Chapter6_SM

# Chapter6_SM - Chapter 6 Viscous Flow in Ducts P6.1 An...

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Chapter 6 Viscous Flow in Ducts P6.1 An engineer claims that flow of SAE 30W oil, at 20 ° C, through a 5-cm-diameter smooth pipe at 1 million N/h, is laminar. Do you agree? A million newtons is a lot, so this sounds like an awfully high flow rate. Solution : For SAE 30W oil at 20 ° C (Table A.3), take ρ = 891 kg/m 3 and μ = 0.29 kg/m-s. Convert the weight flow rate to volume flow rate in SI units: ) ( / 29 . 0 ) 05 . 0 )( / 2 . 16 )( / 891 ( Re 2 . 16 , ) 05 . 0 ( 4 0318 . 0 ) / 81 . 9 )( / 891 ( ) / 3600 / 1 )( / 6 1 ( 3 2 3 2 3 al transition s m kg m s m m kg VD Calculate s m V solve V m s m s m m kg s h h N E g w Q D 2500 = = = = = = = π ± This is not high, but not laminar . Ans . With careful inlet design, low disturbances, and a very smooth wall, it might still be laminar, but No , this is transitional , not definitely laminar. 6.2 Air at approximately 1 atm flows through a horizontal 4-cm-diameter pipe. (a) Find a formula for Q max, the maximum volume flow for which the flow remains laminar, and plot Q max versus temperature in the range 0 ° C T 500 ° C. (b) Is your plot linear? If not, explain. Solution: (a) First convert the Reynolds number from a velocity form to a volume flow form: 2 4 , therefore Re 2300 for laminar flow (/ 4 ) d QV d Q V d d ρρ μπ == = Maximum laminar volume flow is given by . (a) Ans max 2300 Q 4 = πμ d With d = 0.04 m = constant, get and for air from Table A-2 and plot Q max versus T ° C:

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Chapter 6 Viscous Flow in Ducts 435 Fig. P6.2 The curve is not quite linear because ν = μ / ρ is not quite linear with T for air in this range. Ans. (b) 6.3 For a thin wing moving parallel to its chord line, transition to a turbulent boundary layer occurs at a “local” Reynolds number Re x , where x is the distance from the leading edge of the wing. The critical Reynolds number depends upon the intensity of turbulent fluctuations in the stream and equals 2.8E6 if the stream is very quiet. A semiempirical correlation for this case [Ref. 3 of Ch. 6] is crit 1/2 2 1 (1 13.25 ) Re 0.00392 x ζ −+ + 21 / 2 where is the tunnel-turbulence intensity in percent. If V = 20 m/s in air at 20 ° C, use this formula to plot the transition position on the wing versus stream turbulence for between 0 and 2 percent. At what value of is x crit decreased 50 percent from its value at = 0? Solution: This problem is merely to illustrate the strong effect of stream turbulence on the transition point. For air at 20 ° C, take = 1.2 kg/m 3 and = 1.8E 5 kg/m s. Compute Rex,crit from the correlation and plot xtr = Rex/[ (20 m/s)] versus percent turbulence:
436 Solutions Manual Fluid Mechanics, Fifth Edition Fig. P6.3 The value of xcrit decreases by half (to 1.07 meters) at ζ 0.42% . Ans . 6.4 For flow of SAE 30 oil through a 5-cm-diameter pipe, from Fig. A.1, for what flow rate in m 3 /h would we expect transition to turbulence at (a) 20 ° C and (b) 100 ° C? Solution: For SAE 30 oil take and take μ = 0.29 kg/m s at 20 ° C (Table A.3) and 0.01 kg/m-s at 100 ° C (Fig A.1). Write the critical Reynolds number in terms of flow rate Q : 3 891 kg/m ρ = 3 4 4(891 / ) (a) Re 2300 , VD Q kg m Q m s ρρ ==== 33 (0.29 / )(0.05 ) solve 0.0293 . (a) crit Dk g m s m QA n s μπ π == m 106 h 3 44 ( 8 9 1 / ) (b) Re 2300 , crit VD Q kg m Q m s 3 (0.010 / )(0.05 ) solve 0.00101 . (b) g m s m n s 3

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Chapter6_SM - Chapter 6 Viscous Flow in Ducts P6.1 An...

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