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610 Chapter 11 ■ Compressible Flow Insulated wall Adiabatic flow Section (1) Section (2) Control volume F I G U R E 11.15 area flow. Adiabatic constant or ȟ V 2 2 ȟ 0 constant where h 0 is the stagnation enthalpy. For an ideal gas we gather from Eq. 11.9 that ȟ ȟ 0 c p 1T T 0 2 (11.72) (11.73) so that by combining Eqs. 11.72 and 11.73 we get or T V 2 2c p T 0 constant T 1rV22 2c p r 2 T 0 constant By substituting the ideal gas equation of state 1Eq. 11.12 into Eq. 11.74 we obtain T 1rV22 T 2 2c p 1p 2 R 2 2 T 0 constant (11.74) (11.75) From the continuity equation 1Eq. 11.402 we can conclude that the density–velocity product, rV, is constant for a given Fanno flow since the area, A, is constant. Also, for a particular Fanno flow, the stagnation temperature, T 0 , is fixed. Thus, Eq. 11.75 allows us to calculate values of fluid temperature corresponding to values of fluid pressure in the Fanno flow. We postpone our discussion of how pressure is determined until later. As with earlier discussions in this chapter, it is helpful to describe Fanno flow with a temperature–entropy diagram. From the second T ds relationship, an expression for entropy variation was already derived 1Eq. 11.222. If the temperature, T 1 , pressure, p 1 , and entropy, at the entrance of the Fanno flow duct are considered as reference values, then Eq. 11.22 yields s s 1 c p ln T T 1 R ln p p 1 s 1 , (11.76) Entropy increases in Fanno flows because of wall friction. Equations 11.75 and 11.76 taken together result in a curve with T–s coordinates as is illustrated in Fig. 11.16. This curve involves a given gas 1c p and R2 with fixed values of stagnation temperature, density–velocity product, and inlet temperature, pressure, and entropy. Curves like the one sketched in Fig. 11.16 are called Fanno lines. T Fanno line Constant entropy line p a a T a s F I G U R E 11.16 diagram for Fanno flow. The T–s
11.5 Nonisentropic Flow of an Ideal Gas 611 E XAMPLE 11.11 GIVEN Air 1k 1.42 enters [section 112] an insulated, constant cross-sectional area duct with the following properties: T 0 518.67 °R T 1 514.55 °R p 1 14.3 psia Compressible Flow with Friction (Fanno Flow) FIND For Fanno flow, determine corresponding values of fluid temperature and entropy change for various values of downstream pressures and plot the related Fanno line. SOLUTION To plot the Fanno line we can use Eq. 11.75 and Eq. 11.76 to construct a table of values of temperature and entropy change corresponding to different levels of pressure in the Fanno flow. We need values of the ideal gas constant and the specific heat at constant pressure to use in Eqs. 1 and 2. From Table 1.7 we read for air or From Eq. 11.14 we obtain From Eqs. 11.1 and 11.69 we obtain and rV is constant for this flow But and from Eq. 11.56 Thus, with T 1rV22 T 2 2c p p 2 R 2 T 0 constant s s 1 c p ln T T 1 R ln p p 1 R 1716 1ft # lb21slug # °R2 53.3 1ft # lb21lbm # °R2 1112 fts c p Rk k 1 c p 353.3 1ft # lb21lbm # °R2411.42 1.4 1 187 1ft # lb21lbm # °R2 rV p RT Ma1RTk rV r 1 V 1 p 1 RT 1 Ma 1 1RT 1 k T 1 514.55 °F T 0 518.67 °R 0.992 Ma 1 A a 1 0.992 1 b .02 0.2 2RT 1 k 211.42353.3 1ft # lb21lbm # °R241514.55 °R2 196 31ft # lb2lbm4 1 2 3132.2 lbm # fts 2 2lb4 1 2 (1) (2) (3) (4) Eq. 4 becomes or For p 7 psia we have from Eq. 1 or Thus, since 1 lb 32.2 lbm # fts 2 we obtain Hence, where T is in °R. From Eq. 2, we obtain or rV 114.3 psia21144 in.2 ft 2 20.211112 fts2 353.3 1ft # lb21lbm # °R241514.55 °R2 316.7 lbm1ft 2 # s24 2 T 2 T 17 psia2 2 1144 in. 2 ft 2 2 2 23187 1ft # lb21lbm # °R24 353.3 1ft # lb21lbm # °R24 2 518.67 °R rV 16.7 lbm1ft 2 # s2 2.08 10 3 31lbm # fts 2 21lb # °R24 T 2 T 518.67 °R 0 6.5 10 5 T 2 T 518.67 0 T 502.3 °R s s 1 3187 1ft # lb21lbm # 502.3 °R °R24 ln a 514.55 °R b 353.3 1ft # lb21lbm # °R24 ln a 7 psia 14.3 psia b s s 1 33.6 1ft # lb21lbm # °R2 (Ans) (Ans) Proceeding as outlined above, we construct the table of values shown below and graphed as the Fanno line in Fig. E11.11. The T, °R 550 500 450 400 350 300 35 40 45 50 55 s – s 1 , ________ (ft • lb) (lbm•°R) F I G U R E E11.11
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Achieve Positive Learning Outcomes
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WileyPLUS combines robust course ma
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Sixth Edition F undamentals of Flui
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About the Authors vii Bruce R. Muns
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P reface This book is intended for
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Preface xi Life Long Learning Probl
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Preface xiii WileyPLUS offers today
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Featured in this Book xv LEARNING O
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C ontents 1 INTRODUCTION 1 Learning
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Contents xix 6.4.3 Irrotational Flo
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12.6 Axial-Flow and Mixed-Flow Pump
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10 8 Jupiter red spot diameter 2 Ch
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4 Chapter 1 ■ Introduction and do
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6 Chapter 1 ■ Introduction up the
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8 Chapter 1 ■ Introduction the SI
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10 Chapter 1 ■ Introduction E XAM
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12 Chapter 1 ■ Introduction TABLE
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16 Chapter 1 ■ Introduction Crude
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20 Chapter 1 ■ Introduction Kinem
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22 Chapter 1 ■ Introduction As th
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24 Chapter 1 ■ Introduction Boili
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28 Chapter 1 ■ Introduction The r
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30 Chapter 1 ■ Introduction fluid
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32 Chapter 1 ■ Introduction Secti
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34 Chapter 1 ■ Introduction avail
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2 Fluid Statics CHAPTER OPENING PHO
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40 Chapter 2 ■ Fluid Statics in a
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42 Chapter 2 ■ Fluid Statics p Fo
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44 Chapter 2 ■ Fluid Statics the
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58 Chapter 2 ■ Fluid Statics Free
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60 Chapter 2 ■ Fluid Statics The
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64 Chapter 2 ■ Fluid Statics h 1
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66 Chapter 2 ■ Fluid Statics SOLU
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68 Chapter 2 ■ Fluid Statics COMM
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70 Chapter 2 ■ Fluid Statics V2.8
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92 Chapter 2 ■ Fluid Statics 2.12
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94 Chapter 3 ■ Elementary Fluid D
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100 Chapter 3 ■ Elementary Fluid
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148 Chapter 4 ■ Fluid Kinematics
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264 Chapter 6 ■ Differential Anal
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7 Dimensional Analysis, Similitude,
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334 Chapter 7 ■ Dimensional Analy
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384 Chapter 8 ■ Viscous Flow in P
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WARNING Stand clear of Hazard areas
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462 Chapter 9 ■ Flow over Immerse
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∋ 530 Chapter 9 ■ Flow over Imm
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10 Open-Channel Flow CHAPTER OPENIN
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536 Chapter 10 ■ Open-Channel Flo
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660 Chapter 12 ■ Turbomachines 50
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V 1 W 1 = W 2 U 676 Chapter 12 ■
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702 Appendix A ■ Computational Fl
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A ppendix B Physical Properties of
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716 Appendix B ■ Physical Propert
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718 Appendix B ■ Physical Propert
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720 Appendix C ■ Properties of th
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722 Appendix D ■ Compressible Flo
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724 Appendix D ■ Compressible Flo
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Answers to Selected Even-Numbered H
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I ndex Absolute pressure, 13, 48 Ab
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Index I-3 guidelines for selection,
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Index I-5 hydrostatic: on curved su
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Index I-7 piezometer tube, 50, 105,
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Index I-9 Pressure force, 40 Pressu
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Index I-11 Stress field, 277 Strouh
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Index I-13 Weber, M., 29, 350 Weber
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V4.8 Jupiter red spot V4.9 Streamli
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V10.3 Water strider V10.13 Triangul
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■ TABLE 1.4 Conversion Factors fr
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■ TABLE 1.7 Approximate Physical
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