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For fully developed flow u uy only which satisfies

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Unformatted text preview: dx h y h gz) (6.80) This may be nondimensionalized as a friction factor: f 8w V2 24 Vh 24 Reh (6.81) These are exact analytic laminar-flow results, so there is no reason to resort to the hydraulic-diameter concept. However, if we did use Dh, a discrepancy would arise. The hydraulic diameter of a wide channel is b→∞ y = +h y 2h u ( y) x Y u max | v v Fig. 6.14 Fully developed flow between parallel plates. | e-Text Main Menu y=–h | Textbook Table of Contents | Study Guide Chapter 6 Viscous Flow in Ducts 4A Dh lim b→ 4(2bh) 2b 4h 4h (6.82) or twice the distance between the plates. Substituting into Eq. (6.81), we obtain the interesting result Parallel plates: 96 V(4h) flam 96 ReDh (6.83) Thus, if we could not work out the laminar theory and chose to use the approximation f 64/ReDh, we would be 33 percent low. The hydraulic-diameter approximation is relatively crude in laminar flow, as Eq. (6.76) states. Just as in circular-pipe flow, the laminar solution above becomes unstable at about ReDh 2000; transition occurs and turbul...
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