are rearranged to define constant . = 3 3 + 3 − = − 3 2 + 3 − By setting the two equations equal to each other, the constant iseliminated. 3 3 = − 3 2 Rearranging the equation allows for the solution of constant . 109
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= − 2 The time of bubble departure is replaced by the Rayleigh equation. = Substitution into the equation for constant results in the followingequation. 1 2 = − By averaging results for multiple experimental data sets, a value forconstant is defined. 1 2 − , = =1 Constant is solved by substitution of constant into the equation below. = 3 3 + 3 − This equation is rewritten as a function of departure radius by utilizing theRayleigh equation. 3 − = 3 + A single constant is determined by taking the average of multipleexperimental data sets. 3 − , 3 + , = =1 The same procedure is utilized for the Plesse-Zwick equation. Boundaryconditions are defined as follows. 110
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− = = = −1 =− − = = By substation of the appropriate equations into the boundary conditionsdefined above, the following system of equations is created. 1 3 1 2 = −3 − + 1 2 3 −1 2 = −3 − + 2 These equations are rewritten to solve for constant . = 3 3 2 + 3 − 3 2 1 − = −3 4 + 3 − 2 The system of equations is combined. 3 2 3 3 2 1 = − −3 4 2 By rearranging the equation above, it is possible to solve for the constant. = − 2 3 2 The time at departure is replaced using the Plesset-Zwick equation. 2 = The resulting equation for the constant is a function of variable anddeparture radius. 2 3 = − 2 111
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By averaging results for multiple experimental data sets, a value forconstant is defined. 2 , 3 − 2 = =1 Substitution of the constant into the equation below leads to the definingof constant . = 3 3 2 + 3 − Like before, this equation is rewritten by replacing departure time usingthe Plesset-Zwick equation. 3 − = 3 + 2 2 Averaging results of experimental data results in the following equation. 3 − , 3 + , 2 2 = =1 Finally, this method is utilized to determine the value of the constants forthe general solution using the MRG equation. This begins by again defining theboundary conditions. = = = −1 =− = = By substation of the appropriate equations into the boundary conditionsdefined above, the following system of equations is created. 112
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3 2 3 2 22 2 2 1 3 +1 − 2 − 1 = −3 − + 3 2 1 2 1 2 2 2 2 3 +1 − 2 = −3 − + 2 These equations is rewritten to solve for constant . 3 3 2 3 2 3 22 2 2 = +1 − 2 −1 + 3 − 3 2 1 2 1 2 3 2 2 2 = − 3 2 +1 − 2 + 3 − 2 The system of equations is combined. 3 3 2 3 2 3 22 2 2 +1 − 2 −1 3 2 1 2 1 2 3 2 2 2 = − 3 2 +1 − 2 2 This relationship is rewritten as follows. 3 2 3 −3 2 = By rearranging the equation above, it is possible to solve for the constant. 3 2 3 2 2 2 2 +1 − 2 −1 44 2 = − 93 2 1 2 2 1 2 +1 − 2 2 113
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If the radial velocity is known at the point of departure, the constant maybe more easily solve using the following equation 2 = − Because departure time cannot be isolated in the MRG equation, thevalue of constant must be determined using time. Averaging results fromanalysis using experimental data sets results in the following equation. 3 2 3 2 2 2 2 +1 − 2 , −1 4 4 2 , − 9 3 2 1 2 2 1 2 +1 − 2 , 2 , = =1 The constant is determined by substitution of constant into theequation below. 3 3 2 3 2 3 22 2 2 = +1 − 2 −1 + 3 − 3 2 Averaging results for multiple experimental data sets results in thefollowing equation. 3 3 2 3 2 3 2 2 2 2 +1 − 2 −1 + 3 − , 3 2 , = =1 APPENDIX C: DERIVATION OF ENTROPY GENERATOIN RATE (NET FORCE METHOD) 114
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The first law of thermodynamics is utilized to derive a relationship for theRate of Heat Transfer to a spherical bubble undergoing growth in pool boiling. d − = − dt The rate of work performed by the bubble requires a definition for totalwork at a given radius . = Integration by successive parts leads to the following. 1 1 2 2 1 3 3 = − + − + ⋯ 2 6 2 24 3 If the effect of changes in net force are ignored the equation is reduced tothe following. = The rate of work performed by the bubble at a radius is determined bytaking the derivative of the series above with respect to time. 1 2 2 1 3 3 1 4 4 = + − 1 − + − + + ⋯ 2 2 6 3 24 4 The assumption that changes in net force are negligible leads to thefollowing equation = The net force can be defined as the sum of the following forces asdescribed in Chapter 1. = + − − − 115
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Substitution of the appropriate equations results in the following. 43 11 43 = − + sin − − 2 sin 3 16 3 − 4 A non-dimensional scaling factor is introduced to replace base radius andcontact angle. = sin Substitution of the non-dimensional scaling factor and execution of thederivative within the net force equation lead to the following equation for netforce. 4 11 = − 3 − − 32 2 + 3 − 3 12 4 Solving for the rate of work requires the derivative with respect to time ofthe net force equation. The first five derivatives are shown below. 4 3 11 = 3 − 2 − − − 63 + 92 + 3 3 12 − 2 + 4 2 4 2 2 2 3 2 = 6 + 3 − − 6 − − 2 3 2 11 4 − 64 + 362 + 92 2 + 122 + 3 4 12 − 3 + 4 116
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1 2 2 1 3 3 1 4 4 = + − 1 − + − + + ⋯ 2 2 6 3 24 4 42 + −3 3 1 2 2 1 3 3 1 4 4 + − 1 − + − +⋯ 4 2 20 3 120 4 If vapor pressure is assumed constant and the affects of changes in netforce are neglected, the equation above reduces to the following. = − 4 2 To solve for entropy generation, a definition for rate of entropyaccumulation is required. This begins by defining the total entropy at a givenradius . The work performed to derive a relationship for rate of entropyaccumulation in Chapter 2 lead to the development of the following relationship. 4 2 = 3 3 2 2 1 3 3 + − 1 − + − 2 20 3 1 4 4 − + ⋯ 120 4 The entropy transfer rate is defined using the following equation. − 1 1 2 2 1 3 3 1 4 4 = + − + − + + ⋯ 2 2 6 3 24 4 42 + −3 3 1 2 2 1 3 1 4 4 + − 1 − + 3 − +⋯ 4 2 20 3 120 4 124
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The final piece of the second law of thermodynamics required to solve forthe entropy generation rate is the net entropy flow rate at the boundaries of thebubble. This was previously defined in Chapter 2 with the following equation. 4 2 = 3 3 2 2 1 3 3 + − 1 − + − 2 20 3 1 4 4 − + ⋯ 120 4 Substitution of the equations defined above into the second law ofthermodynamics allows for the entropy generation rate to be defined. 125
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APPENDIX D: MATLAB PROGRAMS The following MatLab program has been developed to predict bubbledeparture radius using the equation derived by modeling rate of work using boththe pressure method and net force method.function Bubble()%Define Variables of Analysis%**************************************************************************%Define Input Variables%A Variable Defined in Rayleigh Equation%B Variable Defined in Plesset-Zwick Equation%equation Defines correlation equation%method Defines Scaling Factor Method%model Defines model used for analysis%Rexp Departure Radius from Experimental Data%Define Constants%a Scaling Constant for Net Force Method%b System constant%g Gravitational Acceleration [m/s^2]%n Number of experimental data points for analysis%m Number of points used in analysis%tmax Maximum time value for analysis%Define Calculated Variables%aL Thermal Diffusivity of Liquid%C C Constant for Pressure Method Solution%CpL Specific Heat of Liquid (Constant Pressure)%CvL Specific Heat of Liquid (Constant Volume)%D D Constant for Pressure Method Solution%DenL Liquid Density [kg/m^3]%DenV Vapor Density [kg/m^3]%DenWork Expanded Vapor Density Vector [kg/m^3]%dDenV Derivative of Density Vapor [kg/s*m^3]%d2DenV 2nd Derivative of Density Vapor [kg/s^2*m^3%d3DenV 3rd Derivative of Density Vapor [kg/s^3*m^3]%d4DenV 4th Derivative of Density Vapor [kg/s^4*m^3]%d5DenV 5th Derivative of Density Vapor [kg/s^5*m^3]%ErrR Percent error of predicted radius%F Net Force Acting on Bubble [N]%dF Rate of Change of Net Force [N/s]%d2F 2nd Derivative of Net Force [N/s^2]%d3F 3rd Derivative of Net Force [N/s^3]%d4F 4th Derivative of Net Force [N/s^4] 128
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%d5F 5th Derivative of Net Force [N/s^5]%hfg Specific Enthalpy of Vaporization%Ja Jakob Number%Jastar Modified Jakob Number%kL Thermal Conductivity of Liquid [W/m-K]%Pbulk Bulk Liquid Pressure [MPa]%Pvap Constnat Vapor Pressure [MPa]%P Variable Vapor Pressure [MPa]%dP Rate of Change of Vapor Pressure [MPa/s]%d2P 2nd Derivative of Vapor Pressure [MPa/s^2]%d3P 3rd Derivative of Vapor Pressure [MPa/s^3]%d4P 4th Derivative of Vapor Pressure [MPa/s^4]%d5P 5th Derivative of Vapor Pressure [MPa/s^5]%dr Radius Interval for Analysis%R Bubble Radius [m]%dR Radial Velocity of Bubble Boundary [m/s]%d2R Radial Acceleration of Bubble Boundary [m/s^2]%d3R 3rd Derivative of Bubble Radius [m/s^3]%d4R 4th Derivative of Bubble Radius [m/s^4]%d5R 5th Derivative of Bubble Radius [m/s^5]%d6R 6th Derivative of Bubble Radius [m/s^6]%d7R 7th Derivative of Bubble Radius [m/s^2]%Res Defines Residual of equation%Rpre Predicted Radius%Rworking Radius for use within program%S Scaling Factor%StL Surface Tension of Liquid [N/m]%t Time matrix for analysis%dt Interval size for Time matrix%tMikic Departure Time Predicted from Solution Derived usingMickic%Tbulk Bulk Liquid Temperature [K]%Tsupw Wall Superheat [K]%Tvap Vapor Temperature [K]%Twall Wall Temperature [K]%VisL Viscosity of Liquid%Define Time Interval for Analysis%**************************************************************************n=1001;tmax=0.3000;dt=tmax/(n-1);t=0:dt:tmax;t(1)=dt/100;%rmax=1.000;%dr=rmax/(n-1);%r=0:dr:rmax;%Define Constants for Anlaysis%**************************************************************************g=9.81;a=45;%Define Experimental Data Sets 129
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%**************************************************************************Data=menu(Select Data Set for Analysis,Van Stralen, Cole, Sluyter,and Sohal (1975),Ellion (1954),Cole and Schulman (1966));if Data==1 %Data of Van Stralen, Cole, Sluyter, and Sohal Pbulk=[.1013,.02672,0.02028,0.01321,0.00788,0.00408]; Tbulk=[373.517,340.808,334.31,325.411,315.274,304.754]; Twall=[394.617,351.808,348.71,344.211,342.674,337.354]; Rexp=[0.00092,0.0079,0.0119,0.0136,0.0268,0.0415]; b=pi/7;elseif Data==2 %Experimental Data of Ellion Pbulk=.101325*ones(1,10);Tbulk=[289.8166667,298.7055556,317.5944444,325.3722222,328.7055556,353.7055556,330.3722222,330.3722222,330.3722222,330.3722222];Twall=[408.1243,407.5687444,405.9020778,404.2354111,403.6798556,394.2354111,399.2354111,400.6243,404.2354111,405.9020778];Rexp=[0.00035052,0.00037592,0.00043688,0.0004953,0.0004699,0.0005588,0.0004953,0.0004699,0.00046228,0.0004445]; b=pi/7;elseif Data==3 %Experimental Data of Cole and SchulmenPbulk=[.047996,.047996,.047996,.025998,.025998,.013066,.013066,.013066,.013066,.013066,.013066,.013066,.013066,.013066,.013066,.006666,.006666,.006666];Tbulk=[353.4512,353.4512,353.4512,338.9906,338.9906,324.2882,324.2882,324.2882,324.2882,324.2882,324.2882,324.2882,324.2882,324.2882,324.2882,311.2438,311.2438,311.2438];Twall=[368.4512,368.4512,368.4512,357.3239,357.3239,339.2882,339.2882,339.2882,339.2882,339.2882,339.2882,339.2882,339.2882,339.2882,339.2882,365.1327,365.1327,365.1327];Rexp=[.009,.00775,.0065,.00925,.008,.019,.015,.01275,.013,.00925,.01275,.01175,.011,.01025,.0095,.02075,.02,.019]; b=pi/7;end%Initialize System Property Vectors for Analysis%**************************************************************************m=max(size(Pbulk));Tvap=zeros(1,m);Pvap=zeros(1,m);P=zeros(m,n);dP=zeros(m,n);d2P=zeros(m,n);d3P=zeros(m,n);d4P=zeros(m,n);d5P=zeros(m,n); 130
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CpL=zeros(1,m);CvL=zeros(1,m);DenL=zeros(1,m);DenV=zeros(m,n);dDenV=zeros(m,n);d2DenV=zeros(m,n);d3DenV=zeros(m,n);d4DenV=zeros(m,n);d5DenV=zeros(m,n);DenWork=zeros(m,n+6);kL=zeros(1,m);St=zeros(1,m);aL=zeros(1,m);VisL=zeros(1,m);hfg=zeros(1,m);Tsupw=zeros(1,m);Ja=zeros(1,m);A=zeros(1,m);B=zeros(1,m);Jastar=zeros(1,m);S=zeros(1,m);Rc=zeros(1,m);R=zeros(m,n);dR=zeros(m,n);d2R=zeros(m,n);d3R=zeros(m,n);d4R=zeros(m,n);d5R=zeros(m,n);d6R=zeros(m,n);d7R=zeros(m,n);Rworking=zeros(m,n);Res=zeros(m,n);Rint=zeros(m,3);F=zeros(m,n);dF=zeros(m,n);d2F=zeros(m,n);d3F=zeros(m,n);d4F=zeros(m,n);d5F=zeros(m,n);Rpre=zeros(1,m);ErrR=zeros(1,m);Sgen=zeros(m,n);%Calculate for System Properties Using IAPWS Equations%**************************************************************************for j=1:m %Define Vapor State Tvap(j)=SatTemp(Pbulk(j)); Pvap(j)=SatPress(Tbulk(j)); %Define Bulk Liquid Properties [CpL(j),CvL(j)]=SpecHeatLiq(Tbulk(j),Pvap(j)*1E-9); CpL(j)=1E3*CpL(j); CvL(j)=1E3*CvL(j); [DenL(j)]=DenLiq(Tbulk(j),Pvap(j)); [kL(j)]=ThermCond(Tbulk(j),Pvap(j))/1000; 131
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%Define Variable Vapor Pressure Behavior (Equation of Motion)%******************************************************************P(j,i)=(Pbulk(j)*1E6)+(2*St(j)/R(j,i))+DenL(j)*(3*dR(j,i)^2/2+R(j,i)*d2R(j,i)); dP(j,i)=(-2*St(j)*dR(j,i)/R(j,i)^2+DenL(j)*(4*dR(j,i)*d2R(j,i)+R(j,i)*d3R(j,i))); d2P(j,i)=(4*St(j)*dR(j,i)^2/R(j,i)^3-2*St(j)*d2R(j,i)/R(j,i)^2+DenL(j)*(4*d2R(j,i)^2+5*dR(j,i)*d3R(j,i)+R(j,i)*d4R(j,i))); d3P(j,i)=(-12*St(j)*dR(j,i)^3/R(j,i)^4+12*St(j)*dR(j,i)*d2R(j,i)/R(j,i)^3-2*St(j)*d3R(j,i)/dR(j,i)^2+DenL(j)*(13*d2R(j,i)*d3R(j,i)+6*dR(j,i)*d4R(j,i)+R(j,i)*d5R(j,i)));d4P(j,i)=(4*pi*g/3)*((36*dR(j,i)^2*d2R(j,i)+18*R(j,i)*d2R(j,i)^2+24*R(j,i)*dR(j,i)*d3R(j,i)+3*R(j,i)^2*d4R(j,i))*(DenL(j)-DenV(j,i))-(24*dR(j,i)^3+72*R(j,i)*dR(j,i)*d2R(j,i)+12*R(j,i)^2*d3R(j,i))*dDenV(j,i)-(36*R(j,i)*dR(j,i)^2+18*R(j,i)^2*d2R(j,i))*d2DenV(j,i)-12*R(j,i)^2*dR(j,i)*d3DenV(j,i)-R(j,i)^3*d4DenV(j,i))-pi*St(j)*(S(j)*d4R(j,i))-(11*pi/12)*DenL(j)*(270*dR(j,i)^2*d2R(j,i)^2+120*dR(j,i)^3*d3R(j,i)+90*R(j,i)*d2R(j,i)^3+360*R(j,i)*dR(j,i)*d2R(j,i)*d3R(j,i)+30*R(j,i)^2+d3R(j,i)^2+90*R(j,i)*dR(j,i)^2*d4R(j,i)+45*R(j,i)^2*d2R(j,i)*d4R(j,i)+18*R(j,i)^2*dR(j,i)*d5R(j,i)+R(j,i)^3*d6R(j,i))-(pi/4)*a*VisL(j)*(10*d2R(j,i)*d3R(j,i)+5*dR(j,i)*d4R(j,i)+R(j,i)*d5R(j,i));d5P(j,i)=(4*pi*g/3)*((90*dR(j,i)*d2R(j,i)^2+60*dR(j,i)^2*d3R(j,i)+60*R(j,i)*d2R(j,i)*d3R(j,i)+30*R(j,i)*dR(j,i)*d4R(j,i)+3*R(j,i)^2*d5R(j,i))*(DenL(j)-DenV(j,i)))-pi*St(j)*S(j)*d5R(j,i)-(11*pi/12)*DenL(j)*(630*dR(j,i)*d2R(j,i)^3+1260*dR(j,i)^2*d2R(j,i)*d3R(j,i)+210*dR(j,i)^3*d4R(j,i)+630*R(j,i)*d2R(j,i)^2*d3R(j,i)+420*R(j,i)*dR(j,i)*d3R(j,i)^2+120*R(j,i)^2*d3R(j,i)*d4R(j,i)+660*R(j,i)*dR(j,i)*d2R(j,i)*d4R(j,i)+126*R(j,i)*dR(j,i)^2*d5R(j,i)+78*R(j,i)^2*d2R(j,i)*d5R(j,i)+21*R(j,i)^2*dR(j,i)*d6R(j,i)+R(j,i)^3*d7R(j,i))-(pi/4)*a*VisL(j)*(10*d3R(j,i)^3+15*d2R(j,i)*d4R(j,i)+6*dR(j,i)*d5R(j,i)+R(j,i)*d6R(j,i)); if P(j,i)>=Pvap(j)*1E6 P(j,i)=Pvap(j)*1E6; dP(j,i)=0; d2P(j,i)=0; d3P(j,i)=0; d4P(j,i)=0; d5P(j,i)=0; end endend%Define Method of Analysis (Pressure Method or Net Force Method)%**************************************************************************model=menu(Select a Model for Analysis,Pressure Method,Net ForceMethod); 133
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%**************************************************************************%Pressure Method%**************************************************************************if model==1 %Define Entropy Generation Rate%********************************************************************** for j=1:m for i=1:n Sgen(j,i)=(4*pi/Twall(j))*Pbulk(j)*1E6*R(j,i)^2*dR(j,i); end figure axis auto plot(R(j,:),Sgen(j,:)) xlabel(Radius, m) ylabel(Entropy Generation Rate, W) title(Entropy Generation Rate vs. Bubble Radius) set(gcf,color,w) end %Define Pressure Method Equation (Direct Sub or Gen Solution)%********************************************************************** equation=menu(Select Equation,Direct Substitution,GeneralSolution);%********************************************************************** %Pressure Method-Direct Substitution%********************************************************************** if equation==1 fprintf( THE FOLLOWING RESULTS ARE OBTAINED USING THEPRESSURE METHODn) fprintf( WITH DIRECT SUBSTITUTIONn) if Data==1 fprintf(n Data of Van Stralen, Cole, Sluyter,and Sohal (1975) n)fprintf(**************************************************************************n) elseif Data==2 fprintf(n Data of Ellion (1954)n)fprintf(**************************************************************************n) elseif Data==3 fprintf(n Data of Cole and Shulman(1966) n) 134
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fprintf(**************************************************************************n) end for j=1:m %Solve Second Order, Non-linear Differential Equation%************************************************************** for i=1:n Res(j,i)=R(j,i)*d2R(j,i)+2*dR(j,i)^2; end %Plot Results%************************************************************** figure plot(t,Res(j,:)) axix auto title(Residual vs. Time) xlabel(Time, sec) ylabel(Residual) set(gcf,color,w) %Print Results%************************************************************** fprintf(Bubble Number:%1.0fnn,j); fprintf(Liquid Pressure: %f[MPa]n,Pbulk(j)) fprintf(Liquid Temperature: %f[K]n,Tbulk(j)) fprintf(Liquid Density: %f[kg/m^3]n,DenL(j)) fprintf(Liquid Surface Tension: %f[N/m]n,St(j)) fprintf(Liquid Viscosity: %E[Ns/m^2]n,VisL(j)) fprintf(Liquid Specific Heat (Const Pressure): %f[kJ/kg]n,CpL(j)*1E-3) fprintf(Liquid Specific Heat (Const Volume): %f[kJ/kg]n,CvL(j)*1E-3) fprintf(Liquid Thermal Conductivity: %f[W/mK]n,kL(j)) fprintf(Liquid Thermal Diffusivity: %E[m^2/s]n,aL(j)) fprintf(Specific Enthalpy of Vaporization: %f[kJ/kg]nn,hfg(j)*1E-3) fprintf(Vapor Pressure: %f[MPa]n,Pvap(j)) fprintf(Vapor Temperature: %f[K]n,Tvap(j)) fprintf(Vapor Density: %f[kg/m^3]nn,DenV(j)) fprintf(Wall Temperature: %f[K]n,Twall(j)) 135
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fprintf(Wall Superheat: %f[K]nn,Tsupw(j)) fprintf(Ja: %fn,Ja(j)) fprintf(A:%fn,A(j)); fprintf(B:%fnn,B(j));fprintf(**************************************************************************n) end fprintf(n);%********************************************************************** %Pressure Method-General Solution%********************************************************************** elseif equation==2; %Define Method of Analysis%****************************************************************** method=menu(Select Method of Analysis,Uniform Constants Cand D,System Dependant Constants C and D); submethod=menu(Select Equation,Rayleigh Equation,Plesset-Zwick Equation,Mikic Equation); %Initiate Matrices%****************************************************************** C=zeros(1,m); D=zeros(1,m); Rworking=zeros(m,3); ErrR=zeros(1,m);%****************************************************************** %Uniform Constants C and D%****************************************************************** if method==1 %Define Constants%************************************************************** if submethod==1 C=ones(1,max(size(A)))*10.129130; D=ones(1,max(size(A)))*1.683431E-05; elseif submethod==2 C=ones(1,max(size(A)))*6.784382; D=ones(1,max(size(A)))*3.149594E-04; elseif submethod==3 C=ones(1,max(size(A)))*6.666687; 136
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D=ones(1,max(size(A)))*3.912008E-04; end%****************************************************************** %System Dependant Constants C and D%****************************************************************** elseif method==2 %Define Constants%************************************************************** if submethod==1 for j=1:max(size(A)); C(j)=7.459635*log(A(j))+2.607226; D(j)=2.278040E-11*Ja(j)^2+6.485067E-09*Ja(j)-3.367751E-07; end elseif submethod==2 for j=1:max(size(A)); C(j)=-0.894132*log(B(j))+4.010944; D(j)=4.127304E-03*exp(-1.036544*A(j)); end elseif submethod==3 for j=1:max(size(A)); C(j)=5.814845E-2*A(j)^2+8.891619E-1*A(j)+3.399097; D(j)=-1.957951E-12*B(j)^4/A(j)^2+1.124843E-7*B(j)^2/A(j)+7.128086E-4; end end end %Print Details of Analysis%******************************************************************fprintf(***************************************************************************n) fprintf( PREDICTED RADII USING GENERALSOLUTIONn) if method==1 fprintf( USINIG UNIFORM CONSTANTS C ANDDn); elseif method==2 fprintf( USING SYSTEM DEPENDANT CONSTANTS CAND Dn); end if Data==1 fprintf(n Data of Van Stralen, Cole, Sluyter,and Sohal (1975) n)fprintf(**************************************************************************n) elseif Data==2 137
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fprintf(n Data of Ellion (1954)n)fprintf(**************************************************************************n) elseif Data==3 fprintf(n Data of Cole and Shulman(1966) n)fprintf(**************************************************************************n) end %Print System Properties%****************************************************************** for j=1:max(size(A)) fprintf(Bubble Number:%1.0fnn,j); fprintf(Liquid Pressure: %f[MPa]n,Pbulk(j)) fprintf(Liquid Temperature: %f[K]n,Tbulk(j)) fprintf(Liquid Density: %f[kg/m^3]n,DenL(j)) fprintf(Liquid Surface Tension: %f[N/m]n,St(j)) fprintf(Liquid Viscosity: %E[Ns/m^2]n,VisL(j)) fprintf(Liquid Specific Heat (Const Pressure): %f[kJ/kg]n,CpL(j)*1E-3) fprintf(Liquid Specific Heat (Const Volume): %f[kJ/kg]n,CvL(j)*1E-3) fprintf(Liquid Thermal Conductivity: %f[W/mK]n,kL(j)) fprintf(Liquid Thermal Diffusivity: %E[m^2/s]n,aL(j)) fprintf(Specific Enthalpy of Vaporization: %f[kJ/kg]nn,hfg(j)*1E-3) fprintf(Vapor Pressure: %f[MPa]n,Pvap(j)) fprintf(Vapor Temperature: %f[K]n,Tvap(j)) fprintf(Vapor Density: %f[kg/m^3]nn,DenV(j)) fprintf(Wall Temperature: %f[K]n,Twall(j)) fprintf(Wall Superheat: %f[K]nn,Tsupw(j)) fprintf(Ja: %fn,Ja(j)) fprintf(A:%fn,A(j)); fprintf(B:%fnn,B(j)); %Define Polynomial Equations for Rayleigh and Plesset-Zwick 138
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%************************************************************** if submethod==1 Rworking=[1,0,3*exp(-C(j))/A(j),-D(j)]; elseif submethod==2 Rworking=[1,3*exp(-C(j))/B(j)^2,0,-D(j)]; end %Determine Solutions to Polynomial Equations%************************************************************** if submethod<=2 Rint(j,:)=roots(Rworking); Rpre(j)=Rint(j,3); elseif submethod==3 Rpre(j)=0.00000005; dr=0.00000005; test=1; step=1; while test==1 Res=(2*B(j)^2/(3*A(j)))*((-A(j)^2*(Rpre(j))^3/(3*B(j)^2*exp(-C(j)))+A(j)^2*D(j)/(3*B(j)^2*exp(-C(j)))+1)^1.5-(-A(j)^2*(Rpre(j))^3/(B(j)^2*3*exp(-C(j)))+A(j)^2*D(j)/(B(j)^2*3*exp(-C(j))))^1.5-1)-(Rpre(j)); if Res<0 test=2; else Rpre(j)=Rpre(j)+dr; test=1; end step=step+1; end end %Calculated Error of Predicted Radii%************************************************************** ErrR(j)=100*(Rpre(j)-Rexp(j))/Rexp(j); %Print Results%************************************************************** fprintf(Experimental Departure Radius:%fnn,Rexp(j)); fprintf(Predicted Departure Radius:%fn,Rpre(j)); fprintf(Error:%fnn,ErrR(j)); end fprintf(n); end%**************************************************************************%Net Force Method%************************************************************************** 139
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elseif model==2 %Define Net Force Equation for Analysis%********************************************************************** equation=menu(Select Equation,Constant Vapor Pressure &Negligible Net Force Derivatives,Complete Equation); %Define Scaling Factor Method%********************************************************************** method=menu(Select Scaling Factor,Fritz Based Equation,ColeBased Equation,Cole & Rohsenow Based Equation); %Define Scaling Factor%********************************************************************** for j=1:m if method==1 beta=30; S(j)=(0.0208*beta)^2/6; elseif method==2 S(j)=(1/6)*(0.04*Ja(j))^2; elseif method==3 Jastar(j)=DenL(j)*CpL(j)*Tvap(j)/(DenV(j)*hfg(j)); S(j)=((1.5E-4)^2/6)*Jastar(j)^(5/2); end end%********************************************************************** %Constant Vapor Pressure and Negligible Net Force Derivatives%********************************************************************** if equation==1 fprintf( THE FOLLOWING RESULTS ARE OBTAINED USING THE NETFORCE METHODn) fprintf( WITH SIMPLIFIED SOLUTIONn) if Data==1 fprintf(n Data of Van Stralen, Cole, Sluyter,and Sohal (1975) n)fprintf(**************************************************************************n) elseif Data==2 fprintf(n Data of Ellion (1954)n)fprintf(**************************************************************************n) elseif Data==3 fprintf(n Data of Cole and Shulman(1966) n)fprintf(**************************************************************************n) 140
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%********************************************************** Sgen(j,i)=-F(j,i)*dR(j,i)/Twall(j)+(4*pi/Twall(j))*Pbulk(j)*1E6*R(j,i)^2*dR(j,i); %Define Residual of Entropy Generation Rate Derivative%********************************************************** Rworking(j,i)=-F(j,i)+4*pi*(2*R(j,i)*dR(j,i)^2/d2R(j,i)+R(j,i)^2)*Pbulk(j)*1E6; end figure axis auto plot(R(j,:),Sgen(j,:)) xlabel(Radius, m) ylabel(Entropy Generation Rate, W) title(Entropy Generation Rate vs. Bubble Radius) set(gcf,color,w) Rpre(j)=interp1(Rworking(j,:),R(j,:),0,spline); %Calculated Error of Predicted Radii%************************************************************** ErrR(j)=100*(Rpre(j)-Rexp(j))/Rexp(j); %Print Results%************************************************************** fprintf(Experimental Departure Radius:%fnn,Rexp(j)); fprintf(Predicted Departure Radius:%fn,Rpre(j)); fprintf(Error:%fnn,ErrR(j));fprintf(***************************************************************************n) end%********************************************************************** %Complete Equation%********************************************************************** elseif equation==2 fprintf( THE FOLLOWING RESULTS ARE OBTAINED USING THE NETFORCE METHODn) fprintf( WITH COMPLETE SOLUTIONn) if Data==1 fprintf(n Data of Van Stralen, Cole, Sluyter,and Sohal (1975) n)fprintf(**************************************************************************n) 142
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elseif Data==2 fprintf(n Data of Ellion (1954)n)fprintf(**************************************************************************n) elseif Data==3 fprintf(n Data of Cole and Shulman(1966) n)fprintf(**************************************************************************n) end for j=1:m DenV(j,:)=DenVap(Tvap(j),Pbulk(j))*ones(1,n); fprintf(Bubble Number:%1.0fnn,j); fprintf(Liquid Pressure: %f[MPa]n,Pbulk(j)) fprintf(Liquid Temperature: %f[K]n,Tbulk(j)) fprintf(Liquid Density: %f[kg/m^3]n,DenL(j)) fprintf(Liquid Surface Tension: %f[N/m]n,St(j)) fprintf(Liquid Viscosity: %E[Ns/m^2]n,VisL(j)) fprintf(Liquid Specific Heat (Const Pressure): %f[kJ/kg]n,CpL(j)*1E-3) fprintf(Liquid Specific Heat (Const Volume): %f[kJ/kg]n,CvL(j)*1E-3) fprintf(Liquid Thermal Conductivity: %f[W/mK]n,kL(j)) fprintf(Liquid Thermal Diffusivity: %E[m^2/s]n,aL(j)) fprintf(Specific Enthalpy of Vaporization: %f[kJ/kg]nn,hfg(j)*1E-3) fprintf(Vapor Pressure:Variablen) fprintf(Vapor Temperature: %f[K]n,Tvap(j)) fprintf(Vapor Density:Variablen) fprintf(Wall Temperature: %f[K]n,Twall(j)) fprintf(Wall Superheat: %f[K]nn,Tsupw(j)) fprintf(Ja: %fn,Ja(j)) fprintf(A:%fn,A(j)); fprintf(B:%fnn,B(j)); for i=1:n 143
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%Define Variable Vapor Temperature%********************************************************** Tvap(j,i)=SatTemp(P(j,i)/1E6); %Define Variable Vapor Properties%********************************************************** DenV(j,i)=DenVap(Tvap(j,i),P(j,i)/1E6); end %Define Vapor Density Behavior%************************************************************** for i=1:n DenWork(j,i+3)=DenV(j,i); end DenWork(j,1)=interp1(t,DenV(j,:),-3*dt,spline,extrap); DenWork(j,2)=interp1(t,DenV(j,:),-2*dt,spline,extrap); DenWork(j,3)=interp1(t,DenV(j,:),-dt,spline,extrap);DenWork(j,n+4)=interp1(t,DenV(j,:),tmax+dt,spline,extrap);DenWork(j,n+5)=interp1(t,DenV(j,:),tmax+2*dt,spline,extrap);DenWork(j,n+6)=interp1(t,DenV(j,:),tmax+3*dt,spline,extrap); for i=1:n k=i+3; dDenV(j,i)=(-DenWork(j,k+2)+8*DenWork(j,k+1)-8*DenWork(j,k-1)+DenWork(j,k-1))/(12*dt); d2DenV(j,i)=(-DenWork(j,k+2)+16*DenWork(j,k+1)-30*DenWork(j,k)+16*DenWork(j,k-1)-DenWork(j,k-2))/(12*dt^2); d3DenV(j,i)=(-DenWork(j,k+3)+8*DenWork(j,k+2)-13*DenWork(j,k+1)+13*DenWork(j,k-1)-8*DenWork(j,k-2)+DenWork(j,k+3))/(8*dt^3); d4DenV(j,i)=(-DenWork(j,k+3)+12*DenWork(j,k+2)+39*DenWork(j,k+1)+56*DenWork(j,k)-39*DenWork(j,k-1)-8*DenWork(j,k-2)+DenWork(j,k+3))/(6*dt^4); d5DenV(j,i)=0; end for i=1:n %Define Net Force Behavior%********************************************************** F(j,i)=(4*pi*g/3)*R(j,i)^3*(DenL(j)-DenV(j,i))-pi*St(j)*S(j)*R(j,i)-(11*pi/12)*DenL(j)*(3*R(j,i)^2*dR(j,i)^2+R(j,i)^3*d2R(j,i))-(pi/4)*a*VisL(j)*R(j,i)*dR(j,i); dF(j,i)=(4*pi*g/3)*(3*R(j,i)^2*dR(j,i)*(DenL(j)-DenV(j,i))-R(j,i)^3*dDenV(j,i))-pi*St(j)*S(j)*dR(j,i)-(11*pi/12)*DenL(j)*(6*R(j,i)*dR(j,i)^3+9*R(j,i)^2*dR(j,i)*d2R(j,i)+R(j,i)^3*d3R(j,i))-(pi/4)*a*VisL(j)*(R(j,i)*d2R(j,i)+dR(j,i)^2);d2F(j,i)=(4*pi*g/3)*((6*(R(j,i)*dR(j,i)^2+3*R(j,i)^2*d2R(j,i)))*(DenL(j)-DenV(j,i))-6*R(j,i)^2*dR(j,i)*dDenV(j,i)-R(j,i)^3*d2DenV(j,i))-pi*St(j)*S(j)*d2R(j,i)- 144
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(11*pi/12)*DenL(j)*(6*dR(j,i)^4+36*R(j,i)*dR(j,i)^2*d2R(j,i)+9*R(j,i)^2*d2R(j,i)^2+12*R(j,i)^2*dR(j,i)*d3R(j,i)+R(j,i)^3*d4R(j,i))-(pi/4)*a*VisL(j)*(3*dR(j,i)*d2R(j,i)+R(i)*d3R(j,i));d3F(j,i)=(4*pi*g/3)*((6*dR(j,i)^3+18*R(j,i)*dR(j,i)*d2R(j,i)+3*R(j,i)^2*d3R(j,i))*(DenL(j)-DenV(j,i))-(18*R(j,i)*dR(j,i)^2+9*R(j,i)^2*d2R(j,i))*dDenV(j,i)-9*R(j,i)^2*dR(j,i)*d2DenV(j,i)-R(j,i)^3*d3DenV(j,i))-pi*St(j)*(S(j)*d3R(j,i))-(11*pi/12)*DenL(j)*(60*dR(j,i)^3*d2R(j,i)+90*R(j,i)*dR(j,i)*d2R(j,i)^2+60*R(j,i)*dR(j,i)^2*d3R(j,i)+30*R(j,i)^2*d2R(j,i)*d3R(j,i)+15*R(j,i)^2*dR(j,i)*d4R(j,i)+R(j,i)^3*d5R(j,i))-(pi/4)*a*VisL(j)*(3*d2R(j,i)^2+4*dR(j,i)*d3R(j,i)+R(j,i)*d4R(j,i));d4F(j,i)=(4*pi*g/3)*((36*dR(j,i)^2*d2R(j,i)+18*R(j,i)*d2R(j,i)^2+24*R(j,i)*dR(j,i)*d3R(j,i)+3*R(j,i)^2*d4R(j,i))*(DenL(j)-DenV(j,i))-(24*dR(j,i)^3+72*R(j,i)*dR(j,i)*d2R(j,i)+12*R(j,i)^2*d3R(j,i))*dDenV(j,i)-(36*R(j,i)*dR(j,i)^2+18*R(j,i)^2*d2R(j,i))*d2DenV(j,i)-12*R(j,i)^2*dR(j,i)*d3DenV(j,i)-R(j,i)^3*d4DenV(j,i))-pi*St(j)*(S(j)*d4R(j,i))-(11*pi/12)*DenL(j)*(270*dR(j,i)^2*d2R(j,i)^2+120*dR(j,i)^3*d3R(j,i)+90*R(j,i)*d2R(j,i)^3+360*R(j,i)*dR(j,i)*d2R(j,i)*d3R(j,i)+30*R(j,i)^2+d3R(j,i)^2+45*R(j,i)*dR(j,i)^2*d4R(j,i)+60*R(j,i)^2*d2R(j,i)*d4R(j,i)+18*R(j,i)^2*dR(j,i)*d5R(j,i)+R(j,i)^3*d6R(j,i))-(pi/4)*a*VisL(j)*(10*d2R(j,i)*d3R(j,i)+5*dR(j,i)*d4R(j,i)+R(j,i)*d5R(j,i));d5F(j,i)=(4*pi*g/3)*((90*dR(j,i)*d2R(j,i)^2+60*dR(j,i)^2*d3R(j,i)+60*R(j,i)*d2R(j,i)*d3R(j,i)+30*R(j,i)*dR(j,i)*d4R(j,i)+3*R(j,i)^2*d5R(j,i))*(DenL(j)-DenV(j,i))-(180*dR(j,i)^2*d2R(j,i)+90*R(j,i)*d2R(j,i)^2+120*R(j,i)*dR(j,i)*d3R(j,i)+15*R(j,i)^2*d4R(j,i)*dDenV(j,i)-(60*dR(j,i)^3+180*R(j,i)*dR(j,i)*d2R(j,i)+30*R(j,i)^2*d3R(j,i))*d2DenV(j,i)-(60*R(j,i)*dR(j,i)^2+30*R(j,i)^2*d2R(j,i))*d3DenV(j,i)-15*R(j,i)^2*dR(j,i)*d4DenV(j,i)-R(j,i)^3*d5DenV(j,i))-pi*St(j)*S(j)*d5R(j,i)-(11*pi/12)*DenL(j)*(630*dR(j,i)*d2R(j,i)^3+1260*dR(j,i)^2*d2R(j,i)*d3R(j,i)+120*dR(j,i)^3*d4R(j,i)+630*R(j,i)*d2R(j,i)^2*d3R(j,i)+420*R(j,i)*dR(j,i)*d3R(j,i)^2+630*R(j,i)*dR(j,i)*d2R(j,i)*d4R(j,i)+90*dR(j,i)^3*d4R(j,i)+105*R(j,i)^2*d3R(j,i)*d4R(j,i)+63*R(j,i)^2*d2R(j,i)*d5R(j,i)+126*R(j,i)*dR(j,i)^2*d5R(j,i)+21*R(j,i)^2*dR(j,i)*d6R(j,i)+R(j,i)^3*d7R(j,i))-(pi/4)*a*VisL(j)*(10*d3R(j,i)^3+15*d2R(j,i)*d4R(j,i)+5*dR(j,i)*d5R(j,i)+6*dR(j,i)*d5R(j,i)+dR(j,i)*d6R(j,i))); if R(j,i)<=Rc(j) F(j,i)=0; dF(j,i)=0; d2F(j,i)=0; d3F(j,i)=0; d4F(j,i)=0; d5F(j,i)=0; end %Define Entropy Generation Rate%********************************************************** Sgen(j,i)=-F(j,i)*dR(j,i)/Twall(j)-((dR(j,i)-1)/Twall(j))*(-dF(j,i)*R(j,i)+(1/2)*d2F(j,i)*R(j,i)^2- 145
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(1/6)*d3F(j,i)^R(j,i)^3+(1/24)*d4F(j,i)*R(j,i)^4)+(4*pi*R(j,i)^2/(3*Twall(j)))*(-3*P(j,i)*dR(j,i)+(dR(j,i)-1)*(R(j,i)*dP(j,i)-(1/4)*R(j,i)^2*d2P(j,i)+(1/20)*R(j,i)^3*d3P(j,i)-(1/120)*R(j,i)^4*d4P(j,i))); %Define the Residual of Entropy Generation RateDerivative%********************************************************** Rworking(j,i)=(-1/Twall(j))*(-d2R(j,i)*F(j,i)+(R(j,i)*d2R(j,i)+R(j,i)^2-2*dR(j,i))*dF(j,i)+(2*R(j,i)*dR(j,i)-(1/2)*R(j,i)^2*d2R(j,i)-R(j,i)*d2R(j,i)-R(j,i))*d2F(j,i)+((1/6)*R(j,i)^3*d2R(j,i)+(1/2)*R(j,i)^2*dR(j,i)^2+(1/2)*R(j,i)^2-R(j,i)^2*dR(j,i))*d3F(j,i)+((1/3)*R(j,i)^3*dR(j,i)-(1/24)*R(j,i)^4*d2R(j,i)-(1/6)*R(j,i)^3*dR(j,i)^2-(1/6)*R(j,i)^3)*d4F(j,i)+((1/24)*R(j,i)^4-(1/24)*R(j,i)^4*dR(j,i))*d5F(j,i)-(4*pi/3)*((-6*R(j,i)*dR(j,i)^2-3*R(j,i)^2*d2R(j,i))*P(j,i)+(-6*R(j,i)^2*dR(j,i)+3*R(j,i)^2*dR(j,i)^2+R(j,i)^3*d2R(j,i))*dP(j,i)+(-R(j,i)^3+2*R(j,i)^3*dR(j,i)-R(j,i)^3*dR(j,i)^2-(1/4)*R(j,i)^4*d2R(j,i))*d2P(j,i)+((1/4)*R(j,i)-(1/2)*R(j,i)^4*dR(j,i)+(2/5)*R(j,i)^4*dR(j,i)^2+(1/20)*R(j,i)^5*d2R(j,i))*d3P(j,i)+(-(1/20)*R(j,i)^5+(3/20)*R(j,i)^5*dR(j,i)-(1/20)*R(j,i)^5*dR(j,i)^2-(1/120)*R(j,i)^6*d2R(j,i))*d4P(j,i)+((1/120)*R(j,i)^6-(1/120)*R(j,i)^6*dR(j,i)))); end figure axis auto plot(R(j,:),Sgen(j,:)) xlabel(Radius, m) ylabel(Entropy Generation Rate, W) title(Entropy Generation Rate vs. Bubble Radius) set(gcf,color,w) for i=1:n Rpre(j)=interp1(Rworking(j,:),R(j,:),0,spline); end %Calculated Error of Predicted Radii%************************************************************** ErrR(j)=100*(Rpre(j)-Rexp(j))/Rexp(j); %Print Results%************************************************************** fprintf(Experimental Departure Radius:%fnn,Rexp(j)); fprintf(Predicted Departure Radius:%fn,Rpre(j)); fprintf(Error:%fnn,ErrR(j)); 146
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fprintf(***************************************************************************n) end endendfprintf(n);%Plot Results%**************************************************************************figureaxis autoplot(Ja,Rexp,*r,Ja,Rpre,ob)xlabel(Jakob Number)ylabel(Radius, m)title(Departure Radius vs. Jakob Number)legend(Experimental,Predicted,location,Best)set(gcf,color,w)figureaxis squareplot(Rexp,Rexp,-k,Rexp,0.85*Rexp,--k,Rexp,0.7*Rexp,--k,0.85*Rexp,Rexp,--k,0.7*Rexp,Rexp,--k,Rexp,Rpre,*b)xlabel(Departure Radius, m)ylabel(Predicted Departure Radius, m)title(Error Analysis)set(gcf,color,w) Thermal properties of the fluid and vapor have been solved for using thefollowing programs derived using the IAWPS standards for water and steamproperties. 147
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function [Tsat]=SatTemp(P)%Revised Release on the IAPWS Industrial Formulation 1997 for the%Thermodynamic Properties of Water and Steam (The revision only relatesto%the extension of region 5 to 50 MPa)%%August 2007%%Section 8.3 - The Saturation-Temperature Equation (Backward Equation)Pstar=1; %Reference Pressure, MPaTstar=1; %Reference Temperature, Kn=[0.11670521452767E4,-0.72421316703206E6,-0.17073846940092E2,0.12020824702470E5,-0.32325550322333E7,0.14915108613530E2,-0.48232657361591E4,0.40511340542057E6,-0.23855557567849,0.65017534844798E3];beta=(P/Pstar)^(1/4);E=beta^2+n(3)*beta+n(6);F=n(1)*beta^2+n(4)*beta+n(7);G=n(2)*beta^2+n(5)*beta+n(8);D=2*G/(-F-(F^2-4*E*G)^(1/2));Tsat=Tstar*((n(10)+D-((n(10)+D)^2-4*(n(9)+n(10)*D))^(1/2))/2); 148
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function [Psat]=SatPress(Tamb)%Revised Release on the IAPWS Industrial Formulation 1997 for the%Thermodynamic Properties of Water and Steam (The revision only relatesto%the extension of region 5 to 50 MPa)%%August 2007%%Section 8.3 - The Saturation-Temperature Equation (Backward Equation)Pstar=1; %Reference Pressure, MPaTstar=1; %Reference Temperature, Kn=[0.11670521452767E4,-0.72421316703206E6,-0.17073846940092E2,0.12020824702470E5,-0.32325550322333E7,0.14915108613530E2,-0.48232657361591E4,0.40511340542057E6,-0.23855557567849,0.65017534844798E3];Nu=(Tamb/Tstar)+n(9)/((Tamb/Tstar)-n(10));A=Nu^2+n(1)*Nu+n(2);B=n(3)*Nu^2+n(4)*Nu+n(5);C=n(6)*Nu^2+n(7)*Nu+n(8);Psat=Pstar*(2*C/(-B+(B^2-4*A*C)^(1/2)))^4; 149
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function [Cp,Cv]=SpecHeatLiq(T,P)%Revised Release on the IAPWS Industrial Formulation 1997 for the%Thermodynamic Properties of Water and Steam (The revision only relatesto%the extension of region 5 to 50 MPa)%%August 2007%%Section 8.3 - The Saturation-Temperature Equation (Backward Equation)Pstar=16.53; %Reference Pressure, MPaTstar=1386; %Reference Temperature, KR=0.461526; %Gas Constant, kJ/kg-KPI=P/Pstar;Tau=Tstar/T;Table2=[0,-2,0.14632971213167;0,-1,-0.84548187169114;0,0,-0.37563603672040E1;0,1,0.33855169168385E1;0,2,-0.95791963387872;0,3,0.15772038513228;0,4,-0.16616417199501E-1;0,5,0.81214629983568E-3;1,-9,0.28319080123804E-3;1,-7,-0.60706301565874E-3;1,-1,-0.18990068218419E-1;1,0,-0.32529748770505E-1;1,1,-0.21841717175414E-1;1,3,-0.52838357969930E-4;2,-3,-0.47184321073267E-3;2,0,-0.30001780793026E-3;2,1,0.47661393906987E-4;2,3,-0.44141845330846E-5;2,17,-0.72694996297594E-15;3,-4,-0.31679644845054E-4;3,0,-0.28270797985312E-5;3,6,-0.85205128120103E-9;4,-5,-0.22425281908000E-5;4,-2,-0.65171222895601E-6;4,10,-0.14341729937924E-12;5,-8,-0.40516996860117E-6;8,-11,-0.12734301741641E-8;8,-6,-0.17424871230634E-9;21,-29,-0.68762131295531E-18;23,-31,0.14478307828521E-19;29,-38,0.26335781662795E-22;30,-39,-0.11947622640071E-22;31,-40,-.18228094581404E-23;32,-41,-0.93537087292458E-25];I=Table2(:,1);J=Table2(:,2);n=Table2(:,3);Gamma=0;GammaPI=0;GammaPIPI=0;GammaTau=0;GammaTauTau=0;GammaPITau=0;for i=1:34 Gamma=Gamma+n(i)*(7.1-PI)^I(i)*(Tau-1.222)^J(i); GammaPI=GammaPI-n(i)*I(i)*(7.1-PI)^(I(i)-1)*(Tau-1.222)^J(i); GammaPIPI=GammaPIPI+n(i)*I(i)*(I(i)-1)*(7.1-PI)^(I(i)-2)*(Tau-1.222)^J(i); GammaTau=GammaTau+n(i)*(7.1-PI)^I(i)*J(i)*(Tau-1.222)^(J(i)-1); GammaTauTau=GammaTauTau+n(i)*(7.1-PI)^I(i)*J(i)*(J(i)-1)*(Tau-1.222)^(J(i)-2); GammaPITau=GammaPITau-n(i)*I(i)*(7.1-PI)^(I(i)-1)*J(i)*(Tau-1.222)^(J(i)-1);endCp=R*(-Tau^2)*GammaTauTau; 150
165.
function [Cp,Cv]=SpecHeatVap(T,P)%Revised Release on the IAPWS Industrial Formulation 1997 for the%Thermodynamic Properties of Water and Steam (The revision only relatesto%the extension of region 5 to 50 MPa)%%August 2007%%Section 6 - Equations for Region 2Tstar=540; %Reference Temperature, KPstar=1; %Reference Pressure, MPaR=0.461526; %Gas Constant, kJ/kg-KPI=P/Pstar;Tau=Tstar/T;Table10=[0,-.96927686500217E1;1,0.10086655968018E2;-5,-0.56087911283020E-2;-4,0.71452738081455E-1;-3,-0.40710498223928;-2,0.14240819171444E1;-1,-0.43839511319450E1;2,-0.28408632460772;3,0.21268463753307E-1];Table11=[1,0,-0.17731742473213E-2;1,1,-0.17834862292358E-1;1,2,-0.45996013696365E-1;1,3,-0.57581259083432E-1;1,6,-0.50325278727930E-1;2,1,-0.33032641670203E-4;2,2,-0.18948987516315E-3;2,4,-0.39392777243355E-2;2,7,-0.43797295650573E-1;2,36,-0.26674547914087E-4;3,0,0.20481737692309E-7;3,1,0.43870667284435E-6;3,3,-0.32277677238570E-4;3,6,-0.15033924542148E-2;3,35,-0.40668253562649E-1;4,1,-0.78847309559367E-9;4,2,0.12790717852285E-7;4,3,0.48225372718507E-6;5,7,0.22922076337661E-5;6,3,-0.16714766451061E-10;6,16,-0.21171472321355E-2;6,35,-0.23895741934104E2;7,0,-0.59059564324270E-17;7,11,-0.12621808899101E-5;7,25,-0.38946842435739E-1;8,8,.11256211360459E-10;8,36,-0.82311340897998E1;9,13,0.19809712802088E-7;10,4,0.10406965210174E-18;10,10,-0.10234747095929E-12;10,14,-0.10018179379511E-8;16,29,-0.80882908646985E-10;16,50,0.10693031879409;18,57,-0.33662250574171;20,20,0.89185845355421E-24;20,35,0.30629316876232E-12;20,48,-0.42002467698208E-5; 21,21,-0.59056029685639E-25;22,53,0.37826947613457E-5;23,39,-0.12768608934681E-14;24,26,0.73087610595061E-28;24,40,0.55414715350778E-16;24,58,-0.94369707241210E-6];J0=Table10(:,1);n0=Table10(:,2);I=Table11(:,1);J=Table11(:,2);n=Table11(:,3);Gamma0=0;GammaTau0=0;GammaTauTau0=0;GammaPI0=1/PI;GammaPIPI0=-1/PI^2;for i=1:9 Gamma0=Gamma0+n0(i)*Tau^J0(i); GammaTau0=GammaTau0+n0(i)*J0(i)*Tau^(J0(i)-1); 152
167.
function [dl]=DenLiq(T,P)%Revised Release on the IAPWS Industrial Formulation 1997 for the%Thermodynamic Properties of Water and Steam (The revision only relatesto%the extension of region 5 to 50 MPa)%%August 2007%%Section 8.3 - The Saturation-Temperature Equation (Backward Equation)Pstar=16.53; %Reference Pressure, MPaTstar=1386; %Reference Temperature, KR=0.461526; %Gas Constant, kJ/kg-KPI=P/Pstar;Tau=Tstar/T;Table2=[0,-2,0.14632971213167;0,-1,-0.84548187169114;0,0,-0.37563603672040E1;0,1,0.33855169168385E1;0,2,-0.95791963387872;0,3,0.15772038513228;0,4,-0.16616417199501E-1;0,5,0.81214629983568E-3;1,-9,0.28319080123804E-3;1,-7,-0.60706301565874E-3;1,-1,-0.18990068218419E-1;1,0,-0.32529748770505E-1;1,1,-0.21841717175414E-1;1,3,-0.52838357969930E-4;2,-3,-0.47184321073267E-3;2,0,-0.30001780793026E-3;2,1,0.47661393906987E-4;2,3,-0.44141845330846E-5;2,17,-0.72694996297594E-15;3,-4,-0.31679644845054E-4;3,0,-0.28270797985312E-5;3,6,-0.85205128120103E-9;4,-5,-0.22425281908000E-5;4,-2,-0.65171222895601E-6;4,10,-0.14341729937924E-12;5,-8,-0.40516996860117E-6;8,-11,-0.12734301741641E-8;8,-6,-0.17424871230634E-9;21,-29,-0.68762131295531E-18;23,-31,0.14478307828521E-19;29,-38,0.26335781662795E-22;30,-39,-0.11947622640071E-22;31,-40,-.18228094581404E-23;32,-41,-0.93537087292458E-25];I=Table2(:,1);J=Table2(:,2);n=Table2(:,3);Gamma=0;GammaPI=0;GammaPIPI=0;GammaTau=0;GammaTauTau=0;GammaPITau=0;for i=1:34 Gamma=Gamma+n(i)*(7.1-PI)^I(i)*(Tau-1.222)^J(i); GammaPI=GammaPI-n(i)*I(i)*(7.1-PI)^(I(i)-1)*(Tau-1.222)^J(i); GammaPIPI=GammaPIPI+n(i)*I(i)*(I(i)-1)*(7.1-PI)^(I(i)-2)*(Tau-1.222)^J(i); GammaTau=GammaTau+n(i)*(7.1-PI)^I(i)*J(i)*(Tau-1.222)^(J(i)-1); GammaTauTau=GammaTauTau+n(i)*(7.1-PI)^I(i)*J(i)*(J(i)-1)*(Tau-1.222)^(J(i)-2); GammaPITau=GammaPITau-n(i)*I(i)*(7.1-PI)^(I(i)-1)*J(i)*(Tau-1.222)^(J(i)-1);endvl=R*T*PI*GammaPI/(1000*P); 154
169.
function [dv]=DenVap(T,P)%Revised Release on the IAPWS Industrial Formulation 1997 for the%Thermodynamic Properties of Water and Steam (The revision only relatesto%the extension of region 5 to 50 MPa)%%August 2007%%Section 6 - Equations for Region 2Tstar=540; %Reference Temperature, KPstar=1; %Reference Pressure, MPaR=0.461526; %Gas Constant, kJ/kg-KPI=P/Pstar;Tau=Tstar/T;Table10=[0,-.96927686500217E1;1,0.10086655968018E2;-5,-0.56087911283020E-2;-4,0.71452738081455E-1;-3,-0.40710498223928;-2,0.14240819171444E1;-1,-0.43839511319450E1;2,-0.28408632460772;3,0.21268463753307E-1];Table11=[1,0,-0.17731742473213E-2;1,1,-0.17834862292358E-1;1,2,-0.45996013696365E-1;1,3,-0.57581259083432E-1;1,6,-0.50325278727930E-1;2,1,-0.33032641670203E-4;2,2,-0.18948987516315E-3;2,4,-0.39392777243355E-2;2,7,-0.43797295650573E-1;2,36,-0.26674547914087E-4;3,0,0.20481737692309E-7;3,1,0.43870667284435E-6;3,3,-0.32277677238570E-4;3,6,-0.15033924542148E-2;3,35,-0.40668253562649E-1;4,1,-0.78847309559367E-9;4,2,0.12790717852285E-7;4,3,0.48225372718507E-6;5,7,0.22922076337661E-5;6,3,-0.16714766451061E-10;6,16,-0.21171472321355E-2;6,35,-0.23895741934104E2;7,0,-0.59059564324270E-17;7,11,-0.12621808899101E-5;7,25,-0.38946842435739E-1;8,8,.11256211360459E-10;8,36,-0.82311340897998E1;9,13,0.19809712802088E-7;10,4,0.10406965210174E-18;10,10,-0.10234747095929E-12;10,14,-0.10018179379511E-8;16,29,-0.80882908646985E-10;16,50,0.10693031879409;18,57,-0.33662250574171;20,20,0.89185845355421E-24;20,35,0.30629316876232E-12;20,48,-0.42002467698208E-5; 21,21,-0.59056029685639E-25;22,53,0.37826947613457E-5;23,39,-0.12768608934681E-14;24,26,0.73087610595061E-28;24,40,0.55414715350778E-16;24,58,-0.94369707241210E-6];J0=Table10(:,1);n0=Table10(:,2);I=Table11(:,1);J=Table11(:,2);n=Table11(:,3);Gamma0=0;GammaTau0=0;GammaTauTau0=0;GammaPI0=1/PI;GammaPIPI0=-1/PI^2;for i=1:9 Gamma0=Gamma0+n0(i)*Tau^J0(i); GammaTau0=GammaTau0+n0(i)*J0(i)*Tau^(J0(i)-1); 156
171.
function [hl]=EnthLiq(T,P)%Revised Release on the IAPWS Industrial Formulation 1997 for the%Thermodynamic Properties of Water and Steam (The revision only relatesto%the extension of region 5 to 50 MPa)%%August 2007%%Section 8.3 - The Saturation-Temperature Equation (Backward Equation)Pstar=16.53; %Reference Pressure, MPaTstar=1386; %Reference Temperature, KR=0.461526; %Gas Constant, kJ/kg-KPI=P/Pstar;Tau=Tstar/T;Table2=[0,-2,0.14632971213167;0,-1,-0.84548187169114;0,0,-0.37563603672040E1;0,1,0.33855169168385E1;0,2,-0.95791963387872;0,3,0.15772038513228;0,4,-0.16616417199501E-1;0,5,0.81214629983568E-3;1,-9,0.28319080123804E-3;1,-7,-0.60706301565874E-3;1,-1,-0.18990068218419E-1;1,0,-0.32529748770505E-1;1,1,-0.21841717175414E-1;1,3,-0.52838357969930E-4;2,-3,-0.47184321073267E-3;2,0,-0.30001780793026E-3;2,1,0.47661393906987E-4;2,3,-0.44141845330846E-5;2,17,-0.72694996297594E-15;3,-4,-0.31679644845054E-4;3,0,-0.28270797985312E-5;3,6,-0.85205128120103E-9;4,-5,-0.22425281908000E-5;4,-2,-0.65171222895601E-6;4,10,-0.14341729937924E-12;5,-8,-0.40516996860117E-6;8,-11,-0.12734301741641E-8;8,-6,-0.17424871230634E-9;21,-29,-0.68762131295531E-18;23,-31,0.14478307828521E-19;29,-38,0.26335781662795E-22;30,-39,-0.11947622640071E-22;31,-40,-.18228094581404E-23;32,-41,-0.93537087292458E-25];I=Table2(:,1);J=Table2(:,2);n=Table2(:,3);Gamma=0;GammaPI=0;GammaPIPI=0;GammaTau=0;GammaTauTau=0;GammaPITau=0;for i=1:34 Gamma=Gamma+n(i)*(7.1-PI)^I(i)*(Tau-1.222)^J(i); GammaPI=GammaPI-n(i)*I(i)*(7.1-PI)^(I(i)-1)*(Tau-1.222)^J(i); GammaPIPI=GammaPIPI+n(i)*I(i)*(I(i)-1)*(7.1-PI)^(I(i)-2)*(Tau-1.222)^J(i); GammaTau=GammaTau+n(i)*(7.1-PI)^I(i)*J(i)*(Tau-1.222)^(J(i)-1); GammaTauTau=GammaTauTau+n(i)*(7.1-PI)^I(i)*J(i)*(J(i)-1)*(Tau-1.222)^(J(i)-2); GammaPITau=GammaPITau-n(i)*I(i)*(7.1-PI)^(I(i)-1)*J(i)*(Tau-1.222)^(J(i)-1);endhl=R*T*Tau*GammaTau; 158
173.
function [hv]=EnthVap(T,P)%Revised Release on the IAPWS Industrial Formulation 1997 for the%Thermodynamic Properties of Water and Steam (The revision only relatesto%the extension of region 5 to 50 MPa)%%August 2007%%Section 6 - Equations for Region 2Tstar=540; %Reference Temperature, KPstar=1; %Reference Pressure, MPaR=0.461526; %Gas Constant, kJ/kg-KPI=P/Pstar;Tau=Tstar/T;Table10=[0,-.96927686500217E1;1,0.10086655968018E2;-5,-0.56087911283020E-2;-4,0.71452738081455E-1;-3,-0.40710498223928;-2,0.14240819171444E1;-1,-0.43839511319450E1;2,-0.28408632460772;3,0.21268463753307E-1];Table11=[1,0,-0.17731742473213E-2;1,1,-0.17834862292358E-1;1,2,-0.45996013696365E-1;1,3,-0.57581259083432E-1;1,6,-0.50325278727930E-1;2,1,-0.33032641670203E-4;2,2,-0.18948987516315E-3;2,4,-0.39392777243355E-2;2,7,-0.43797295650573E-1;2,36,-0.26674547914087E-4;3,0,0.20481737692309E-7;3,1,0.43870667284435E-6;3,3,-0.32277677238570E-4;3,6,-0.15033924542148E-2;3,35,-0.40668253562649E-1;4,1,-0.78847309559367E-9;4,2,0.12790717852285E-7;4,3,0.48225372718507E-6;5,7,0.22922076337661E-5;6,3,-0.16714766451061E-10;6,16,-0.21171472321355E-2;6,35,-0.23895741934104E2;7,0,-0.59059564324270E-17;7,11,-0.12621808899101E-5;7,25,-0.38946842435739E-1;8,8,.11256211360459E-10;8,36,-0.82311340897998E1;9,13,0.19809712802088E-7;10,4,0.10406965210174E-18;10,10,-0.10234747095929E-12;10,14,-0.10018179379511E-8;16,29,-0.80882908646985E-10;16,50,0.10693031879409;18,57,-0.33662250574171;20,20,0.89185845355421E-24;20,35,0.30629316876232E-12;20,48,-0.42002467698208E-5; 21,21,-0.59056029685639E-25;22,53,0.37826947613457E-5;23,39,-0.12768608934681E-14;24,26,0.73087610595061E-28;24,40,0.55414715350778E-16;24,58,-0.94369707241210E-6];J0=Table10(:,1);n0=Table10(:,2);I=Table11(:,1);J=Table11(:,2);n=Table11(:,3);Gamma0=0;GammaTau0=0;GammaTauTau0=0;GammaPI0=1/PI;GammaPIPI0=-1/PI^2;for i=1:9 Gamma0=Gamma0+n0(i)*Tau^J0(i); GammaTau0=GammaTau0+n0(i)*J0(i)*Tau^(J0(i)-1); 160
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GammaTauTau0=GammaTauTau0+n0(i)*J0(i)*(J0(i)-1)*Tau^(J0(i)-2);endGamma0=log(PI)+Gamma0;GammaR=0;GammaPITau0=0;GammaPIR=0;GammaPIPIR=0;GammaTauR=0;GammaTauTauR=0;GammaPITauR=0;for i=1:43 GammaR=GammaR+n(i)*PI^I(i)*(Tau-0.5)^J(i); GammaPIR=GammaPIR+n(i)*I(i)*PI^(I(i)-1)*(Tau-0.5)^J(i); GammaPIPIR=GammaPIPIR+n(i)*I(i)*(I(i)-1)*PI^(I(i)-2)*(Tau-0.5)^J(i); GammaTauR=GammaTauR+n(i)*PI^I(i)*J(i)*(Tau-0.5)^(J(i)-1); GammaTauTauR=GammaTauTauR+n(i)*PI^I(i)*J(i)*(J(i)-1)*(Tau-0.5)^(J(i)-2); GammaPITauR=GammaPITauR+n(i)*I(i)*PI^(I(i)-1)*J(i)*(Tau-0.5)^(J(i)-1);endhv=R*T*(Tau*(GammaTau0+GammaTauR));function [hfg]=LatHeatVap(T,P)%Revised Release on the IAPWS Industrial Formulation 1997 for the%Thermodynamic Properties of Water and Steam (The revision only relatesto%the extension of region 5 to 50 MPa)%%August 2007%%Section 6 - Equations for Region 2Tstar=540; %Reference Temperature, KPstar=1; %Reference Pressure, MPaR=0.461526; %Gas Constant, kJ/kg-KPI=P/Pstar;Tau=Tstar/T;%Enthalpy for VaporTable10=[0,-.96927686500217E1;1,0.10086655968018E2;-5,-0.56087911283020E-2;-4,0.71452738081455E-1;-3,-0.40710498223928;-2,0.14240819171444E1;-1,-0.43839511319450E1;2,-0.28408632460772;3,0.21268463753307E-1];Table11=[1,0,-0.17731742473213E-2;1,1,-0.17834862292358E-1;1,2,-0.45996013696365E-1;1,3,-0.57581259083432E-1;1,6,-0.50325278727930E-1;2,1,-0.33032641670203E-4;2,2,-0.18948987516315E-3;2,4,-0.39392777243355E-2;2,7,-0.43797295650573E-1;2,36,-0.26674547914087E-4;3,0,0.20481737692309E-7;3,1,0.43870667284435E-6;3,3,-0.32277677238570E-4;3,6,-0.15033924542148E-2;3,35,-0.40668253562649E-1;4,1,-0.78847309559367E-9;4,2,0.12790717852285E-7;4,3,0.48225372718507E-6;5,7,0.22922076337661E-5;6,3,-0.16714766451061E-10;6,16,-0.21171472321355E-2;6,35,- 161
177.
function [ST]=SurfTen(T)%Surface Temperature Tension of Ordinary Water Substance%%September 1994Tc=647.096; %Reference TemperatureB=235.8;b=-0.625;u=1.256;Tau=1-(T/Tc);ST=B*(Tau*u)*(1+b*Tau)/1000; 164
178.
function [k]=ThermCond(T,d)%Revised Release on the IAPS Formulation 1985 for the ThermalConductivity%of Ordinary Water Substance%%September 2008Tstar=647.26; %Reference Temperature, Kdstar=317.7; %Reference Density, kg/m^3kstar=1; %Reference Thermal Conductivity, W/m-KTbar=T/Tstar; %Dimensionless Temperaturedbar=d/dstar; %Dimensionless Densitya=[0.0102811,0.0299621,0.0156146,-0.00422464];b=[-0.397070,0.400302,1.060000];B=[-0.171587,2.392190];d=[0.0701309,0.0118520,0.00169937,-1.0200];C=[0.642857,-4.11717,-6.17937,0.00308976,0.0822994,10.0932];dTbar=abs(Tbar-1)+C(4);Q=2+C(5)/dTbar^(3/5);if Tbar>=1 S=1/dTbar;else S=C(6)/dTbar^(3/5);endk0=0;for i=1:4 k0=k0+Tbar^(1/2)*a(i)*Tbar^(i-1);endk1=b(1)+b(2)*dbar+b(3)*exp(B(1)*(dbar+B(2))^2);k2=((d(1)/Tbar^10)+d(2))*dbar^(9/5)*exp(C(1)*(1-dbar^(14/5)))+d(3)*S*dbar^Q*exp((Q/(1+Q))*(1-dbar^(1+Q)))+d(4)*exp(C(2)*Tbar^(3/2)+C(3)/dbar^5);k=1000*kstar*(k0+k1+k2); 165
179.
function [V]=VisLiq(T,D)%Release on the IAPWS Formulation 2008 for the Viscosity of Ordinary%Water Substance%%September 2007Tstar=647.096; %Reference Temperature, KDstar=322.0; %Reference Density, kg/m^3Vstar=1E-6; %Reference Viscosity, Pa-sTbar=T/Tstar;Dbar=D/Dstar;Table1=[1.67752,2.20462,0.6366564,-0.241605];Table2=[5.20094E-1,8.50895E-2,-1.08374,-2.89555E-1,0,0;2.22531E-1,9.99115E-1,1.88797,1.26613,0,1.20573E-1;-2.81378E-1,-9.06851E-1,-7.72479E-1,-4.89837E-1,-2.57040E-1,0;1.61913E-1,2.57399E-1,0,0,0,0;-3.25372E-2,0,0,6.98452E-2,0,0;0,0,0,0,8.72102E-3,0;0,0,0,-4.35673E-3,0,-5.93264E-4];%Solve for Vbar0A=zeros(1,4);for i=1:4 A(i)=Table1(i)/Tbar^(i-1);endAt=sum(A);Vbar0=100*sqrt(Tbar)/At;%Solve for Vbar1B=0;for i=1:6 for j=1:7 B=B+Dbar*(1/Tbar-1)^(i-1)*Table2(j,i)*(Dbar-1)^(j-1); endendVbar1=exp(B);%Solve for Vbar2Vbar2=1;%Solve for VbarVbar=Vbar0*Vbar1*Vbar2;%Solve for ViscosityV=Vbar*Vstar; 166
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