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426 Chapter 8 ■ Viscous Flow in Pipes A = cross-sectional area z z V = u(y,z) y x P = perimeter of pipe D h = 4A/P = hydraulic diameter (a) F I G U R E 8.34 Noncircular duct. (b) The hydraulic diameter is used for noncircular duct calculations. cross section, as is shown in Fig. 8.34, the velocity profile is a function of both y and z 3V u1y, z2î 4. This means that the governing equation from which the velocity profile is obtained 1either the Navier–Stokes equations of motion or a force balance equation similar to that used for circular pipes, Eq. 8.62 is a partial differential equation rather than an ordinary differential equation. Although the equation is linear 1for fully developed flow the convective acceleration is zero2, its solution is not as straightforward as for round pipes. Typically the velocity profile is given in terms of an infinite series representation 1Ref. 172. Practical, easy-to-use results can be obtained as follows. Regardless of the cross-sectional shape, there are no inertia effects in fully developed laminar pipe flow. Thus, the friction factor can be written as f CRe h , where the constant C depends on the particular shape of the duct, and Re h is the Reynolds number, Re h rVD hm, based on the hydraulic diameter. The hydraulic diameter defined as D h 4AP is four times the ratio of the cross-sectional flow area divided by the wetted perimeter, P, of the pipe as is illustrated in Fig. 8.34. It represents a characteristic length that defines the size of a cross section of a specified shape. The factor of 4 is included in the definition of D h so that for round pipes the diameter and hydraulic diameter are equal 3D h 4AP 41pD 2 421pD2 D4. The hydraulic diameter is also used in the definition of the friction factor, h L f 1/D h 2V 2 2g, and the relative roughness, eD h . The values of C f Re h for laminar flow have been obtained from theory andor experiment for various shapes. Typical values are given in Table 8.3 along with the hydraulic diameter. Note ■ TABLE 8.3 Friction Factors for Laminar Flow in Noncircular Ducts (Data from Ref. 18) Shape Parameter C f Re h I. Concentric Annulus D h = D 2 – D 1 D 1D 2 0.0001 71.8 0.01 80.1 0.1 89.4 0.6 95.6 1.00 96.0 D 1 II. Rectangle D h = _____ 2ab a + b a D 2 b ab 0 96.0 0.05 89.9 0.10 84.7 0.25 72.9 0.50 62.2 0.75 57.9 1.00 56.9
8.4 Dimensional Analysis of Pipe Flow 427 The Moody chart, developed for round pipes, can also be used for noncircular ducts. that the value of C is relatively insensitive to the shape of the conduit. Unless the cross section is very “thin” in some sense, the value of C is not too different from its circular pipe value, C 64. Once the friction factor is obtained, the calculations for noncircular conduits are identical to those for round pipes. Calculations for fully developed turbulent flow in ducts of noncircular cross section are usually carried out by using the Moody chart data for round pipes with the diameter replaced by the hydraulic diameter and the Reynolds number based on the hydraulic diameter. Such calculations are usually accurate to within about 15%. If greater accuracy is needed, a more detailed analysis based on the specific geometry of interest is needed. E XAMPLE 8.7 Noncircular Conduit GIVEN Air at a temperature of 120 ºF and standard pressure flows from a furnace through an 8-in.-diameter pipe with an average velocity of 10 ft/s. It then passes through a transition section similar to the one shown in Fig. E8.7 and into a square duct whose side is of length a. The pipe and duct surfaces are smooth 1e 02. The head loss per foot is to be the same for the pipe and the duct. FIND Determine the duct size, a. SOLUTION We first determine the head loss per foot for the pipe, h L/ 1 fD2 V 2 2g, and then size the square duct to give the same value. For the given pressure and temperature we obtain 1from Table B.32 n 1.89 10 4 ft 2 s so that With this Reynolds number and with eD 0 we obtain the friction factor from Fig. 8.20 as f 0.022 so that Thus, for the square duct we must have where is the velocity in the duct. By combining Eqs. 1 and 2 we obtain or Re VD n 110 ft s21 8 12 ft2 1.89 10 4 ft 2 s 35,300 h L / 0.022 1 8 12 ft2 0.0512 f a 110 fts2 2 2132.2 fts 2 2 0.0512 h L / f V 2 s D h 2g 0.0512 D h 4AP 4a 2 4a a and p V s Q A 4 a 8 2 12 ftb 110 fts2 3.49 a 2 a 2 a 1.30 f 1 5 13.49a 2 2 2 2132.22 (1) (2) (3) F I G U R E E8.7 V where a is in feet. Similarly, the Reynolds number based on the hydraulic diameter is Re h V sD h n 13.49 a 2 2a 1.85 104 4 1.89 10 a We have three unknowns 1a, f, and Re h 2 and three equations— Eqs. 3, 4, and either in graphical form the Moody chart 1Fig. 8.202 or the Colebrook equation (Eq. 8.35a). If we use the Moody chart, we can use a trial and error solution as follows. As an initial attempt, assume the friction factor for the duct is the same as for the pipe. That is, assume f 0.022. From Eq. 3 we obtain a 0.606 ft, while from Eq. 4 we have Re h 3.05 10 4 . From Fig. 8.20, with this Reynolds number and the given smooth duct we obtain f 0.023, which does not quite agree with the assumed value of f. Hence, we do not have the solution. We try again, using the latest calculated value of f 0.023 as our guess. The calculations are repeated until the guessed value of f agrees with the value obtained from Fig. 8.20. The final result 1after only two iterations2 is f 0.023, Re h 3.03 10 4 , and a 0.611 ft 7.34 in. (4) (Ans) COMMENTS Alternatively, we can use the Colebrook equation (rather than the Moody chart) to obtain the solution as
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About the Authors vii Bruce R. Muns
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Preface xi Life Long Learning Probl
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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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14 Chapter 1 ■ Introduction TABLE
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16 Chapter 1 ■ Introduction Crude
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18 Chapter 1 ■ Introduction Visco
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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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26 Chapter 1 ■ Introduction E XAM
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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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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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10 Open-Channel Flow CHAPTER OPENIN
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646 Chapter 12 ■ Turbomachines Ex
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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-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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