Basics of Fluid Mechanics, 2014a

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11.7. FANNO FLOW 451 11.7.5.1 Maximum Length for the Supersonic Flow It has to be noted and recognized that as opposed to subsonic branch the supersonic branch has a limited length. It also must be recognized that there is a maximum length for which only supersonic flow can exist 27 . These results were obtained from the mathematical derivations but were verified by numerous experiments 28 . The maximum length of the supersonic can be evaluated when M = ∞ as follows: 4fL D 4 fL max D = 1 − M 2 kM 2 + k +1 k+1 2k ln 2 M 2 2 ( 1+ k−1 2 M 2) = −∞ (M →∞) ∼ k ×∞ + k +1 (k +1)∞ ln 2 k (k − 1)∞ = −1 k + k +1 (k +1) ln 2 k (k − 1) = 4fL D (M →∞,k =1.4) = 0.8215 4 fL max D = 4fL D (M →∞,k =1.4) = 0.8215 (11.203) The maximum length of the supersonic flow is limited by the above number. From the above analysis, it can be observed that no matter how high the entrance Mach number will be the tube length is limited and depends only on specific heat ratio, k. 11.7.6 Working Conditions It has to be recognized that there are two regimes that can occur in Fanno flow model one of subsonic flow and the other supersonic flow. Even the flow in the tube starts as a supersonic in parts of the tube can be transformed into the subsonic branch. A shock wave can occur and some portions of the tube will be in a subsonic flow pattern. The discussion has to differentiate between two ways of feeding the tube: converging nozzle or a converging-diverging nozzle. Three parameters, the dimensionless friction, 4 fL , the entrance D Ì ¼ Ì ½ ¡× ¼ Ä ½ ½ ¡× ¼ Ä ¾ ÄÖÖ µ Ä Fig. -11.23. The effects of increase of on the Fanno line. 4 fL D Mach number, M 1 , and the pressure ratio, P 2 /P 1 are controlling the flow. Only a combination of these two parameters is truly independent. However, all the three parameters can be varied and some are discussed separately here. 27 Many in the industry have difficulties in understanding this concept. The author seeks for a nice explanation of this concept for non–fluid mechanics engineers. This solicitation is about how to explain this issue to non-engineers or engineer without a proper background. 28 If you have experiments demonstrating this point, please provide to the undersign so they can be added to this book. Many of the pictures in the literature carry copyright statements and thus can be presented here. ×
452 CHAPTER 11. COMPRESSIBLE FLOW ONE DIMENSIONAL 11.7.6.1 Variations of The Tube Length ( 4fL D ) Effects In the analysis of this effect, it should be assumed that back pressure is constant and/or low as possible as needed to maintain a choked flow. First, the treatment of the two branches are separated. Fanno Flow Subsonic branch For converging nozzle feeding, increasing the tube length results in increasing the exit Mach number (normally denoted herein as M 2 ). Once the Mach number reaches maximum (M =1), no further increase of the exit Mach number can be achieved with same pressure ratio mass flow rate. For increase in the pipe length results in mass flow rate decreases. It is worth noting that entrance Mach number is reduced (as some might explain it to reduce the flow rate). The entrance temperature increases as can be seen from Figure (11.24). T 0 T constant pressure lines Fanno lines 1’’ 1’ 1 s 2 2’ 2’’ Fig. -11.24. The effects of the increase of on the Fanno Line. 4 fL D The velocity therefore must decrease because the loss of the enthalpy (stagnation temperature) is “used.” The density decrease because ρ = P RT and when pressure is remains almost constant the density decreases. Thus, the mass flow rate must decrease. These results are applicable to the converging nozzle. In the case of the converging–diverging feeding nozzle, increase of the dimensionless friction, 4fL D , results in a similar flow pattern as in the converging nozzle. Once the flow becomes choked a different flow pattern emerges. 11.7.6.2 Fanno Flow Supersonic Branch There are several transitional points that change the pattern of the flow. Point a is the choking point (for the supersonic branch) in which the exit Mach number reaches to one. Point b is the maximum possible flow for supersonic flow and is not dependent on the nozzle. The next point, referred here as the critical point c, is the point in which no supersonic flow is possible in the tube i.e. the shock reaches to the nozzle. There is another point d, in which no supersonic flow is possible in the entire nozzle–tube system. Between these transitional points the effect parameters such as mass flow rate, entrance and exit Mach number are discussed. Å Ñ Å ¾ all supersonic flow Ñ ÓÒ×Ø Å ½ a Å ½ b mixed supersonic with subsonic flow with a shock the nozzle between is still choked Ä Fig. -11.25. The Mach numbers at entrance and exit of tube and mass flow rate for Fanno Flow as a function of the 4fL . D c Å ½
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Basics of Fluid Mechanics Genick Ba
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iii ‘We are like dwarfs sitting o
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CONTENTS Nomenclature xxiii GNU Fre
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CONTENTS vii 4 Fluids Statics 69 4.
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CONTENTS ix 9.2.1 Construction of t
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CONTENTS xi 12.5 The Working Equati
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LIST OF FIGURES 1.1 Diagram to expl
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LIST OF FIGURES xv 4.32 Polynomial
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LIST OF FIGURES xvii (a) Streamline
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LIST OF FIGURES xix 12.22The angles
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LIST OF TABLES 1 Books Under Potto
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NOMENCLATURE ¯R Universal gas cons
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LIST OF TABLES xxv w Work per unit
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The Book Change Log Version 0.3.4.0
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LIST OF TABLES xxix ˆ Add discussi
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LIST OF TABLES xxxi ˆ Add the open
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Notice of Copyright For This Docume
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GNU FREE DOCUMENTATION LICENSE xxxv
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GNU FREE DOCUMENTATION LICENSE xxxv
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GNU FREE DOCUMENTATION LICENSE 7. A
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CONTRIBUTOR LIST How to contribute
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CREDITS Typo corrections and other
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About This Author Genick Bar-Meir i
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Prologue For The POTTO Project This
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CREDITS xlix process. Someone has t
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CREDITS li have a new version every
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Prologue For This Book Version 0.3.
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VERSION 0.1 APRIL 22, 2008 lv and t
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How This Book Was Written This book
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Preface "In the beginning, the POTT
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To Do List and Road Map This book i
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CHAPTER 1 Introduction to Fluid Mec
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1.2. BRIEF HISTORY 3 lay is one of
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1.3. KINDS OF FLUIDS 5 results. The
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1.4. SHEAR STRESS 7 For cases where
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1.5. VISCOSITYVISCOSITY 9 The veloc
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1.5. VISCOSITYVISCOSITY 11 always p
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2.5 2.0 1.5 1.0 0.5 0 100 200 300 4
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1.5. VISCOSITYVISCOSITY 15 Chemical
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1.5. VISCOSITYVISCOSITY 17 2 Reduce
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1.5. VISCOSITYVISCOSITY 19 In very
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1.6. FLUID PROPERTIES 21 The total
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1.6. FLUID PROPERTIES 23 straight.
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1.6. FLUID PROPERTIES 25 Table -1.5
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1.6. FLUID PROPERTIES 27 Two layers
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1.6. FLUID PROPERTIES 29 1.6.2.1 Bu
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1.7. SURFACE TENSION 31 terials. Th
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1.7. SURFACE TENSION 33 or Or after
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1.7. SURFACE TENSION 35 The maximum
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1.7. SURFACE TENSION 37 kipenii,”
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1.7. SURFACE TENSION 39 Equation (1
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1.7. SURFACE TENSION 41 Example 1.1
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1.7. SURFACE TENSION 43 Table -1.7.
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CHAPTER 2 Review of Thermodynamics
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2.1. BASIC DEFINITIONS 47 Since the
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2.1. BASIC DEFINITIONS 49 when the
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2.1. BASIC DEFINITIONS 51 Utilizing
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CHAPTER 3 Review of Mechanics This
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3.2. CENTER OF MASS 55 3.2 Center o
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3.3. MOMENT OF INERTIA 57 The momen
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3.3. MOMENT OF INERTIA 59 Equation
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3.3. MOMENT OF INERTIA 61 Example 3
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3.3. MOMENT OF INERTIA 63 or x (1 a
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3.5. ANGULAR MOMENTUM AND TORQUE 65
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3.5. ANGULAR MOMENTUM AND TORQUE 67
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CHAPTER 4 Fluids Statics 4.1 Introd
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4.3. PRESSURE AND DENSITY IN A GRAV
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4.4. FLUID IN A ACCELERATED SYSTEM
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4.5. FLUID FORCES ON SURFACES 101 T
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4.5. FLUID FORCES ON SURFACES 103 x
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4.5. FLUID FORCES ON SURFACES 107 I
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4.5. FLUID FORCES ON SURFACES 111 C
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4.5. FLUID FORCES ON SURFACES 113 A
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4.5. FLUID FORCES ON SURFACES 115 F
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4.6. BUOYANCY AND STABILITY 117 In
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4.6. BUOYANCY AND STABILITY 119 ρ
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4.6. BUOYANCY AND STABILITY 121 ( )
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4.6. BUOYANCY AND STABILITY 123 The
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4.6. BUOYANCY AND STABILITY 125 on
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4.6. BUOYANCY AND STABILITY 127 A w
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4.6. BUOYANCY AND STABILITY 129 It
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4.6. BUOYANCY AND STABILITY 131 Thu
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4.6. BUOYANCY AND STABILITY 133 The
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4.6. BUOYANCY AND STABILITY 135 A n
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4.6. BUOYANCY AND STABILITY 137 of
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4.7. RAYLEIGH-TAYLOR INSTABILITY 13
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4.7. RAYLEIGH-TAYLOR INSTABILITY 14
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4.8. QUALITATIVE QUESTIONS 143 can
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Part I Integral Analysis 145
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148 CHAPTER 5. MASS CONSERVATION di
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150 CHAPTER 5. MASS CONSERVATION It
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152 CHAPTER 5. MASS CONSERVATION 5.
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154 CHAPTER 5. MASS CONSERVATION Th
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156 CHAPTER 5. MASS CONSERVATION Wh
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158 CHAPTER 5. MASS CONSERVATION Th
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162 CHAPTER 5. MASS CONSERVATION ri
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166 CHAPTER 5. MASS CONSERVATION En
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168 CHAPTER 5. MASS CONSERVATION Ex
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170 CHAPTER 5. MASS CONSERVATION So
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172 CHAPTER 5. MASS CONSERVATION In
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174 CHAPTER 6. MOMENTUM CONSERVATIO
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198 CHAPTER 7. ENERGY CONSERVATION
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CHAPTER 8 Differential Analysis 8.1
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8.2. MASS CONSERVATION 229 The firs
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8.2. MASS CONSERVATION 231 Combinin
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8.2. MASS CONSERVATION 233 Equation
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8.2. MASS CONSERVATION 235 The dens
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8.2. MASS CONSERVATION 237 Example
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8.3. CONSERVATION OF GENERAL QUANTI
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8.4. MOMENTUM CONSERVATION 241 8.4
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8.4. MOMENTUM CONSERVATION 243 The
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8.5. DERIVATIONS OF THE MOMENTUM EQ
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8.6. BOUNDARY CONDITIONS AND DRIVIN
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8.6. BOUNDARY CONDITIONS AND DRIVIN
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8.7. EXAMPLES FOR DIFFERENTIAL EQUA
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CHAPTER 9 Dimensional Analysis This
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9.1. INTRODUCTORY REMARKS 275 the r
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9.1. INTRODUCTORY REMARKS 277 End S
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9.1. INTRODUCTORY REMARKS 279 Deriv
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9.2. BUCKINGHAM-π-THEOREM 281 para
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9.2. BUCKINGHAM-π-THEOREM 283 magn
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9.2. BUCKINGHAM-π-THEOREM 285 Tabl
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9.2. BUCKINGHAM-π-THEOREM 287 End
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9.2. BUCKINGHAM-π-THEOREM 289 Equa
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9.2. BUCKINGHAM-π-THEOREM 291 the
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9.2. BUCKINGHAM-π-THEOREM 293 than
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9.2. BUCKINGHAM-π-THEOREM 295 Unde
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9.2. BUCKINGHAM-π-THEOREM 297 Solu
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9.3. NUSSELT’S TECHNIQUE 299 The
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9.3. NUSSELT’S TECHNIQUE 301 boun
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9.3. NUSSELT’S TECHNIQUE 303 Noti
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9.3. NUSSELT’S TECHNIQUE 305 In t
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9.3. NUSSELT’S TECHNIQUE 307 Wher
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9.4. SUMMARY OF DIMENSIONLESS NUMBE
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9.5. SUMMARY 319 It can be noticed
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9.6. APPENDIX SUMMARY OF DIMENSIONL
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BIBLIOGRAPHY [1] Buckingham, E. On
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CHAPTER 10 Inviscid Flow or Potenti
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10.1. INTRODUCTION 327 where in thi
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10.1. INTRODUCTION 329 With the ide
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10.1. INTRODUCTION 331 The partial
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10.2. POTENTIAL FLOW FUNCTION 333 T
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10.2. POTENTIAL FLOW FUNCTION 335 t
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10.2. POTENTIAL FLOW FUNCTION 341 9
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10.3. POTENTIAL FLOW FUNCTIONS INVE
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10.4. CONFORMING MAPPING 369 flow d
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10.4. CONFORMING MAPPING 371 The un
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10.5. UNSTEADY STATE BERNOULLI IN A
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10.6. QUESTIONS 375 Table -10.2. Di
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CHAPTER 11 Compressible Flow One Di
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11.3. SPEED OF SOUND 379 to be cont
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11.3. SPEED OF SOUND 381 Solution T
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11.3. SPEED OF SOUND 383 consider a
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11.4. ISENTROPIC FLOW 385 Now, subs
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11.4. ISENTROPIC FLOW 387 Static Pr
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11.4. ISENTROPIC FLOW 389 that the
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11.4. ISENTROPIC FLOW 391 Using the
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11.4. ISENTROPIC FLOW 393 Solution
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11.4. ISENTROPIC FLOW 395 Expressin
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11.4. ISENTROPIC FLOW 397 The tempe
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11.4. ISENTROPIC FLOW 399 Table -11
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Page 486 and 487: 11.6. ISOTHERMAL FLOW 423 Rearrangi
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Page 492 and 493: 11.6. ISOTHERMAL FLOW 429 The fact
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Page 500 and 501: 11.7. FANNO FLOW 437 The energy con
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Page 524 and 525: 11.7. FANNO FLOW 461 The solution i
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Page 542 and 543: 11.10. INTRODUCTION 479 Example 11.
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12.2. OBLIQUE SHOCK 501 or explicit
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12.2. OBLIQUE SHOCK 503 Fig. -12.9.
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12.2. OBLIQUE SHOCK 505 Table -12.1
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12.2. OBLIQUE SHOCK 507 Example 12.
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12.2. OBLIQUE SHOCK 509 In the obli
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12.2. OBLIQUE SHOCK 511 M x M ys M
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12.2. OBLIQUE SHOCK 513 M x M y T y
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12.2. OBLIQUE SHOCK 515 M x M yw θ
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12.2. OBLIQUE SHOCK 517 The pressur
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12.2. OBLIQUE SHOCK 519 From these
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12.3. PRANDTL-MEYER FUNCTION 521 as
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12.3. PRANDTL-MEYER FUNCTION 523 Th
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12.3. PRANDTL-MEYER FUNCTION 525 No
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1 2 12.7. FLAT BODY WITH AN ANGLE O
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12.8. EXAMPLES FOR PRANDTL-MEYER FU
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12.9. COMBINATION OF THE OBLIQUE SH
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12.9. COMBINATION OF THE OBLIQUE SH
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CHAPTER 13 Multi-Phase Flow 13.1 In
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13.4. KIND OF MULTI-PHASE FLOW 537
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13.5. CLASSIFICATION OF LIQUID-LIQU
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13.5. CLASSIFICATION OF LIQUID-LIQU
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13.6. MULTI-PHASE FLOW VARIABLES DE
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13.6. MULTI-PHASE FLOW VARIABLES DE
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13.7. HOMOGENEOUS MODELS 547 13.7 H
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13.7. HOMOGENEOUS MODELS 549 There
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13.8. SOLID-LIQUID FLOW 551 where
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13.8. SOLID-LIQUID FLOW 553 So far
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13.9. COUNTER-CURRENT FLOW 555 The
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13.9. COUNTER-CURRENT FLOW 561 For
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13.10. MULTI-PHASE CONCLUSION 565 1
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APPENDIX A The Mathematics Backgrou
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A.1. VECTORS 569 where θ is the an
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A.1. VECTORS 571 Divergence The sam
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A.1. VECTORS 573 R · S =(xî + y 2
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A.1. VECTORS 575 The curl is writte
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A.1. VECTORS 577 The divergence of
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A.2. ORDINARY DIFFERENTIAL EQUATION
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A.3. PARTIAL DIFFERENTIAL EQUATIONS
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A.4. TRIGONOMETRY 595 A.4 Trigonome
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SUBJECTS INDEX 597 Subjects Index S
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SUBJECTS INDEX 599 Ideal gas, 91 Re
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SUBJECTS INDEX 601 Slip condition r
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AUTHORS INDEX 603 Authors Index B B
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