Chapt06

# 3 assume that eq 621 correlates the local mean

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Unformatted text preview: alize that V/u* is directly related to the Darcy friction factor | e-Text Main Menu | Textbook Table of Contents | Study Guide 6.4 Flow in a Circular Pipe 345 V V2 1/2 8 1/2 (6.51) u* f w Moreover, the argument of the logarithm in (6.50) is equivalent to 1 Ru* 1 f 1/2 2 Vd u* Red (6.52) V 2 8 Introducing (6.52) and (6.51) into Eq. (6.50), changing to a base-10 logarithm, and rearranging, we obtain 1 1.99 log (Red f 1/2) 1.02 (6.53) f 1/2 In other words, by simply computing the mean velocity from the logarithmic-law correlation, we obtain a relation between the friction factor and Reynolds number for turbulent pipe flow. Prandtl derived Eq. (6.53) in 1935 and then adjusted the constants slightly to fit friction data better 1 2.0 log (Red f 1/2) 0.8 (6.54) f 1/2 This is the accepted formula for a smooth-walled pipe. Some numerical values may be listed as follows: Red 4000 104 105 106 107 108 f 0.0399 0.0309 0.0180 0.0116 0.0081 0.0059 Thus f drops by only a factor of 5 over a 10,000-fold increase in Reynolds number. Equation (6.54) is cumbersome to so...
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