In 1961, Ascher Shapiro founded the National Committee for Fluid Mechanics Films (NCFMF) in cooperation with the Education Development Center and released a series of 39 videos and accompanying texts which revolutionized the teaching of fluid mechanics. MIT's iFluids program has made a number of the films from this series available on the web. (Download / Purchase information.) The preface to Illustrated Experiments in Fluid Mechanics: The NCFMF Book of Film Notes can be found below. | (Click images to enlarge) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Preface “Since things in motion sooner catch the eye than what stirs not.” Troilus and Cressida This volume contains text and photographic material related to the sound films prepared under the direction of the National Committee for Fluid Mechanics Films (NCFMF). The films, and the related text material herein, cover nearly all of the fundamental phenomena of fluid motions. The work of the U.S. National Committee for Fluid Mechanics Films is now well We hope that the related written materials presented in this volume will be received with equal pleasure by those already familiar with the films, and, further, that the book will augment the film audience. Ascher H. Shapiro | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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- 2Flow regime
- 3Choosing a formula
- 4Approximations of the Colebrook equation
Notation[edit]
- The Reynolds number Re is taken to be Re = VD / ν, where V is the mean velocity of fluid flow, D is the pipe diameter, and where
- ν is the kinematic viscosity μ / ρ, with μ the fluid's viscosity, and ρ the fluid's density.
- The pipe's relative roughness ε / D, where ε is the pipe's effective roughness height and D the pipe (inside) diameter.
- f stands for the Darcy friction factor. Its value depends on the flow's Reynolds number Re and on the pipe's relative roughness ε / D.
- The log function is understood to be base-10 (as is customary in engineering fields): if x = log(y), then y = 10x.
- The ln function is understood to be base-e: if x = ln(y), then y = ex.
Flow regime[edit]
- Laminar flow
- Transition between laminar and turbulent flow
- Fully turbulent flow in smooth conduits
- Fully turbulent flow in rough conduits
- Free surface flow.
Transition flow[edit]
Turbulent flow in smooth conduits[edit]
Turbulent flow in rough conduits[edit]
Free surface flow[edit]
Choosing a formula[edit]
- Required accuracy
- Speed of computation required
- Available computational technology:
- calculator (minimize keystrokes)
- spreadsheet (single-cell formula)
- programming/scripting language (subroutine).
Colebrook–White equation[edit]
- Hydraulic diameter, (m, ft) – For fluid-filled, circular conduits, = D = inside diameter
- Hydraulic radius, (m, ft) – For fluid-filled, circular conduits, = D/4 = (inside diameter)/4
Solving[edit]
Expanded forms[edit]
- where:
- 1.7384... = 2 log (2 × 3.7) = 2 log (7.4)
- 18.574 = 2.51 × 3.7 × 2
- where:
- or
- where:
- 1.1364... = 1.7384... − 2 log (2) = 2 log (7.4) − 2 log (2) = 2 log (3.7)
- 9.287 = 18.574 / 2 = 2.51 × 3.7.
- where:
Free surface flow[edit]
Approximations of the Colebrook equation[edit]
Haaland equation[edit]
Swamee–Jain equation[edit]
Serghides's solution[edit]
Goudar–Sonnad equation[edit]
Brkić solution[edit]
Blasius correlations[edit]
- .
- ,
- Pipe diameter, D (m, ft)
- Curve radius, R (m, ft)
- Helicoidal pitch, H (m, ft)
- Reynolds number, Re (dimensionless)
- Retr < Re < 105
- 6.7 < 2Rc/D < 346.0
- 0 < H/D < 25.4
Table of Approximations[edit]
Equation | Author | Year | Range | Ref |
---|---|---|---|---|
Moody | 1947 | |||
| Wood | 1966 | ||
Eck | 1973 | |||
Swamee and Jain | 1976 | |||
Churchill | 1973 | Not specified | ||
Jain | 1976 | |||
| Churchill | 1977 | ||
Chen | 1979 | |||
Round | 1980 | |||
Barr | 1981 | |||
| Zigrang and Sylvester | 1982 | ||
Haaland[10] | 1983 | |||
| Serghides | 1984 | ||
if then and if then | Tsal | 1989 | ||
Manadilli | 1997 | |||
Romeo, Royo, Monzon | 2002 | |||
| Goudar, Sonnad | 2006 | ||
| Vatankhah, Kouchakzadeh | 2008 | ||
| Buzzelli | 2008 | ||
where | Cheng | 2008 | all flow regimes | |
Avci, Kargoz | 2009 | |||
Evangelides, Papaevangelou, Tzimopoulos | 2010 | |||
Fang | 2011 | |||
, | Brkić | 2011 | ||
| S.Alashkar | 2012 | ||
where | Bellos, Nalbantis, Tsakiris | 2018 | all flow regimes |
References[edit]
- ^Manning, Francis S.; Thompson, Richard E. (1991). Oilfield Processing of Petroleum. Vol. 1: Natural Gas. PennWell Books. ISBN978-0-87814-343-6., 420 pages. See page 293.
- ^Colebrook, C. F.; White, C. M. (1937). 'Experiments with Fluid Friction in Roughened Pipes'. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences. 161 (906): 367–381. Bibcode:1937RSPSA.161..367C. doi:10.1098/rspa.1937.0150.
Often erroneously cited as the source of the Colebrook-White equation. This is partly because Colebrook (in a footnote in his 1939 paper) acknowledges his debt to White for suggesting the mathematical method by which the smooth and rough pipe correlations could be combined.
- ^Colebrook, C F (1939). 'TURBULENT FLOW IN PIPES, WITH PARTICULAR REFERENCE TO THE TRANSITION REGION BETWEEN THE SMOOTH AND ROUGH PIPE LAWS'. Journal of the Institution of Civil Engineers. 11 (4): 133–156. doi:10.1680/ijoti.1939.13150. ISSN0368-2455.
- ^VDI Gesellschaft (2010). VDI Heat Atlas. Springer. ISBN978-3-540-77876-9.
- ^More, A. A. (2006). 'Analytical solutions for the Colebrook and White equation and for pressure drop in ideal gas flow in pipes'. Chemical Engineering Science. 61 (16): 5515–5519. doi:10.1016/j.ces.2006.04.003.
- ^Brkić, D. (2012). 'Lambert W Function in Hydraulic Problems'(PDF). Mathematica Balkanica. 26 (3–4): 285–292.
- ^Keady, G. (1998). 'Colebrook-White Formula for Pipe Flows'. Journal of Hydraulic Engineering. 124 (1): 96–97. CiteSeerX10.1.1.1027.8918. doi:10.1061/(ASCE)0733-9429(1998)124:1(96).
- ^Bellos, Vasilis; Nalbantis, Ioannis; Tsakiris, George (December 2018). 'Friction Modeling of Flood Flow Simulations'. Journal of Hydraulic Engineering. 144 (12): 04018073. doi:10.1061/(asce)hy.1943-7900.0001540. ISSN0733-9429.
- ^Haaland, SE (1983). 'Simple and Explicit Formulas for the Friction Factor in Turbulent Flow'. Journal of Fluids Engineering. 105 (1): 89–90. doi:10.1115/1.3240948.
- ^ abMassey, Bernard Stanford (1989). Mechanics of fluids. Chapman & Hall. ISBN978-0-412-34280-6.
- ^Swamee, P.K.; Jain, A.K. (1976). 'Explicit equations for pipe-flow problems'. Journal of the Hydraulics Division. 102 (5): 657–664.
- ^T.K, Serghides (1984). 'Estimate friction factor accurately'. Chemical Engineering Journal. 91 (5): 63–64. ISSN0009-2460.
- ^Goudar, C. T; Sonnad, J. R. (2008). 'Comparison of the iterative approximations of the Colebrook-White equation: Here's a review of other formulas and a mathematically exact formulation that is valid over the entire range of Re values'. Hydrocarbon Processing. 87 (8).
- ^Brkić, Dejan (2011). 'An Explicit Approximation of Colebrook's equation for fluid flow friction factor'(PDF). Petroleum Science and Technology. 29 (15): 1596–1602. doi:10.1080/10916461003620453.
- ^Massey, B. S. (2006). Mechanics of Fluids (8th ed.). Chapter 7 eq 7.5: Taylor & Francis. p. 254. ISBN978-0-415-36205-4.
- ^Trinh, Khanh Tuoc (2010), On the Blasius correlation for friction factors, arXiv:1007.2466, Bibcode:2010arXiv1007.2466T
- ^Bejan, Adrian; Kraus, Allan D. (2003). Heat Transfer Handbook. John Wiley & Sons. ISBN978-0-471-39015-2.
- ^Beograd, Dejan Brkić (March 2012). 'Determining Friction Factors in Turbulent Pipe Flow'. Chemical Engineering: 34–39.(subscription required)
- ^Churchill, S.W. (November 7, 1977). 'Friction-factor equation spans all fluid-flow regimes'. Chemical Engineering: 91–92.
- ^Cheng, Nian-Sheng (September 2008). 'Formulas for Friction Factor in Transitional Regimes'. Journal of Hydraulic Engineering. 134 (9): 1357–1362. doi:10.1061/(asce)0733-9429(2008)134:9(1357). ISSN0733-9429.
- ^Bellos, Vasilis; Nalbantis, Ioannis; Tsakiris, George (December 2018). 'Friction Modeling of Flood Flow Simulations'. Journal of Hydraulic Engineering. 144 (12): 04018073. doi:10.1061/(asce)hy.1943-7900.0001540. ISSN0733-9429.
Further reading[edit]
- Moody, L.F. (1944). 'Friction Factors for Pipe Flow'. Transactions of the ASME. 66 (8): 671–684.
- Brkić, Dejan (2011). 'Review of explicit approximations to the Colebrook relation for flow friction'(PDF). Journal of Petroleum Science and Engineering. 77 (1): 34–48. doi:10.1016/j.petrol.2011.02.006.
- Brkić, Dejan (2011). 'W solutions of the CW equation for flow friction'. Applied Mathematics Letters. 24 (8): 1379–1383. doi:10.1016/j.aml.2011.03.014.
- Brkić, Dejan; Ćojbašić, Žarko (2017). 'Evolutionary Optimization of Colebrook's Turbulent Flow Friction Approximations'. Fluids. 2 (2): 15. doi:10.3390/fluids2020015. ISSN2311-5521.