Chapter07

Mechanics of Fluids

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121 CHAPTER 7 Internal Flows 7.1 Re . = = × - VD n 1 10 6 a) 2000 2 1 10 0 001 6 = × × = - V V . . . m / s b) 2000 02 1 10 0 1 6 = × × = - V V . . . . m / s c) 002 1 10 1 0 6 = × × = - V V . . . . m / s 7.2 Re . = = × - Vh n 1 10 6 a) 2 2 22 00 11 ( ) ( ) ( ). 44 kk d pp u r r r rr d xL mm = - =- b) 6 1 150 0 . 0.0015 m/s. 1 10 V V - × = ∴= × c) 6 0.3 150 0 . 0.005 m/s. 1 10 V V - × = × 7.3 Re .5 . / . . = = × × = - n 1 2 12 1 4 10 1790 5 Using R crit , the flow is turbulent = 1500 . 7.4 Re ( / ) . = = × = - n 1 2 1 4 10 700 6 000. Very turbulent 7.5 Re . = n a) V D = × = × = - Re . . . n 2000 10 0 02 0 1 6 m / s b) V D = × = × = - Re . . n 40 10 0 02 2 6 000 m / s 7.6 L D V E = = = × = 0 065 0002 02 1592 2 . Re Re . . . . m / s. n p a) L E = × × × = - . . . . . . . 065 1592 04 131 10 04 12 6 6 m b) L E = × × × = - . . . . . . . 065 1592 04 1007 10 04 16 4 6 m c) L E = × × × = - . . . . . . . 065 1592 04 661 10 04 25 0 6 m d) L E = × × × = - . . . . . . . 065 1592 04 367 10 04 45 1 6 m
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122 7.7 a) V D = × = × × = - Re .51 . . . n 1000 1 10 0 04 0 378 5 m / s L D E = × = × × = 0 065 0 065 1000 0 04 2 6 . Re . . . . m L L i E 2245 = = / . / . . 4 2 6 4 0 65 m b) V D = × = × × = - Re .51 . . . n 80 1 10 0 04 30 2 5 000 m / s L D E 2245 = × = 120 120 0 04 4 8 . . . m L D i 2245 = × = 10 10 0 04 0 4 . . . m 7.8 5 26 0.02 5 8.8 4 0.06 8.84. R e 5. 3 10. 0.0 3 1.00 7 10 V p - × == = ×× Turbulent. = × = L E 120 06 7 2 . . m. Developed . 7.9 V Q A = = × × = ( / ) / . . 18 1000 2 3600 0 001 0 796 2 p m / s Re . . . . . = = × × = - VD n 0 796 0 002 114 10 139 6 5 laminar. L D E = × = × × = 0 065 0 065 139 6 0 002 0 0181 . Re . . . . m. negligible 7.10 0.04R e 0.0 4 770 0 .01 2 3.7 m. E Lh = × =××= ( 29 L E min . . . . = × × = 04 1500 012 0 72 m 7.11 ( 29 L D E lam m = = × × × = - 0065 0 065 5 06 155 10 06 755 5 . Re . . . . . . ( 29 L E turb m = × = 120 06 7 2 . . . (Re = 32 300) 7.12 Re . . . = = × = - n 0 2 0 04 10 8000 6 a) If laminar, L D E = × × = × × = 0 065 0 065 8000 0 04 20 8 . Re . . . m L L i E = = = / . / . 4 20 8 4 5 2 m b) This is a low Reynolds number turbulent flow. A minimum entrance length would be L D E = = × = 120 120 0 04 4 8 . with a minimum inviscid core length of L D i = = × = 10 10 0 04 0 4 . . . m
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123 7.13 Σ F p r r x x = - = - p t p 0 2 0 0 2 mom . out in & & = + p x r x 2 0 0 t & For developed flow p x r = 2 0 0 t since mom = Const. & From the velocity distribution in an entrance (see Fig. 7.1) it is obvious that ( ) ( ) t t 0 0 entrance developed since u y wall is greater in the entrance. Also, & x 0 since the momentum flux increases from the inlet to the developed flow. Hence, p x p x . 7.14 a) For a high Re flow transition to turbulence occurs near the origin. In the entrance region the velocity gradient u y / at the wall is very large resulting in a large wall shear. This large wall shear requires a large pressure gradient. In addition, the momentum flux is increasing in the x -direction, also requiring an increased pressure gradient (see the solution to 7.13 for more detail). b) For a low Re turbulent flow, the flow is laminar through much of the entrance region, up to about L d (see Fig. 7.2). The laminar flow results in a much smaller velocity gradient at the wall compared with that of the turbulent flow of part (a) requiring a much smaller pressure gradient. This results in the lower distribution of Fig. 7.3. c) The pressure distribution must move from the lower distribution to the higher distribution of Fig. 7.3 as the Re increases. This occurs at an intermediate Re when transition occurs near L i . Research that gives accurate data for such low turbulent Re transition does not exist.
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Chapter07 - CHAPTER 7 Internal Flows VD VD V 2 = . a) 2000...

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