Basics of Fluid Mechanics, 2014a

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7.4. ENERGY EQUATION IN ACCELERATED SYSTEM 213 Subtracting equation (7.84) from equation (7.75) results in change in pressure energy change change in kinetic in potential energy {(∫ }} ∫ ){ { }} { dP 0=w shaft + dP ρ ∣ − U 2 energy 2 2 − U { }} { 1 2 ρ ∣ + + g (z 2 − z 1 ) (7.85) 1 2 Equation (7.85) for constant density is 0=w shaft + P 2 − P 1 ρ For no shaft work equation (7.86) reduced to 0= P 2 − P 1 ρ + U 2 2 − U 1 2 2 + U 2 2 − U 1 2 2 7.4 Energy Equation in Accelerated System + g (z 2 − z 1 ) (7.86) + g (z 2 − z 1 ) (7.87) In the discussion so far, it was assumed that the control volume is at rest. The only acceptation to the above statement, is the gravity that was compensated by the gravity potential. In building the gravity potential it was assumed that the gravity is a conservative force. It was pointed earlier in this book that accelerated forces can be translated to potential force. In many cases, the control volume is moving in accelerated coordinates. These accelerations will be translated to potential energy. The accelerations are referring to two kinds of acceleration, linear and rotational. There is no conceptional difference between these two accelerations. However, the mathematical treatment is somewhat different which is the reason for the separation. General Acceleration can be broken into a linear acceleration and a rotating acceleration. 7.4.1 Energy in Linear Acceleration Coordinate The potential is defined as ∫ 2 P.E. = − ref F · dl (7.88) In Chapter 3 a discussion about gravitational energy potential was presented. For example, for the gravity force is F = − GMm r 2 (7.89) Where G is the gravity coefficient and M is the mass of the Earth. r and m are the distance and mass respectively. The gravity potential is then PE gravity = − ∫ r ∞ − GMm r 2 dr (7.90)
214 CHAPTER 7. ENERGY CONSERVATION The reference was set to infinity. The gravity force for fluid element in small distance then is gdzdm. The work this element moving from point 1 to point 2 is ∫ 2 1 gdzdm= g (z 2 − z 1 ) dm (7.91) The total work or potential is the integral over the whole mass. 7.4.2 Linear Accelerated System The acceleration can be employed in similar fashion as the gravity force. The linear acceleration “creates” a conservative force of constant force and direction. The “potential” of moving the mass in the field provides the energy. The Force due to the acceleration of the field can be broken into three coordinates. Thus, the element of the potential is dPE a = a · dldm (7.92) The total potential for element material PE a = ∫ (1) (0) a · dldm=(a x (x 1 − x 0 ) a y (y 1 − y 0 ) a z (z 1 − z 0 )) dm (7.93) At the origin (of the coordinates) x =0, y =0, and z =0. Using this trick the notion of the a x (x 1 − x 0 ) can be replaced by a x x. The same can be done for the other two coordinates. The potential of unit material is ∫ PE atotal = (a x x + a y y + a z z) ρdV (7.94) sys The change of the potential with time is D Dt PE atotal = D ∫ (a x x + a y y + a z z) dm (7.95) Dt sys Equation can be added to the energy equation as ˙Q − Ẇ = D ∫ [E u + U 2 ] Dt sys 2 + a x x + a y y +(a z + g)z ρdV (7.96) The Reynolds Transport Theorem is used to transferred the calculations to control volume as Energy Equation in Linear Accelerated Coordinate ˙Q − Ẇ = d ∫ [E u + U 2 ] dt cv 2 + a x x + a y y +(a z + g)z ρdV ∫ + (h + U 2 ) cv 2 + a x x + a y y +(a z + g)z U rn ρdA ∫ + PU bn dA cv (7.97)
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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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160 CHAPTER 5. MASS CONSERVATION Wh
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Page 290 and 291: CHAPTER 8 Differential Analysis 8.1
Page 292 and 293: 8.2. MASS CONSERVATION 229 The firs
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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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11.4. ISENTROPIC FLOW 405 Example 1
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11.5. NORMAL SHOCK 407 In a shock w
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11.5. NORMAL SHOCK 409 Energy equat
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11.5. NORMAL SHOCK 411 The density
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11.5. NORMAL SHOCK 413 or in a dime
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11.5. NORMAL SHOCK 415 which differ
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11.5. NORMAL SHOCK 417 is different
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11.5. NORMAL SHOCK 419 Table -11.3.
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11.6. ISOTHERMAL FLOW 421 Table -11
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11.6. ISOTHERMAL FLOW 423 Rearrangi
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11.6. ISOTHERMAL FLOW 425 dT 0 T 0
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11.6. ISOTHERMAL FLOW 427 10 2 0 1
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11.6. ISOTHERMAL FLOW 429 The fact
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11.6. ISOTHERMAL FLOW 431 Calculati
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11.6. ISOTHERMAL FLOW 433 maximum M
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11.6. ISOTHERMAL FLOW 435 From the
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11.7. FANNO FLOW 437 The energy con
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11.7. FANNO FLOW 439 Differentiatin
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11.7. FANNO FLOW 441 11.7.3 The Mec
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11.7. FANNO FLOW 443 pressure on th
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11.7. FANNO FLOW 445 10 2 0 1 2 3 4
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11.7. FANNO FLOW 447 End Solution A
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11.7. FANNO FLOW 449 M x M y T y T
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11.7. FANNO FLOW 451 11.7.5.1 Maxim
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11.7. FANNO FLOW 453 At the startin
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11.7. FANNO FLOW 455 Fanno Flow 5 4
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11.7. FANNO FLOW 457 11.7.7.1 Choki
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11.7. FANNO FLOW 459 back pressure
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11.7. FANNO FLOW 461 The solution i
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11.7. FANNO FLOW 463 11.7.8 The Pra
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11.7. FANNO FLOW 465 0.9 0.8 0.7 0.
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11.7. FANNO FLOW 467 3.0 2.5 2.0 Co
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11.8. THE TABLE FOR FANNO FLOW 469
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11.9. RAYLEIGH FLOW 471 Table -11.6
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11.10. INTRODUCTION 473 or in simpl
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11.10. INTRODUCTION 475 or explicit
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11.10. INTRODUCTION 477 Table -11.7
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11.10. INTRODUCTION 479 Example 11.
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11.10. INTRODUCTION 481 Example 11.
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11.10. INTRODUCTION 483 what the ma
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CHAPTER 12 Compressible Flow 2-Dime
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12.2. OBLIQUE SHOCK 487 12.2 Obliqu
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12.2. OBLIQUE SHOCK 489 The relatio
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12.2. OBLIQUE SHOCK 491 Where S = 3
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12.2. OBLIQUE SHOCK 493 by evaluati
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12.2. OBLIQUE SHOCK 495 Figure (12.
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12.2. OBLIQUE SHOCK 497 be made. Th
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12.2. OBLIQUE SHOCK 499 demonstrate
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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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