Modern Engineering Thermodynamics

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16.6 Choked Flow 665 T a a p a = constant T T b T a′ b a ′ p exit = p b = constant FIGURE 16.14 Isentropic expansion of a vapor near its saturation at state a through a nozzle to a nonequilibrium supersaturated state at a’. This is followed by an irreversible nucleation and condensation process to equilibrium state b. s 16.6 CHOKED FLOW The mass flow rate per unit area in an isentropic nozzle can be determined for an ideal gas from Eqs. (16.11), (16.12), and (16.13) and the ideal gas formula, ρ = p/RT, as _m /A = ρV = ðp/RTÞðMcÞ pffiffiffiffiffiffiffiffiffiffiffiffi = pMð kg c RTÞðT os /T os Þ 1/2 ðp os /p os Þ/RT = p os Mðkg c /RT os Þ 1/2 ðT os /TÞ 1/2 ðp/p os Þ (16.21) = p os M kg 1/2 c 1 + k − 1 ðk+1Þ/2ð1−kÞ RT os 2 M2 Figure 16.15 is a schematic of the variation in _m /A with the ratio of the back pressure p B to the upstream stagnation pressure p os .WhenM throat = M = 1:0, the nozzle is passing the maximum possible flow _m max : Since we always have M throat ≤ 1:0, clearly the maximum nozzle mass flow rate occurs when the throat velocity is sonic. Then _m = _m max ,M throat = M = 1:0, A throat = A , and Eq. (16.21) becomes 1/2 _m max /A kg = p c k + 1 ðk+1Þ/2ð1−kÞ os (16.22a) 2 RT os Note that the value of _m max /A depends only on the upstream isentropic stagnation properties and is completely independent of the downstream conditions. For air, k = 1.40 and, in <strong>Engineering</strong> English units, this equation reduces to ð _m max /A Þ air = 0:532 lbm pffiffiffi . R os pffiffiffiffiffiffi (16.22b) lbf .s T os where _m max must be in lbm/s, A must be in in 2 , p os must be in lbf/in 2 ,andT os must be in R. This is called Fliegner’s (or sometimes Zeuner’s) formula and was experimentally discovered in the 1870s. In metric SI units, Eq. (16.22a) reduces to ð _m max /A Þ air = 0:0404 kg pffiffiffi . K os pffiffiffiffiffiffi (16.22c) N.s T os where _m max must be in kg/s, A must be in m 2 , p os must be in N/m 2 ,andT os must be in K. m/A m max /A* Choked flow region M throat = 1 0 p B /p os M throat < 1 FIGURE 16.15 The relative variation in dimensionless isentropic converging nozzle air mass flow rate with increasing back pressure to upstream stagnation pressure ratio. Since the Mach number in a converging nozzle cannot exceed 1.0, the maximum flow rate through the nozzle occurs when the Mach number at the throat is 1.0. 1.0
666 CHAPTER 16: Compressible Fluid Flow EXAMPLE 16.7 Automobile safety equipment now includes air bags, which are essentially large balloons that inflate very quickly on vehicle impact. To investigate the feasibility of using high-pressure compressed air to inflate bags, we want to determine the minimum tube diameter necessary to completely fill a spherical air bag to a diameter of 3.00 feet in 30.0 milliseconds to a pressure of 15.0 psia using air from a compressed air storage tank maintained at p os = 1500. psia and T os = 70.0°F. Solution If the air bags are to be inflated from a high-pressure compressed air storage tank, the maximum air flow rate from the tank corresponds to the minimum required fill-tube diameter operating under choked flow conditions. The average mass flow rate of air into the bag is where _m avg = V bag = πD3 bag 6 Mass of air in the bag Required bag fill time = ρ airV bag t fill = πð3:00 ftÞ3 6 = 14:1ft 3 The temperature of the air entering the bag can be computed from Eq. (16.18) as 2 2 T air = T os = ð70:0 + 459:67 RÞ = 441 R k + 1 1:40 + 1 Consequently, the density of the air entering the bag at 15 psia is ρ air = p air = ð15:0 lbf/ft2 Þð144 in 2 /ft 2 Þ lbm = 0:0918 R air T air ð53:34 ft.lbf/lbm.RÞð441 RÞ ft 3 The minimum diameter of the bag fill tube may now be determined from Eq. (16.22b) by setting _m avg = _m max : or A tube = πD2 tube 4 = _m pffiffiffiffiffiffi max T os 0:532πp os D tube = 4 _m pffiffiffiffiffiffi 1/2 " pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi # 1/2 avg T os 4ð43:4 lbm/sÞ 70:0 + 459:67 R = p ffiffiffi = 1:26 in 0:532πp os ð0:532 lbm. R /lbf .sÞðπÞð1500: lbf/in 2 Þ The graph in Figure 16.16 shows the variation in fill-tube diameter with compressed air pressure. Note that, even with a storage tank at 4000. psia, a ¾-inch diameter fill tube is required. Also, since our analysis was for an isentropic process, losses in the system would necessitate using a larger diameter to achieve the required inflation. Considering the size of the fill tube and the financial cost of this type of inflation system, it is not surprising to find that air bags are actually inflated by a rapid chemical reaction (explosion) that releases a large amount of gas in a short period. 4 Fill-tube diameter (inches) 3 2 1 0 0 500 1000 1500 2000 2500 3000 3500 4000 Fill-tank pressure (psia) FIGURE 16.16 Example 16.7.
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Modern Engineering Thermodynamics
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Modern Engineering Thermodynamics R
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Dedication WHAT IS AN ENGINEER AND
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Contents PREFACE . . . . . . . . .
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Contents ix 7.6 Heat Engines Runnin
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Contents xi 14.13 Air Standard Gas
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Preface TEXT OBJECTIVES This textbo
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Preface xv Step 5. Write down the b
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Acknowledgments I wish to acknowled
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Resources That Accompany This Book
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List of Symbols A a B COP CR c c p
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Prologue PARIS FRANCE, 10:35 AM, AU
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CHAPTER 1 The Beginning CONTENTS 1.
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1.2 Why Is Thermodynamics Important
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1.3 Getting Answers: A Basic Proble
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1.5 How Do We Measure Things? 7 bei
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1.6 Temperature Units 9 THE DEVELOP
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1.7 Classical Mechanical and Electr
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1.7 Classical Mechanical and Electr
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1.9 Modern Units Systems 15 where M
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1.10 Significant Figures 17 CRITICA
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1.10 Significant Figures 19 WHAT AB
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1.11 Potential and Kinetic Energies
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1.11 Potential and Kinetic Energies
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Summary 25 FIGURE 1.19 Case study 1
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Problems 27 Problems (* indicates p
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Problems 29 14. Determine the mass
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Problems 31 49. Using the CGS units
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CHAPTER 2 Thermodynamic Concepts CO
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2.3 Phases of Matter 35 System boun
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2.4 System States and Thermodynamic
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2.6 Thermodynamic Processes 39 WHAT
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2.7 Pressure and Temperature Scales
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2.9 The Continuum Hypothesis 43 Sur
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2.10 The Balance Concept 45 Solutio
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2.11 The Conservation Concept 47 or
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2.11 The Conservation Concept 49 Th
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Summary 51 change in the mass of X
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Problems 53 Problems (* indicates p
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Problems 55 and Death rate = α 2 +
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CHAPTER 3 Thermodynamic Properties
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3.3 Fun with Mathematics 59 CRITICA
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3.4 Some Exciting New Thermodynamic
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3.6 Enthalpy 63 WHO WAS AMALIE EMMY
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3.7 Phase Diagrams 65 WHO WAS EMMY
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Pressure Vapor 3.7 Phase Diagrams 6
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3.7 Phase Diagrams 69 WHAT IS A “
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3.7 Phase Diagrams 71 considerable
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3.8 Quality 73 10 5 300 C Critical
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3.8 Quality 75 Although Eq. (3.26)
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3.9 Thermodynamic Equations of Stat
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3.10 Thermodynamic Tables 85 Critic
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3.12 Thermodynamic Charts 87 vð100
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3.13 Thermodynamic Property Softwar
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Summary 91 and e = E/m = u + V2 2g
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Problems 93 c. For a saturated mixt
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Problems 95 specific enthalpy of sa
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Problems 97 Table 3.23 Problem 65 M
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CHAPTER 4 The First Law of Thermody
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4.3 The First Law of Thermodynamics
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4.3 The First Law of Thermodynamics
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4.4 Energy Transport Mechanisms 105
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4.5 Point and Path Functions 107 4.
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4.6 Mechanical Work Modes of Energy
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4.7 Nonmechanical Work Modes of Ene
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4.9 Work Efficiency 125 In the case
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4.12 Heat Modes of Energy Transport
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4.13 Heat Transfer Modes 129 Table
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4.14 A Thermodynamic Problem Solvin
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4.14 A Thermodynamic Problem Solvin
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4.15 How to Write a Thermodynamics
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4.15 How to Write a Thermodynamics
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Summary 139 The general open system
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Problems 141 11.* Determine the hea
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Problems 143 where K = 0.810 lbf. D
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Problems 145 Table 4.12 Problem 67
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CHAPTER 5 First Law Closed System A
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5.2 Sealed, Rigid Containers 149 In
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5.4 Power Plants 151 Step 7. Calcul
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5.5 Incompressible Liquids 153 The
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1Q 2 − 1 W 2 = mðu 2 − u 1 Þ
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5.8 Closed System Unsteady State Pr
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5.9 The Explosive Energy of Pressur
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Problems 161 Problems (* indicates
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Problems 163 31. A thermoelectric g
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Problems 165 where v, T, and r are
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CHAPTER 6 First Law Open System App
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6.2 Mass Flow Energy Transport 169
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6.3 Conservation of Energy and Cons
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6.4 Flow Stream Specific Kinetic an
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6.5 Nozzles and Diffusers 175 m (a)
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6.5 Nozzles and Diffusers 177 We ar
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6.6 Throttling Devices 179 These as
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6.6 Throttling Devices 181 For an i
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6.7 Throttling Calorimeter 183 The
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6.8 Heat Exchangers 185 Convective
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6.9 Shaft Work Machines 187 6.9 SHA
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6.9 Shaft Work Machines 189 1 Basem
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6.10 Open System Unsteady State Pro
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Problems 197 we have _m R / _m D =
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Problems 201 32.* A commercial slid
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Summary 203 59. Incompressible liqu
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CHAPTER 7 Second Law of Thermodynam
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7.3 The Second Law of Thermodynamic
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7.4 Carnot’s Heat Engine and the
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7.4 Carnot’s Heat Engine and the
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7.5 The Absolute Temperature Scale
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7.5 The Absolute Temperature Scale
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7.6 Heat Engines Running Backward 2
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7.7 Clausius’s Definition of Entr
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7.8 Numerical Values for Entropy 22
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7.10 Differential Entropy Balance 2
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7.11 Heat Transport of Entropy 229
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7.13 Entropy Production Mechanisms
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7.14 Heat Transfer Production of En
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7.15 Work Mode Production of Entrop
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7.15 Work Mode Production of Entrop
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7.16 Phase Change Entropy Productio
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Summary 241 ■ Indirect method inv
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Problems 243 amount of work W irr i
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Problems 245 a. If the heat pump is
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Problems 247 51. Develop a program
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CHAPTER 8 Second Law Closed System
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8.2 Systems Undergoing Reversible P
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8.3 Systems Undergoing Irreversible
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8.4 Diffusional Mixing 271 Exercise
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Summary 273 Note that this example
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Problems 275 15. A 20.0 ft 3 tank c
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Problems 277 46. a. Determine a for
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CHAPTER 9 Second Law Open System Ap
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9.4 Open System Entropy Balance Equ
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9.4 Open System Entropy Balance Equ
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9.5 Nozzles, Diffusers, and Throttl
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9.5 Nozzles, Diffusers, and Throttl
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9.6 Heat Exchangers 289 Exercises 1
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9.6 Heat Exchangers 291 (T H ) (T H
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9.7 Mixing 293 Now, _m air is given
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9.7 Mixing 295 Critical value of y,
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9.9 Unsteady State Processes in Ope
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Problems 309 Multiplying this equat
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Problems 311 entropy production rat
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Problems 313 heat is added to the b
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Problems 315 55.* Determine the max
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Problems 317 The cavitation process
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CHAPTER 10 Availability Analysis CO
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10.3 What Are Conservative Forces?
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10.6 Availability 323 WHAT IS A SYS
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10.6 Availability 325 and the total
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10.7 Closed System Availability Bal
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10.7 Closed System Availability Bal
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10.8 Flow Availability 331 EXAMPLE
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s − s 0 = c ln T T 0 10.8 Flow Av
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10.10 Modified Availability Rate Ba
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10.10 Modified Availability Rate Ba
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10.11 Energy Efficiency Based on th
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Summary 351 In this problem, we hav
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Summary 353 7. The general open sys
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Problems 355 12.5 Btu/hr·ft ·R. I
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Problems 357 flow rate of 15.0 lbm/
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Problems 359 85. Create a specific
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CHAPTER 11 More Thermodynamic Relat
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11.2 Two New Properties: Helmholtz
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11.2 Two New Properties: Helmholtz
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11.4 Maxwell Equations 367 11.4 MAX
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11.4 Maxwell Equations 369 then,
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11.5 The Clapeyron Equation 371 and
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11.6 Determining u, h, and s from p
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11.7 Constructing Tables and Charts
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11.8 Thermodynamic Charts 381 so th
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11.9 Gas Tables 383 where p r is th
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11.10 Compressibility Factor and Ge
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11.11 Is Steam Ever an Ideal Gas? 3
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Summary 399 equation, and a series
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Problems 401 20.* Estimate h fg for
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Problems 403 Design Problems The fo
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CHAPTER 12 Mixtures of Gases and Va
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12.2 Thermodynamic Properties of Ga
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12.3 Mixtures of Ideal Gases 413 Th
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12.3 Mixtures of Ideal Gases 415 Co
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12.4 Psychrometrics 417 Finally, th
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12.4 Psychrometrics 419 EXAMPLE 12.
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12.6 The Sling Psychrometer 421 The
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12.6 The Sling Psychrometer 423 p w
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12.7 Air Conditioning 425 WHAT ENVI
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12.8 Psychrometric Enthalpies 427 E
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12.8 Psychrometric Enthalpies 429 o
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12.9 Mixtures of Real Gases 431 Whe
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12.9 Mixtures of Real Gases 433 and
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12.9 Mixtures of Real Gases 435 The
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12.9 Mixtures of Real Gases 437 Sol
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Summary 439 Last, we combine Dalton
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Problems 443 exhaust gas is an idea
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Problems 445 57.* Cooling towers ar
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CHAPTER 13 Vapor and Gas Power Cycl
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13.2 Part I. Engines and Vapor Powe
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13.4 Rankine Cycle 457 A B Reversib
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13.5 Operating Efficiencies 459 For
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13.5 Operating Efficiencies 461 13.
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13.5 Operating Efficiencies 463 b.
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13.5 Operating Efficiencies 465 Sta
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13.6 Rankine Cycle with Superheat 4
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13.7 Rankine Cycle with Regeneratio
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13.8 The Development of the Steam T
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13.9 Rankine Cycle with Reheat 477
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13.9 Rankine Cycle with Reheat 479
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13.10 Modern Steam Power Plants 481
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13.12 Air Standard Power Cycles 487
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13.13 Stirling Cycle 489 Q H 4 1 4
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13.14 Ericsson Cycle 491 Q H 4 1 3
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13.15 Lenoir Cycle 493 Exercises 28
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13.16 Brayton Cycle 495 Solution Us
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13.16 Brayton Cycle 497 Equation (7
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13.17 Aircraft Gas Turbine Engines
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13.17 Aircraft Gas Turbine Engines
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13.18 Otto Cycle 503 T Q H 1 1 1 v
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13.18 Otto Cycle 505 Exercises 40.
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13.18 Otto Cycle 507 and, since the
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13.20 Miller Cycle 509 13.19.1 Mode
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13.20 Miller Cycle 511 Then, from F
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13.21 Diesel Cycle 513 to stroll on
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13.21 Diesel Cycle 515 Solution a.
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13.22 Modern Prime Mover Developmen
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13.23 Second Law Analysis of Vapor
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Summary 525 Fill port 160 mm End of
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Problems 527 7. The thermal efficie
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efficiency increase if the condense
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Problems 531 horsepower hour. Assum
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Problems 533 72. Determine the valu
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CHAPTER 14 Vapor and Gas Refrigerat
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14.3 Carnot Refrigeration Cycle 537
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14.4 In the Beginning There Was Ice
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14.4 In the Beginning There Was Ice
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14.5 Vapor-Compression Refrigeratio
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14.5 Vapor-Compression Refrigeratio
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14.6 Refrigerants 547 T 3 2s 2 p 3
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14.7 Refrigerant Numbers 549 14.7 R
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14.7 Refrigerant Numbers 551 R-110
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14.8 CFCs and the Ozone Layer 553 H
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14.9 Cascade and Multistage Vapor-C
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14.10 Absorption Refrigeration 561
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14.11 Commercial and Household Refr
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14.14 Reversed Brayton Cycle Refrig
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14.14 Reversed Brayton Cycle Refrig
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14.15 Reversed Stirling Cycle Refri
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14.16 Miscellaneous Refrigeration T
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14.16 Miscellaneous Refrigeration T
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14.18 Second Law Analysis of Refrig
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14.18 Second Law Analysis of Refrig
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2. The coefficient of performance o
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Problems 585 13.* A refrigeration u
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Problems 587 Loop B Station 1B Stat
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Problems 589 under these conditions
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CHAPTER 15 Chemical Thermodynamics
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15.2 Stoichiometric Equations 593 1
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15.2 Stoichiometric Equations 595 C
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15.3 Organic Fuels 597 ANSWERS SOME
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15.4 Fuel Modeling 599 Hydrocarbon
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15.4 Fuel Modeling 601 equation, si
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15.5 Standard Reference State 603 I
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15.6 Heat of Formation 605 and H 2
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15.7 Heat of Reaction 607 EXAMPLE 1
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15.7 Heat of Reaction 609 CRITICAL
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15.7 Heat of Reaction 611 Then, h P
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15.8 Adiabatic Flame Temperature 61
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17.8 Locomotion Transport Number 71
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17.9 Thermodynamics of Aging and De
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17.9 Thermodynamics of Aging and De
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1/3 V most efficient = P o ρAC D S
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Problems 723 a. If the monster cons
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Problems 725 the officer asks Paul
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CHAPTER 18 Introduction to Statisti
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18.3 Kinetic Theory of Gases 729 3.
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U trans = 3 2 NkT (Continued ) 18.3
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18.4 Intermolecular Collisions 733
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18.5 Molecular Velocity Distributio
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18.5 Molecular Velocity Distributio
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18.6 Equipartition of Energy 739 We
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18.7 Introduction to Mathematical P
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18.8 Quantum Statistical Thermodyna
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18.9 Three Classical Quantum Statis
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18.11 Monatomic Maxwell-Boltzmann G
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18.12 Diatomic Maxwell-Boltzmann Ga
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18.12 Diatomic Maxwell-Boltzmann Ga
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18.13 Polyatomic Maxwell-Boltzmann
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Summary 759 In this chapter, we sum
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Problems 761 4. Find the temperatur
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CHAPTER 19 Introduction to Coupled
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19.3 Linear Phenomenological Equati
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19.4 Thermoelectric Coupling 767 Eq
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19.4 Thermoelectric Coupling 769 Th
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19.4 Thermoelectric Coupling 771 wh
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19.4 Thermoelectric Coupling 773 Th
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19.4 Thermoelectric Coupling 775 b.
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19.5 Thermomechanical Coupling 777
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water), it seems reasonable that li
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Problems 785 b. For a Knudson gas,
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Appendix A: Physical Constants and
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Appendix B: Greek and Latin Origins
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Appendix B 791 Table B.4 Plural End
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Index Page numbers followed by f in
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Index 795 E e (specific energy), 10
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Index 797 Isobaric coefficient of v
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Index 799 Ranque, Georges Joseph, 3
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Index 801 William III, King of Engl
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Modern Engineering Thermodynamics

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