2034:
1678:
2557:
4860:. Doubling the diameter of a pipe of a given schedule (say, ANSI schedule 40) roughly doubles the amount of material required per unit length and thus its installed cost. Meanwhile, the head loss is decreased by a factor of 32 (about a 97% reduction). Thus the energy consumed in moving a given volumetric flow of the fluid is cut down dramatically for a modest increase in capital cost.
4255:
3820:
is consistent with the measurements on commercial pipes. Indeed, such pipes are very different from those carefully prepared by
Nikuradse: their surfaces are characterized by many different roughness heights and random spatial distribution of roughness points, while those of Nikuradse have surfaces with uniform roughness height, with the points extremely closely packed.
3645:
3459:
2412:
3256:
3094:
5107:
3796:
4454:
4117:
4816:
are parallel to the length of the pipe, the velocity of the fluid on the inner surface of the pipe is zero due to viscosity, and the velocity in the center of the pipe must therefore be larger than the average velocity obtained by dividing the volumetric flow rate by the wet area. The average kinetic
3819:
is small, it is consistent with smooth pipe flow, when large, it is consistent with rough pipe flow. However its performance in the transitional domain overestimates the friction factor by a substantial margin. Colebrook acknowledges the discrepancy with
Nikuradze's data but argues that his relation
2448:
is significant (typically at high
Reynolds number), the friction factor departs from the smooth pipe curve, ultimately approaching an asymptotic value ("rough pipe" regime). In this regime, the resistance to flow varies according to the square of the mean flow velocity and is insensitive to Reynolds
5425:
435:
and the flow regime (laminar, transitional, or turbulent), tended to obscure its dependence on the quantities in
Weisbach's formula, leading many researchers to derive irrational and dimensionally inconsistent empirical formulas. It was understood not long after Weisbach's work that the friction
2715:
3469:
3283:
2004:
In effect, the friction loss in the laminar regime is more accurately characterized as being proportional to flow velocity, rather than proportional to the square of that velocity: one could regard the Darcy–Weisbach equation as not truly applicable in the laminar flow regime.
562:
2008:
In laminar flow, friction loss arises from the transfer of momentum from the fluid in the center of the flow to the pipe wall via the viscosity of the fluid; no vortices are present in the flow. Note that the friction loss is insensitive to the pipe roughness height
2540:
2190:
4784:
2255:
4097:
3105:
2943:
1937:
1779:
as measured by experimenters for many different fluids, over a wide range of
Reynolds numbers, and for pipes of various roughness heights. There are three broad regimes of fluid flow encountered in these data: laminar, critical, and turbulent.
4990:
4825:, the fluid acquires random velocity components in all directions, including perpendicular to the length of the pipe, and thus turbulence contributes to the kinetic energy per unit volume but not to the average lengthwise velocity of the fluid.
2098:
When the pipe surface is smooth (the "smooth pipe" curve in Figure 2), the friction factor's variation with Re can be modeled by the Kármán–Prandtl resistance equation for turbulent flow in smooth pipes with the parameters suitably adjusted
3669:
4327:
1276:
4250:{\displaystyle {\begin{aligned}R_{*}&={\frac {\varepsilon }{D}}\cdot \mathrm {Re} {\sqrt {f_{\mathrm {D} }}}\cdot {\frac {1}{\sqrt {8}}}\\&={\frac {1}{2}}{\frac {\sqrt {g}}{\nu }}\varepsilon {\sqrt {S}}{\sqrt {D}}\end{aligned}}}
117:(1718-1798), in fact, had published a formula in 1770 that, although referring to open channels (i.e., not under pressure), was formally identical to the one Weisbach would later introduce, provided it was reformulated in terms of the
1563:
909:
4585:
The procedure above is similar for any available
Reynolds number that is an integer power of ten. It is not necessary to remember the value 1000 for this procedure—only that an integer power of ten is of interest for this purpose.
1177:
5320:
3991:
2070:
For
Reynolds number greater than 4000, the flow is turbulent; the resistance to flow follows the Darcy–Weisbach equation: it is proportional to the square of the mean flow velocity. Over a domain of many orders of magnitude of
4558:
If the value of the friction factor is 0.016, then the
Fanning friction factor is plotted in the Moody diagram. Note that the nonzero digits in 0.016 are the numerator in the formula for the laminar Fanning friction factor:
109:
would provide the head losses but in terms of quantities not known a priori, such as pressure. Therefore, empirical relationships were sought to correlate the head loss with quantities like pipe diameter and fluid velocity.
4529:
If the value of the friction factor is 0.064, then the Darcy friction factor is plotted in the Moody diagram. Note that the nonzero digits in 0.064 are the numerator in the formula for the laminar Darcy friction factor:
1652:
4619:
with velocity was lacking, so the Darcy–Weisbach equation was outperformed at first by the empirical Prony equation in many cases. In later years it was eschewed in many special-case situations in favor of a variety of
427:, which had a polynomial form in terms of velocity (often approximated by the square of the velocity), continued to be used. Beyond the historical developments, Weisbach's formula had the objective merit of adhering to
2609:
2025:, the flow is unsteady (varies grossly with time) and varies from one section of the pipe to another (is not "fully developed"). The flow involves the incipient formation of vortices; it is not well understood.
5679:
3640:{\displaystyle {\frac {1}{\sqrt {f_{\mathrm {D} }}}}=-1.93\,\log _{10}\left({\frac {1.91}{\mathrm {Re} {\sqrt {f_{\mathrm {D} }}}}}\left\{1+0.34R_{*}\;\left(1-\exp {\frac {-R_{*}}{26}}\right)\right\}\right),}
3454:{\displaystyle {\frac {1}{\sqrt {f_{\mathrm {D} }}}}=-2.0\,\log _{10}\left({\frac {2.51}{\mathrm {Re} {\sqrt {f_{\mathrm {D} }}}}}\left\{1+0.305R_{*}\;\left(1-\exp {\frac {-R_{*}}{26}}\right)\right\}\right),}
1460:
1847:
469:
370:
191:
4122:
4693:
as the pressure drop per unit length is a constant. To turn the relationship into a proportionality coefficient of dimensionless quantity, we can divide by the hydraulic diameter of the pipe,
2458:
2105:
4644:
Away from the ends of the pipe, the characteristics of the flow are independent of the position along the pipe. The key quantities are then the pressure drop along the pipe per unit length,
2090:). Within the turbulent flow regime, the nature of the flow can be further divided into a regime where the pipe wall is effectively smooth, and one where its roughness height is salient.
2407:{\displaystyle {\frac {1}{\sqrt {f_{\mathrm {D} }}}}={\frac {1.930}{\ln(10)}}W\left(10^{\frac {-0.537}{1.930}}{\frac {\ln(10)}{1.930}}\mathrm {Re} \right)=0.838\ W(0.629\ \mathrm {Re} )}
1340:
4703:
1980:
1062:
5949:
3251:{\displaystyle {\frac {1}{\sqrt {f_{\mathrm {D} }}}}=-1.930\log \left({\frac {1.90}{\mathrm {Re} {\sqrt {f_{\mathrm {D} }}}}}\left(1+0.34R_{*}\exp {\frac {-11}{R_{*}}}\right)\right),}
830:) which itself can be expressed in terms of easily measured or published physical quantities (see section below). Making this substitution the Darcy–Weisbach equation is rewritten as
4017:
3089:{\displaystyle {\frac {1}{\sqrt {f_{\mathrm {D} }}}}=-2\log \left({\frac {2.51}{\mathrm {Re} {\sqrt {f_{\mathrm {D} }}}}}\left(1+0.305R_{*}\exp {\frac {-11}{R_{*}}}\right)\right),}
708:
5102:{\displaystyle {\frac {1}{\sqrt {f_{\mathrm {D} }}}}=-2.00\log \left(2.51{\frac {1}{\mathrm {Re} {\sqrt {f_{\mathrm {D} }}}}}+{\frac {1}{3.7}}{\frac {\varepsilon }{D}}\right)}
1865:
754:
3649:
This function shares the same values for its term in common with the Kármán–Prandtl resistance equation, plus one parameter 0.305 or 0.34 to fit the asymptotic behavior for
3261:
This function shares the same values for its term in common with the Kármán–Prandtl resistance equation, plus one parameter 0.305 or 0.34 to fit the asymptotic behavior for
5936:
2792:
is identically zero: flow is always in the smooth pipe regime. The data for these points lie to the left extreme of the abscissa and are not within the frame of the graph.
4459:
Most charts or tables indicate the type of friction factor, or at least provide the formula for the friction factor with laminar flow. If the formula for laminar flow is
6212:
3791:{\displaystyle {\frac {1}{\sqrt {f_{\mathrm {D} }}}}=-2.00\log \left({\frac {2.51}{\mathrm {Re} {\sqrt {f_{\mathrm {D} }}}}}\left(1+{\frac {R_{*}}{3.3}}\right)\right).}
2033:
5718:
4449:{\displaystyle \Delta p=f_{\mathrm {D} }\cdot {\frac {L}{D}}\cdot {\frac {\rho {\langle v\rangle }^{2}}{2}}=f\cdot {\frac {L}{D}}\cdot {2\rho {\langle v\rangle }^{2}}}
220:
2417:
In this flow regime, many small vortices are responsible for the transfer of momentum between the bulk of the fluid to the pipe wall. As the friction
Reynolds number
1677:
393:
413:
1396:
1199:
798:
648:
619:
1475:
836:
316:
290:
264:
242:
5840:
18. Afzal, Noor (2013) "Roughness effects of commercial steel pipe in turbulent flow: Universal scaling". Canadian
Journal of Civil Engineering 40, 188-193.
1997:
of the pipe, which, for a cylindrical pipe flowing full, equals the inside diameter. In Figures 1 and 2 of friction factor versus Reynolds number, the regime
5420:{\displaystyle {\frac {1}{\sqrt {f_{\mathrm {D} }}}}=2\log \left(\mathrm {Re} {\sqrt {f_{\mathrm {D} }}}\right)-0.8\quad {\text{for }}\mathrm {Re} >3000.}
4305:, so attention must be paid to note which one of these is meant in any "friction factor" chart or equation being used. Of the two, the Darcy–Weisbach factor
1105:
125:(a former student of his) published an account describing his results. It is likely that Weisbach was aware of Chézy's formula through Prony's publications.
3931:
5919:
4868:
The Darcy-Weisbach's accuracy and universal applicability makes it the ideal formula for flow in pipes. The advantages of the equation are as follows:
2210:(called the "friction Reynolds number") can be considered, like the Reynolds number, to be a (dimensionless) parameter of the flow: at fixed values of
113:
Julius Weisbach was certainly not the first to introduce a formula correlating the length and diameter of a pipe to the square of the fluid velocity.
105:. However, the development of these formulas and charts also involved other scientists and engineers over its historical development. Generally, the
5594:
Colebrook, C. F. (February 1939). "Turbulent flow in pipes, with particular reference to the transition region between smooth and rough pipe laws".
2432:
increases, the profile of the fluid velocity approaches the wall asymptotically, thereby transferring more momentum to the pipe wall, as modeled in
1590:
6217:
4813:
4522:
Which friction factor is plotted in a Moody diagram may be determined by inspection if the publisher did not include the formula described above:
5806:
2710:{\displaystyle B(R_{*})={\frac {1}{1.930{\sqrt {f_{\mathrm {D} }}}}}+\log \left({\frac {1.90}{\sqrt {8}}}\cdot {\frac {\varepsilon }{D}}\right)}
1751:
and may be evaluated by the use of various empirical relations, or it may be read from published charts. These charts are often referred to as
4845:(the concomitant power consumption) to be the critical important factors. The practical consequence is that, for a fixed volumetric flow rate
5946:
5828:
5447:
5142:
6222:
5995:
2595:
approaches a constant value. Phenomenological functions attempting to fit these data, including the Afzal and Colebrook–White are shown.
4808:
Note that the dynamic pressure is not the kinetic energy of the fluid per unit volume, for the following reasons. Even in the case of
6181:
5887:
5859:
5175:
1405:
1802:
4260:
we have the two parameters needed to substitute into the Colebrook–White relation, or any other function, for the friction factor
81:. This is also variously called the Darcy–Weisbach friction factor, friction factor, resistance coefficient, or flow coefficient.
4636:, ease of calculation is no longer a major issue, and so the Darcy–Weisbach equation's generality has made it the preferred one.
4318:
by chemical engineers, but care should be taken to identify the correct factor regardless of the source of the chart or formula.
5933:
54:
along a given length of pipe to the average velocity of the fluid flow for an incompressible fluid. The equation is named after
4972:
The data exhibit, however, a systematic departure of up to 50% from the theoretical Hagen–Poiseuille equation in the region of
4685:
Pressure has dimensions of energy per unit volume, therefore the pressure drop between two points must be proportional to the
6186:
4902:
4790:
3829:
757:
557:{\displaystyle {\frac {\Delta p}{L}}=f_{\mathrm {D} }\cdot {\frac {\rho }{2}}\cdot {\frac {{\langle v\rangle }^{2}}{D_{H}}},}
4922:
1793:
981:
5258:
2535:{\displaystyle R_{*}={\frac {1}{\sqrt {8}}}\left(\mathrm {Re} {\sqrt {f_{\mathrm {D} }}}\,\right){\frac {\varepsilon }{D}}}
2195:
The numbers 1.930 and 0.537 are phenomenological; these specific values provide a fairly good fit to the data. The product
2185:{\displaystyle {\frac {1}{\sqrt {f_{\mathrm {D} }}}}=1.930\log \left(\mathrm {Re} {\sqrt {f_{\mathrm {D} }}}\right)-0.537.}
2917:
Afzal's fit to these data in the transition from smooth pipe flow to rough pipe flow employs an exponential expression in
1764:
332:
134:
3852:
1015:) expresses the pressure loss due to friction in terms of the equivalent height of a column of the working fluid, so the
6064:
4917:
4625:
3271:
along with one further parameter, 11, to govern the transition from smooth to rough flow. It is exhibited in Figure 3.
985:
62:. Currently, there is no formula more accurate or universally applicable than the Darcy-Weisbach supplemented by the
6191:
4779:{\displaystyle \Delta p\propto {\frac {L}{D}}q={\frac {L}{D}}\cdot {\frac {\rho }{2}}\cdot {\langle v\rangle }^{2}}
3882:
In typical engineering applications, there will be a set of given or known quantities. The acceleration of gravity
3859:. While the Colebrook–White relation is, in the general case, an iterative method, the Swamee–Jain equation allows
5928:
440:(and thus the velocity) only in the case of rough pipes in a turbulent flow regime (Prandtl-von Kármán equation).
6069:
6059:
1304:
3856:
1952:
1025:
5988:
4676:
4092:{\displaystyle \mathrm {Re} {\sqrt {f_{\mathrm {D} }}}={\frac {1}{\nu }}{\sqrt {2g}}{\sqrt {S}}{\sqrt {D^{3}}}}
416:
6049:
5916:
4897:
6140:
4947:
4797:, the Reynolds number and (outside the laminar regime) the relative roughness of the pipe (the ratio of the
4299:
106:
4793:" or "flow coefficient". This dimensionless coefficient will be a combination of geometric factors such as
6160:
6125:
4907:
4818:
2433:
5776:"Turbulent flow in a machine honed rough pipe for large Reynolds numbers: General roughness scaling laws"
1932:{\displaystyle \mathrm {Re} ={\frac {\rho }{\mu }}\langle v\rangle D={\frac {\langle v\rangle D}{\nu }},}
431:, resulting in a dimensionless friction factor f. The complexity of f, dependent on the mechanics of the
6105:
6084:
6033:
5527:
4834:
2041:
1685:
687:
4798:
730:
5694:
5644:
5482:
3911:
can be calculated directly from the chosen fitting function. Solving the Darcy–Weisbach equation for
940:
715:
428:
4689:
q. We also know that pressure must be proportional to the length of the pipe between the two points
6100:
5981:
4632:, most of which were significantly easier to use in calculations. However, since the advent of the
1986:
419:
in 1848 and soon became well known there. In contrast, it did not initially gain much traction in
326:
However, the friction factor f was expressed by Weisbach through the following empirical formula:
6130:
6120:
5710:
5660:
5635:
Shockling, M. A.; Allen, J. J.; Smits, A. J. (2006). "Roughness effects in turbulent pipe flow".
5498:
4802:
4672:
4621:
2242:
1994:
722:
651:
74:
67:
39:
5544:
1092:
6115:
6074:
5883:
5855:
5824:
5443:
5171:
5148:
5138:
4884:
It is useful in the transition region between laminar flow and fully developed turbulent flow.
1756:
1574:
1271:{\displaystyle S=f_{\text{D}}\cdot {\frac {1}{2g}}\cdot {\frac {{\langle v\rangle }^{2}}{D}}.}
924:
202:
102:
378:
5816:
5787:
5756:
5702:
5652:
5599:
5573:
5490:
4686:
4629:
398:
319:
122:
118:
114:
3659:
along with one further parameter, 26, to govern the transition from smooth to rough flow.
1558:{\displaystyle S=f_{\text{D}}\cdot {\frac {8}{\pi ^{2}g}}\cdot {\frac {Q^{2}}{D_{c}^{5}}}.}
1374:
776:
626:
604:
6227:
6145:
6135:
6110:
6078:
6023:
5953:
5940:
5923:
5810:
5467:
5463:
4603:
2449:
number. Here, it is useful to employ yet another dimensionless parameter of the flow, the
2001:
demonstrates laminar flow; the friction factor is well represented by the above equation.
1857:
1727:
is not a constant: it depends on such things as the characteristics of the pipe (diameter
1689:
904:{\displaystyle {\frac {\Delta p}{L}}={\frac {128}{\pi }}\cdot {\frac {\mu Q}{D_{c}^{4}}},}
802:
437:
415:
depending on the diameter and the type of pipe wall. Weisbach's work was published in the
98:
59:
5745:"Erratum: Friction factor directly from transitional roughness in a turbulent pipe flow"
5698:
5648:
5486:
6165:
6150:
5276:
5261:
from the original on October 20, 2020 – via Defense Technical Information Center.
4960:
4822:
4667:, and the volumetric flow rate. The flow rate can be converted to a mean flow velocity
4595:
4526:
Observe the value of the friction factor for laminar flow at a Reynolds number of 1000.
3835:
2883:
constitutes a transition from one behavior to the other. The data depart from the line
1752:
1172:{\displaystyle S={\frac {\Delta h}{L}}={\frac {1}{\rho g}}\cdot {\frac {\Delta p}{L}},}
997:
432:
424:
301:
275:
249:
227:
31:
17:
6206:
6155:
5898:
5744:
5664:
5561:
4912:
1748:
1666:
711:
591:
63:
5714:
5502:
3986:{\displaystyle {\sqrt {f_{\mathrm {D} }}}={\frac {\sqrt {2gSD}}{\langle v\rangle }}}
5257:. Vol. I. Redondo Beach CA: TRW Systems Group. p. 87, equation 3.9.2.1e.
4809:
1789:
1658:
1581:
769:
3663:
The Colebrook–White relation fits the friction factor with a function of the form
2556:
1193:
Therefore, the Darcy–Weisbach equation can also be written in terms of head loss:
1083:= The head loss due to pipe friction over the given length of pipe (SI units: m);
6028:
5252:
4599:
4314:
is more commonly used by civil and mechanical engineers, and the Fanning factor
3848:
94:
90:
55:
5791:
5562:"Friction Factor Directly From Transitional Roughness in a Turbulent Pipe Flow"
6018:
6004:
5869:
Shah, R. K.; London, A. L. (1978). "Laminar Flow Forced Convection in Ducts".
5706:
5656:
5494:
4927:
4633:
2838:, the data asymptotically approach a horizontal line; they are independent of
5820:
5603:
5314:(8th ed.). John Wiley & Sons. p. 379; Eq. 10:23, 10:24, paragraph 4.
5152:
1647:{\displaystyle \tau ={\frac {1}{8}}f_{\text{D}}\rho {\langle v\rangle }^{2}.}
4841:
within a pipe (that is, its productivity) and the head loss per unit length
2572:. The data fall on a single trajectory when plotted in this way. The regime
2066:. Data are from Nikuradse (1932, 1933), Colebrook (1939), and McKeon (2004).
1714:. Data are from Nikuradse (1932, 1933), Colebrook (1939), and McKeon (2004).
43:
1016:
293:
267:
51:
47:
128:
Weisbach's formula was proposed in 1845 in the form we still use today:
2937:(the transition from the smooth pipe regime to the rough pipe regime):
1796:(which stems from an exact classical solution for the fluid flow) that
1662:
5969:
ThermoTurb – A web application for thermal and turbulent flow analysis
5760:
5577:
1099:
It is useful to present head loss per length of pipe (dimensionless):
5899:"The History of the Darcy-Weisbach Equation for Pipe Flow Resistance"
5807:"The History of the Darcy-Weisbach Equation for Pipe Flow Resistance"
4607:
420:
93:
for calculating head losses in pipes, is traditionally attributed to
5775:
5137:(3rd ed.). Burlington, MA: Butterworth-Heinemann. p. 3.5.
2733:
for the rough pipe data of Nikuradse, Shockling, and Langelandsvik.
1989:. In this expression for Reynolds number, the characteristic length
5964:
Web application with pressure drop calculations for pipes and ducts
4878:
It is useful for any fluid, including oil, gas, brine, and sludges.
980:
Note that this laminar form of Darcy–Weisbach is equivalent to the
5468:"A new friction factor relationship for fully developed pipe flow"
2555:
2032:
2013:: the flow velocity in the neighborhood of the pipe wall is zero.
1759:, and hence the factor itself is sometimes erroneously called the
1676:
943:, used here to measure flow instead of mean velocity according to
928:
595:
4946:
The value of the Darcy friction factor is four times that of the
5958:
5310:
Crowe, Clayton T.; Elger, Donald F.; Robertson, John A. (2005).
4679:
2079:), the friction factor varies less than one order of magnitude (
436:
factor f depended on the flow regime and was independent of the
5977:
5296:(5th ed.). John Wiley & Sons. p. 470 paragraph 3.
1584:
is expressed in terms of the Darcy–Weisbach friction factor as
1455:{\displaystyle Q={\frac {\pi }{4}}D_{c}^{2}\langle v\rangle .}
5973:
3274:
The friction factor for another analogous roughness becomes
1842:{\displaystyle f_{\mathrm {D} }={\frac {64}{\mathrm {Re} }},}
1735:), the characteristics of the fluid (its kinematic viscosity
5968:
3801:
This relation has the correct behavior at extreme values of
2048:
for smooth pipe and a range of values of relative roughness
1767:
friction factor, after the approximate formula he proposed.
1696:
for smooth pipe and a range of values of relative roughness
4602:
of France, and further refined into the form used today by
2237:
can be expressed in closed form as an analytic function of
5963:
5678:
Langelandsvik, L. I.; Kunkel, G. J.; Smits, A. J. (2008).
4881:
It can be derived analytically in the laminar flow region.
4821:, which always exceeds the mean velocity. In the case of
654:
of the pipe (for a pipe of circular section, this equals
463:
and can be characterized by the Darcy–Weisbach equation:
452:, flowing full, the pressure loss due to viscous effects
5878:
Rohsenhow, W. M.; Hartnett, J. P.; Ganić, E. N. (1985).
5543:
In translation, NACA TM 1292. The data are available in
3824:
Calculating the friction factor from its parametrization
5815:. American Society of Civil Engineers. pp. 34–43.
3868:
to be found directly for full flow in a circular pipe.
801:, the friction factor is inversely proportional to the
4789:
The proportionality coefficient is the dimensionless "
5323:
5168:
Oilfield Processing of Petroleum. Vol. 1: Natural Gas
4993:
4706:
4594:
Historically this equation arose as a variant on the
4330:
4120:
4020:
3934:
3672:
3472:
3286:
3108:
2946:
2612:
2461:
2258:
2108:
1955:
1868:
1805:
1593:
1478:
1408:
1377:
1307:
1202:
1108:
1028:
839:
779:
733:
690:
629:
607:
472:
401:
381:
335:
304:
278:
252:
230:
205:
137:
121:. However, Chézy's formula was lost until 1800, when
4697:, which is also constant along the pipe. Therefore,
2736:
In this view, the data at different roughness ratio
365:{\displaystyle f=\alpha +{\beta \over {\sqrt {V}}}}
186:{\displaystyle \Delta H=f\cdot {LV^{2} \over {2gD}}}
6174:
6093:
6042:
6011:
4856:with the inverse fifth power of the pipe diameter,
4837:application, it is typical for the volumetric flow
2582:is effectively that of smooth pipe flow. For large
5947:Pipe pressure drop calculator for two phase flows.
5419:
5166:Manning, Francis S.; Thompson, Richard E. (1991).
5101:
4778:
4448:
4249:
4091:
3985:
3810:, as shown by the labeled curve in Figure 3: when
3790:
3639:
3453:
3250:
3088:
2709:
2599:It is illustrative to plot the roughness function
2563:Roughness function B vs. friction Reynolds number
2534:
2406:
2184:
1974:
1931:
1841:
1646:
1557:
1454:
1390:
1334:
1270:
1171:
1056:
903:
792:
748:
702:
642:
613:
556:
407:
387:
364:
310:
284:
258:
236:
214:
185:
4887:The friction factor variation is well documented.
4624:valid only for certain flow regimes, notably the
5630:
5628:
5589:
5587:
5521:
5519:
89:The Darcy-Weisbach equation, combined with the
3902:is a known quantity, then the friction factor
3855:(upon which the Moody chart is based), or the
5989:
5596:Journal of the Institution of Civil Engineers
5292:Incopera, Frank P.; Dewitt, David P. (2002).
5254:Aerospace Fluid Component Designers' Handbook
4610:in 1845. Initially, data on the variation of
1465:Then the Darcy–Weisbach equation in terms of
1370:In a full-flowing, circular pipe of diameter
8:
5809:. In Rogers, J. R.; Fredrich, A. J. (eds.).
5774:Afzal, Noor; Seena, Abu; Bushra, A. (2013).
5555:
5553:
5305:
5303:
5270:
5268:
4766:
4760:
4435:
4429:
4382:
4376:
3977:
3971:
1914:
1908:
1896:
1890:
1631:
1625:
1446:
1440:
1326:
1320:
1286:The relationship between mean flow velocity
1250:
1244:
697:
691:
529:
523:
4682:of the pipe if the pipe is full of fluid).
3898:. If as well the head loss per unit length
3890:are known, as are the diameter of the pipe
2228:In the Kármán–Prandtl resistance equation,
1335:{\displaystyle Q=A\cdot \langle v\rangle ,}
5996:
5982:
5974:
5959:Open source pipe pressure drop calculator.
5917:The History of the Darcy–Weisbach Equation
5882:(2nd ed.). McGraw–Hill Book Company.
4285:Confusion with the Fanning friction factor
3838:, methods for finding the friction factor
3579:
3393:
1975:{\displaystyle \nu ={\frac {\mu }{\rho }}}
1057:{\displaystyle \Delta p=\rho g\,\Delta h,}
448:In a cylindrical pipe of uniform diameter
5871:Supplement 1 to Advances in Heat Transfer
5812:Environmental and Water Resources History
5403:
5398:
5377:
5376:
5370:
5362:
5335:
5334:
5324:
5322:
5084:
5074:
5059:
5058:
5052:
5044:
5038:
5005:
5004:
4994:
4992:
4770:
4759:
4745:
4732:
4716:
4705:
4485:, and if the formula for laminar flow is
4439:
4428:
4420:
4407:
4386:
4375:
4368:
4355:
4345:
4344:
4329:
4236:
4229:
4214:
4204:
4182:
4170:
4169:
4163:
4155:
4142:
4129:
4121:
4119:
4102:Expressing the roughness Reynolds number
4081:
4075:
4068:
4058:
4048:
4036:
4035:
4029:
4021:
4019:
3954:
3942:
3941:
3935:
3933:
3886:and the kinematic viscosity of the fluid
3764:
3758:
3735:
3734:
3728:
3720:
3714:
3684:
3683:
3673:
3671:
3607:
3597:
3573:
3543:
3542:
3536:
3528:
3522:
3508:
3503:
3484:
3483:
3473:
3471:
3421:
3411:
3387:
3357:
3356:
3350:
3342:
3336:
3322:
3317:
3298:
3297:
3287:
3285:
3227:
3213:
3201:
3171:
3170:
3164:
3156:
3150:
3120:
3119:
3109:
3107:
3065:
3051:
3039:
3009:
3008:
3002:
2994:
2988:
2958:
2957:
2947:
2945:
2692:
2677:
2651:
2650:
2644:
2635:
2623:
2611:
2522:
2516:
2507:
2506:
2500:
2492:
2475:
2466:
2460:
2444:When the pipe surface's roughness height
2393:
2359:
2335:
2319:
2283:
2270:
2269:
2259:
2257:
2162:
2161:
2155:
2147:
2120:
2119:
2109:
2107:
1962:
1954:
1905:
1880:
1869:
1867:
1826:
1821:
1811:
1810:
1804:
1749:high accuracy within certain flow regimes
1635:
1624:
1614:
1600:
1592:
1544:
1539:
1529:
1523:
1508:
1498:
1489:
1477:
1434:
1429:
1415:
1407:
1382:
1376:
1306:
1254:
1243:
1240:
1222:
1213:
1201:
1151:
1133:
1115:
1107:
1044:
1027:
984:, which is analytically derived from the
890:
885:
871:
858:
840:
838:
784:
778:
739:
738:
732:
689:
634:
628:
606:
543:
533:
522:
519:
506:
496:
495:
473:
471:
400:
380:
353:
348:
334:
303:
277:
251:
229:
204:
170:
163:
153:
136:
6213:Dimensionless numbers of fluid mechanics
2763:, demonstrating scaling in the variable
567:where the pressure loss per unit length
5693:. Cambridge University Press: 323–339.
5481:. Cambridge University Press: 429–443.
5466:; Zagarola, M. V; Smits, A. J. (2005).
5122:
4950:, with which it should not be confused.
4939:
73:The Darcy–Weisbach equation contains a
5880:Handbook of Heat Transfer Fundamentals
5442:(3rd ed.). Springer. p. 45.
5294:Fundamentals of Heat and Mass Transfer
3872:Direct calculation when friction loss
2772:. The following features are present:
1739:), and the velocity of the fluid flow
1366:= The cross-sectional wetted area (m).
5780:Journal of Hydro-environment Research
5238:
5226:
5214:
5202:
5190:
3847:include using a diagram, such as the
2825:; flow is in the smooth pipe regime.
2225:, the friction factor is also fixed.
7:
5251:Howell, Glen (1970-02-01). "3.9.2".
5128:
5126:
4289:The Darcy–Weisbach friction factor
3851:, or solving equations such as the
2754:fall together when plotted against
1946:is the viscosity of the fluid and
674:for a pipe of cross-sectional area
5929:Darcy–Weisbach equation calculator
5528:"Strömungsgesetze in rauen Rohren"
5407:
5404:
5378:
5366:
5363:
5336:
5060:
5048:
5045:
5006:
4985:In its originally published form,
4707:
4640:Derivation by dimensional analysis
4510:, it is the Darcy–Weisbach factor
4346:
4331:
4171:
4159:
4156:
4037:
4025:
4022:
3943:
3736:
3724:
3721:
3685:
3544:
3532:
3529:
3485:
3358:
3346:
3343:
3299:
3172:
3160:
3157:
3121:
3010:
2998:
2995:
2959:
2903:very slowly, reach a maximum near
2652:
2508:
2496:
2493:
2397:
2394:
2363:
2360:
2271:
2163:
2151:
2148:
2121:
2021:For Reynolds numbers in the range
1873:
1870:
1830:
1827:
1812:
1763:. It is also sometimes called the
1154:
1118:
1045:
1029:
843:
740:
621:, the density of the fluid (kg/m);
497:
476:
206:
138:
25:
5680:"Flow in a commercial steel pipe"
4976:up to the onset of critical flow.
2926:that ensures proper behavior for
714:, experimentally measured as the
703:{\displaystyle \langle v\rangle }
4598:; this variant was developed by
2913:, then fall to a constant value.
1091:= The local acceleration due to
749:{\displaystyle f_{\mathrm {D} }}
5397:
5170:. PennWell Books. p. 293.
4875:It is dimensionally consistent.
4277:, and the volumetric flow rate
2549:is scaled to the pipe diameter
772:in a circular pipe of diameter
6218:Eponymous equations of physics
5278:Elementary Mechanics of Fluids
4903:Darcy friction factor formulae
4675:of the flow (which equals the
3830:Darcy friction factor formulae
2629:
2616:
2401:
2384:
2350:
2344:
2301:
2295:
760:(also called flow coefficient
77:friction factor, known as the
1:
5934:Pipe pressure drop calculator
5749:Journal of Fluids Engineering
5566:Journal of Fluids Engineering
5440:Applied Hydraulic Transients
5133:Jones, Garr M., ed. (2006).
4872:It is based on fundamentals.
1770:Figure 1 shows the value of
1358:= The volumetric flow (m/s),
6223:Equations of fluid dynamics
5312:Engineering Fluid Mechanics
4481:, it is the Fanning factor
4298:is 4 times larger than the
2805:, the data lie on the line
2545:where the roughness height
2044:versus Reynolds number for
1282:In terms of volumetric flow
320:acceleration due to gravity
6244:
5792:10.1016/j.jher.2011.08.002
5687:Journal of Fluid Mechanics
5637:Journal of Fluid Mechanics
5475:Journal of Fluid Mechanics
3827:
2872:The intermediate range of
1747:. It has been measured to
931:(Pa·s = N·s/m = kg/(m·s));
459:is proportional to length
42:equation that relates the
27:Equation in fluid dynamics
6070:Hydrological optimization
6060:Groundwater flow equation
5707:10.1017/S0022112007009305
5657:10.1017/S0022112006001467
5495:10.1017/S0022112005005501
4923:Hagen–Poiseuille equation
4819:root mean-square velocity
4817:energy then involves the
3894:and its roughness height
2591:, the roughness function
2451:roughness Reynolds number
1792:, it is a consequence of
1294:and volumetric flow rate
982:Hagen–Poiseuille equation
721:per unit cross-sectional
5617:Schlichting, H. (1955).
5604:10.1680/ijoti.1939.14509
5438:Chaudhry, M. H. (2013).
5281:. John Wiley & Sons.
3853:Colebrook–White equation
417:United States of America
215:{\displaystyle \Delta H}
6065:Hazen–Williams equation
6055:Darcy–Weisbach equation
5943:for single phase flows.
5897:Glenn O. Brown (2002).
5572:(10). ASME: 1255–1267.
5535:V. D. I. Forschungsheft
4959:This is related to the
4948:Fanning friction factor
4918:Hazen–Williams equation
4626:Hazen–Williams equation
4300:Fanning friction factor
2241:through the use of the
986:Navier–Stokes equations
388:{\displaystyle \alpha }
36:Darcy–Weisbach equation
18:Darcy-Weisbach equation
5786:(1). Elsevier: 81–90.
5526:Nikuradse, J. (1933).
5421:
5135:Pumping station design
5103:
4780:
4450:
4251:
4093:
3987:
3792:
3641:
3455:
3252:
3090:
2711:
2596:
2536:
2434:Blasius boundary layer
2408:
2186:
2067:
2023:2000 < Re < 4000
1976:
1933:
1843:
1790:laminar (smooth) flows
1715:
1648:
1559:
1456:
1392:
1336:
1272:
1173:
1058:
905:
794:
750:
704:
644:
615:
558:
444:Pressure-loss equation
409:
408:{\displaystyle \beta }
389:
366:
312:
286:
260:
238:
216:
187:
6085:Pipe network analysis
6050:Bernoulli's principle
6034:Hydraulic engineering
5873:. New York: Academic.
5805:Brown, G. O. (2003).
5619:Boundary Layer Theory
5422:
5104:
4898:Bernoulli's principle
4835:hydraulic engineering
4829:Practical application
4791:Darcy friction factor
4781:
4451:
4252:
4094:
3988:
3793:
3642:
3456:
3253:
3091:
2712:
2559:
2537:
2409:
2187:
2042:Darcy friction factor
2036:
1977:
1934:
1844:
1761:Moody friction factor
1731:and roughness height
1686:Darcy friction factor
1680:
1673:Darcy friction factor
1649:
1560:
1457:
1393:
1391:{\displaystyle D_{c}}
1337:
1273:
1174:
1059:
906:
795:
793:{\displaystyle D_{c}}
758:Darcy friction factor
751:
705:
645:
643:{\displaystyle D_{H}}
616:
614:{\displaystyle \rho }
598:) is a function of:
559:
410:
390:
367:
313:
287:
261:
244:: length of the pipe.
239:
217:
188:
85:Historical background
79:Darcy friction factor
5821:10.1061/40650(2003)4
5755:(10). ASME: 107001.
5743:Afzal, Noor (2011).
5560:Afzal, Noor (2007).
5321:
4991:
4704:
4328:
4269:, the flow velocity
4118:
4018:
3932:
3857:Swamee–Jain equation
3670:
3470:
3284:
3106:
2944:
2610:
2459:
2256:
2106:
2077:4000 < Re < 10
2046:1000 < Re < 10
1953:
1866:
1803:
1718:The friction factor
1591:
1476:
1406:
1375:
1305:
1200:
1186:is the pipe length (
1106:
1026:
941:volumetric flow rate
837:
777:
731:
716:volumetric flow rate
688:
627:
605:
470:
429:dimensional analysis
399:
379:
333:
302:
276:
250:
228:
203:
135:
107:Bernoulli's equation
5699:2008JFM...595..323L
5649:2006JFM...564..267S
5487:2005JFM...538..429M
4671:by dividing by the
4622:empirical equations
3996:we can now express
1993:is taken to be the
1987:kinematic viscosity
1549:
1439:
895:
5952:2019-07-13 at the
5939:2019-07-13 at the
5922:2011-07-20 at the
5854:. Addison–Wesley.
5850:De Nevers (1970).
5417:
5275:Rouse, H. (1946).
5099:
4803:hydraulic diameter
4776:
4446:
4247:
4245:
4089:
3983:
3788:
3637:
3451:
3248:
3086:
2707:
2597:
2532:
2404:
2182:
2094:Smooth-pipe regime
2068:
1995:hydraulic diameter
1972:
1929:
1839:
1716:
1694:10 < Re < 10
1644:
1555:
1535:
1452:
1425:
1388:
1332:
1268:
1169:
1054:
901:
881:
790:
746:
700:
652:hydraulic diameter
640:
611:
554:
405:
385:
362:
308:
282:
256:
234:
212:
183:
68:Colebrook equation
6200:
6199:
6075:Open-channel flow
5830:978-0-7844-0650-2
5761:10.1115/1.4004961
5724:on 16 August 2016
5578:10.1115/1.2776961
5449:978-1-4614-8538-4
5401:
5384:
5343:
5342:
5144:978-0-08-094106-6
5092:
5082:
5069:
5066:
5013:
5012:
4753:
4740:
4724:
4415:
4396:
4363:
4241:
4234:
4224:
4220:
4212:
4192:
4191:
4177:
4150:
4087:
4073:
4066:
4056:
4043:
3981:
3969:
3949:
3773:
3745:
3742:
3692:
3691:
3617:
3553:
3550:
3492:
3491:
3431:
3367:
3364:
3306:
3305:
3233:
3181:
3178:
3128:
3127:
3071:
3019:
3016:
2966:
2965:
2700:
2687:
2686:
2661:
2658:
2530:
2514:
2485:
2484:
2440:Rough-pipe regime
2392:
2380:
2357:
2332:
2305:
2278:
2277:
2169:
2128:
2127:
1970:
1924:
1888:
1834:
1617:
1608:
1575:wall shear stress
1569:Shear-stress form
1550:
1518:
1492:
1423:
1263:
1235:
1216:
1164:
1146:
1128:
992:Head-loss formula
925:dynamic viscosity
896:
866:
853:
549:
514:
486:
360:
358:
311:{\displaystyle g}
285:{\displaystyle V}
259:{\displaystyle D}
237:{\displaystyle L}
181:
103:Lewis Ferry Moody
16:(Redirected from
6235:
5998:
5991:
5984:
5975:
5906:
5903:researchgate.net
5893:
5874:
5865:
5835:
5834:
5802:
5796:
5795:
5771:
5765:
5764:
5740:
5734:
5733:
5731:
5729:
5723:
5717:. Archived from
5684:
5675:
5669:
5668:
5632:
5623:
5622:
5614:
5608:
5607:
5591:
5582:
5581:
5557:
5548:
5542:
5532:
5523:
5514:
5513:
5511:
5509:
5472:
5460:
5454:
5453:
5435:
5429:
5426:
5424:
5423:
5418:
5410:
5402:
5399:
5390:
5386:
5385:
5383:
5382:
5381:
5371:
5369:
5344:
5341:
5340:
5339:
5329:
5325:
5315:
5307:
5298:
5297:
5289:
5283:
5282:
5272:
5263:
5262:
5248:
5242:
5236:
5230:
5224:
5218:
5212:
5206:
5200:
5194:
5188:
5182:
5181:
5163:
5157:
5156:
5130:
5111:
5108:
5106:
5105:
5100:
5098:
5094:
5093:
5085:
5083:
5075:
5070:
5068:
5067:
5065:
5064:
5063:
5053:
5051:
5039:
5014:
5011:
5010:
5009:
4999:
4995:
4983:
4977:
4975:
4970:
4964:
4961:piezometric head
4957:
4951:
4944:
4859:
4852:
4848:
4844:
4840:
4812:, where all the
4799:roughness height
4796:
4785:
4783:
4782:
4777:
4775:
4774:
4769:
4754:
4746:
4741:
4733:
4725:
4717:
4696:
4692:
4687:dynamic pressure
4670:
4666:
4665:
4663:
4662:
4657:
4654:
4630:Manning equation
4618:
4580:
4579:
4577:
4576:
4573:
4570:
4554:
4553:
4551:
4550:
4547:
4544:
4518:
4509:
4508:
4506:
4505:
4502:
4499:
4484:
4480:
4479:
4477:
4476:
4473:
4470:
4455:
4453:
4452:
4447:
4445:
4444:
4443:
4438:
4416:
4408:
4397:
4392:
4391:
4390:
4385:
4369:
4364:
4356:
4351:
4350:
4349:
4317:
4313:
4304:
4297:
4280:
4276:
4268:
4256:
4254:
4253:
4248:
4246:
4242:
4237:
4235:
4230:
4225:
4216:
4215:
4213:
4205:
4197:
4193:
4187:
4183:
4178:
4176:
4175:
4174:
4164:
4162:
4151:
4143:
4134:
4133:
4110:
4098:
4096:
4095:
4090:
4088:
4086:
4085:
4076:
4074:
4069:
4067:
4059:
4057:
4049:
4044:
4042:
4041:
4040:
4030:
4028:
4010:
4009:
4008:
3992:
3990:
3989:
3984:
3982:
3980:
3956:
3955:
3950:
3948:
3947:
3946:
3936:
3924:
3923:
3922:
3910:
3901:
3897:
3893:
3889:
3885:
3876:
3867:
3846:
3818:
3809:
3797:
3795:
3794:
3789:
3784:
3780:
3779:
3775:
3774:
3769:
3768:
3759:
3746:
3744:
3743:
3741:
3740:
3739:
3729:
3727:
3715:
3693:
3690:
3689:
3688:
3678:
3674:
3658:
3646:
3644:
3643:
3638:
3633:
3629:
3628:
3624:
3623:
3619:
3618:
3613:
3612:
3611:
3598:
3578:
3577:
3554:
3552:
3551:
3549:
3548:
3547:
3537:
3535:
3523:
3513:
3512:
3493:
3490:
3489:
3488:
3478:
3474:
3460:
3458:
3457:
3452:
3447:
3443:
3442:
3438:
3437:
3433:
3432:
3427:
3426:
3425:
3412:
3392:
3391:
3368:
3366:
3365:
3363:
3362:
3361:
3351:
3349:
3337:
3327:
3326:
3307:
3304:
3303:
3302:
3292:
3288:
3270:
3257:
3255:
3254:
3249:
3244:
3240:
3239:
3235:
3234:
3232:
3231:
3222:
3214:
3206:
3205:
3182:
3180:
3179:
3177:
3176:
3175:
3165:
3163:
3151:
3129:
3126:
3125:
3124:
3114:
3110:
3095:
3093:
3092:
3087:
3082:
3078:
3077:
3073:
3072:
3070:
3069:
3060:
3052:
3044:
3043:
3020:
3018:
3017:
3015:
3014:
3013:
3003:
3001:
2989:
2967:
2964:
2963:
2962:
2952:
2948:
2936:
2925:
2912:
2902:
2882:
2868:
2867:
2865:
2864:
2861:
2858:
2850:
2841:
2837:
2824:
2804:
2791:
2782:
2771:
2762:
2753:
2752:
2750:
2749:
2746:
2743:
2732:
2723:
2716:
2714:
2713:
2708:
2706:
2702:
2701:
2693:
2688:
2682:
2678:
2662:
2660:
2659:
2657:
2656:
2655:
2645:
2636:
2628:
2627:
2602:
2594:
2590:
2581:
2571:
2552:
2548:
2541:
2539:
2538:
2533:
2531:
2523:
2521:
2517:
2515:
2513:
2512:
2511:
2501:
2499:
2486:
2480:
2476:
2471:
2470:
2447:
2431:
2430:
2429:
2413:
2411:
2410:
2405:
2400:
2390:
2378:
2371:
2367:
2366:
2358:
2353:
2336:
2334:
2333:
2328:
2320:
2306:
2304:
2284:
2279:
2276:
2275:
2274:
2264:
2260:
2246:
2240:
2236:
2224:
2223:
2222:
2209:
2208:
2207:
2191:
2189:
2188:
2183:
2175:
2171:
2170:
2168:
2167:
2166:
2156:
2154:
2129:
2126:
2125:
2124:
2114:
2110:
2089:
2078:
2074:
2065:
2064:
2062:
2061:
2058:
2055:
2047:
2029:Turbulent regime
2024:
2012:
2000:
1992:
1985:is known as the
1981:
1979:
1978:
1973:
1971:
1963:
1945:
1938:
1936:
1935:
1930:
1925:
1920:
1906:
1889:
1881:
1876:
1855:
1848:
1846:
1845:
1840:
1835:
1833:
1822:
1817:
1816:
1815:
1794:Poiseuille's law
1778:
1746:
1738:
1734:
1730:
1726:
1713:
1712:
1710:
1709:
1706:
1703:
1695:
1653:
1651:
1650:
1645:
1640:
1639:
1634:
1619:
1618:
1615:
1609:
1601:
1579:
1564:
1562:
1561:
1556:
1551:
1548:
1543:
1534:
1533:
1524:
1519:
1517:
1513:
1512:
1499:
1494:
1493:
1490:
1468:
1461:
1459:
1458:
1453:
1438:
1433:
1424:
1416:
1398:
1397:
1395:
1394:
1389:
1387:
1386:
1364:
1356:
1341:
1339:
1338:
1333:
1297:
1293:
1277:
1275:
1274:
1269:
1264:
1259:
1258:
1253:
1241:
1236:
1234:
1223:
1218:
1217:
1214:
1185:
1178:
1176:
1175:
1170:
1165:
1160:
1152:
1147:
1145:
1134:
1129:
1124:
1116:
1089:
1081:
1063:
1061:
1060:
1055:
1014:
1005:
975:
963:
961:
960:
957:
954:
938:
922:
910:
908:
907:
902:
897:
894:
889:
880:
872:
867:
859:
854:
849:
841:
829:
828:
826:
825:
822:
819:
800:
799:
797:
796:
791:
789:
788:
763:
755:
753:
752:
747:
745:
744:
743:
720:
709:
707:
706:
701:
681:
677:
673:
659:
649:
647:
646:
641:
639:
638:
620:
618:
617:
612:
589:
588:
586:
585:
580:
577:
563:
561:
560:
555:
550:
548:
547:
538:
537:
532:
520:
515:
507:
502:
501:
500:
487:
482:
474:
462:
458:
451:
414:
412:
411:
406:
394:
392:
391:
386:
371:
369:
368:
363:
361:
359:
354:
349:
317:
315:
314:
309:
291:
289:
288:
283:
265:
263:
262:
257:
243:
241:
240:
235:
221:
219:
218:
213:
192:
190:
189:
184:
182:
180:
169:
168:
167:
154:
123:Gaspard de Prony
119:hydraulic radius
21:
6243:
6242:
6238:
6237:
6236:
6234:
6233:
6232:
6203:
6202:
6201:
6196:
6175:Public networks
6170:
6089:
6079:Manning formula
6038:
6024:Hydraulic fluid
6007:
6002:
5954:Wayback Machine
5941:Wayback Machine
5924:Wayback Machine
5913:
5896:
5890:
5877:
5868:
5862:
5852:Fluid Mechanics
5849:
5846:
5844:Further reading
5838:
5831:
5804:
5803:
5799:
5773:
5772:
5768:
5742:
5741:
5737:
5727:
5725:
5721:
5682:
5677:
5676:
5672:
5634:
5633:
5626:
5616:
5615:
5611:
5593:
5592:
5585:
5559:
5558:
5551:
5541:. Berlin: 1–22.
5530:
5525:
5524:
5517:
5507:
5505:
5470:
5462:
5461:
5457:
5450:
5437:
5436:
5432:
5372:
5361:
5357:
5330:
5319:
5318:
5309:
5308:
5301:
5291:
5290:
5286:
5274:
5273:
5266:
5250:
5249:
5245:
5237:
5233:
5225:
5221:
5217:, p. 35-36
5213:
5209:
5205:, p. 36-37
5201:
5197:
5193:, p. 35-36
5189:
5185:
5178:
5165:
5164:
5160:
5145:
5132:
5131:
5124:
5120:
5115:
5114:
5054:
5043:
5034:
5030:
5000:
4989:
4988:
4984:
4980:
4973:
4971:
4967:
4963:along the pipe.
4958:
4954:
4945:
4941:
4936:
4894:
4866:
4857:
4850:
4846:
4842:
4838:
4831:
4794:
4758:
4702:
4701:
4694:
4690:
4677:cross-sectional
4668:
4658:
4655:
4649:
4648:
4646:
4645:
4642:
4617:
4611:
4604:Julius Weisbach
4592:
4574:
4571:
4568:
4567:
4565:
4560:
4548:
4545:
4542:
4541:
4539:
4537:
4531:
4517:
4511:
4503:
4500:
4497:
4496:
4494:
4492:
4486:
4482:
4474:
4471:
4468:
4467:
4465:
4460:
4427:
4374:
4370:
4340:
4326:
4325:
4315:
4312:
4306:
4302:
4296:
4290:
4287:
4278:
4270:
4267:
4261:
4244:
4243:
4195:
4194:
4165:
4135:
4125:
4116:
4115:
4109:
4103:
4077:
4031:
4016:
4015:
4007:
4001:
3999:
3997:
3970:
3937:
3930:
3929:
3921:
3915:
3913:
3912:
3909:
3903:
3899:
3895:
3891:
3887:
3883:
3880:
3874:
3866:
3860:
3845:
3839:
3832:
3826:
3817:
3811:
3808:
3802:
3760:
3751:
3747:
3730:
3719:
3713:
3709:
3679:
3668:
3667:
3662:
3656:
3650:
3647:
3603:
3599:
3584:
3580:
3569:
3559:
3555:
3538:
3527:
3521:
3517:
3504:
3479:
3468:
3467:
3461:
3417:
3413:
3398:
3394:
3383:
3373:
3369:
3352:
3341:
3335:
3331:
3318:
3293:
3282:
3281:
3277:
3268:
3262:
3223:
3215:
3197:
3187:
3183:
3166:
3155:
3149:
3145:
3115:
3104:
3103:
3061:
3053:
3035:
3025:
3021:
3004:
2993:
2987:
2983:
2953:
2942:
2941:
2934:
2927:
2924:
2918:
2910:
2904:
2901:
2894:
2884:
2880:
2873:
2862:
2859:
2856:
2855:
2853:
2852:
2849:
2843:
2839:
2835:
2829:
2823:
2816:
2806:
2802:
2796:
2790:
2784:
2777:
2770:
2764:
2761:
2755:
2747:
2744:
2741:
2740:
2738:
2737:
2731:
2725:
2721:
2720:Figure 3 shows
2676:
2672:
2646:
2640:
2619:
2608:
2607:
2600:
2592:
2589:
2583:
2579:
2573:
2570:
2564:
2550:
2546:
2502:
2491:
2487:
2462:
2457:
2456:
2445:
2442:
2428:
2422:
2420:
2418:
2337:
2321:
2315:
2314:
2310:
2288:
2265:
2254:
2253:
2244:
2238:
2235:
2229:
2221:
2215:
2213:
2211:
2206:
2200:
2198:
2196:
2157:
2146:
2142:
2115:
2104:
2103:
2096:
2087:
2080:
2076:
2072:
2059:
2056:
2053:
2052:
2050:
2049:
2045:
2031:
2022:
2019:
2017:Critical regime
2010:
1998:
1990:
1951:
1950:
1943:
1907:
1864:
1863:
1858:Reynolds number
1853:
1806:
1801:
1800:
1786:
1777:
1771:
1740:
1736:
1732:
1728:
1725:
1719:
1707:
1704:
1701:
1700:
1698:
1697:
1693:
1690:Reynolds number
1675:
1623:
1610:
1589:
1588:
1577:
1571:
1525:
1504:
1503:
1485:
1474:
1473:
1466:
1404:
1403:
1378:
1373:
1372:
1371:
1362:
1354:
1303:
1302:
1295:
1287:
1284:
1242:
1227:
1209:
1198:
1197:
1183:
1153:
1138:
1117:
1104:
1103:
1087:
1076:
1024:
1023:
1013:
1007:
1000:
994:
969:
958:
955:
952:
951:
949:
944:
934:
918:
873:
842:
835:
834:
823:
820:
817:
816:
814:
812:
806:
803:Reynolds number
780:
775:
774:
773:
761:
734:
729:
728:
718:
686:
685:
679:
675:
669:
661:
655:
630:
625:
624:
603:
602:
581:
578:
572:
571:
569:
568:
539:
521:
491:
475:
468:
467:
460:
453:
449:
446:
438:Reynolds number
397:
396:
377:
376:
331:
330:
300:
299:
274:
273:
248:
247:
226:
225:
201:
200:
159:
155:
133:
132:
99:Julius Weisbach
87:
60:Julius Weisbach
28:
23:
22:
15:
12:
11:
5:
6241:
6239:
6231:
6230:
6225:
6220:
6215:
6205:
6204:
6198:
6197:
6195:
6194:
6189:
6184:
6178:
6176:
6172:
6171:
6169:
6168:
6163:
6158:
6153:
6148:
6143:
6138:
6133:
6128:
6123:
6118:
6113:
6108:
6103:
6097:
6095:
6091:
6090:
6088:
6087:
6082:
6072:
6067:
6062:
6057:
6052:
6046:
6044:
6040:
6039:
6037:
6036:
6031:
6026:
6021:
6015:
6013:
6009:
6008:
6003:
6001:
6000:
5993:
5986:
5978:
5972:
5971:
5966:
5961:
5956:
5944:
5931:
5926:
5912:
5911:External links
5909:
5908:
5907:
5894:
5888:
5875:
5866:
5860:
5845:
5842:
5837:
5836:
5829:
5797:
5766:
5735:
5670:
5624:
5621:. McGraw-Hill.
5609:
5583:
5549:
5515:
5455:
5448:
5430:
5428:
5427:
5416:
5413:
5409:
5406:
5396:
5393:
5389:
5380:
5375:
5368:
5365:
5360:
5356:
5353:
5350:
5347:
5338:
5333:
5328:
5299:
5284:
5264:
5243:
5231:
5219:
5207:
5195:
5183:
5176:
5158:
5143:
5121:
5119:
5116:
5113:
5112:
5110:
5109:
5097:
5091:
5088:
5081:
5078:
5073:
5062:
5057:
5050:
5047:
5042:
5037:
5033:
5029:
5026:
5023:
5020:
5017:
5008:
5003:
4998:
4978:
4965:
4952:
4938:
4937:
4935:
4932:
4931:
4930:
4925:
4920:
4915:
4910:
4905:
4900:
4893:
4890:
4889:
4888:
4885:
4882:
4879:
4876:
4873:
4865:
4862:
4830:
4827:
4823:turbulent flow
4787:
4786:
4773:
4768:
4765:
4762:
4757:
4752:
4749:
4744:
4739:
4736:
4731:
4728:
4723:
4720:
4715:
4712:
4709:
4641:
4638:
4615:
4596:Prony equation
4591:
4588:
4583:
4582:
4556:
4535:
4527:
4515:
4490:
4457:
4456:
4442:
4437:
4434:
4431:
4426:
4423:
4419:
4414:
4411:
4406:
4403:
4400:
4395:
4389:
4384:
4381:
4378:
4373:
4367:
4362:
4359:
4354:
4348:
4343:
4339:
4336:
4333:
4310:
4294:
4286:
4283:
4265:
4258:
4257:
4240:
4233:
4228:
4223:
4219:
4211:
4208:
4203:
4200:
4198:
4196:
4190:
4186:
4181:
4173:
4168:
4161:
4158:
4154:
4149:
4146:
4141:
4138:
4136:
4132:
4128:
4124:
4123:
4107:
4100:
4099:
4084:
4080:
4072:
4065:
4062:
4055:
4052:
4047:
4039:
4034:
4027:
4024:
4005:
3994:
3993:
3979:
3976:
3973:
3968:
3965:
3962:
3959:
3953:
3945:
3940:
3919:
3907:
3879:
3870:
3864:
3843:
3836:turbulent flow
3828:Main article:
3825:
3822:
3815:
3806:
3799:
3798:
3787:
3783:
3778:
3772:
3767:
3763:
3757:
3754:
3750:
3738:
3733:
3726:
3723:
3718:
3712:
3708:
3705:
3702:
3699:
3696:
3687:
3682:
3677:
3654:
3636:
3632:
3627:
3622:
3616:
3610:
3606:
3602:
3596:
3593:
3590:
3587:
3583:
3576:
3572:
3568:
3565:
3562:
3558:
3546:
3541:
3534:
3531:
3526:
3520:
3516:
3511:
3507:
3502:
3499:
3496:
3487:
3482:
3477:
3465:
3450:
3446:
3441:
3436:
3430:
3424:
3420:
3416:
3410:
3407:
3404:
3401:
3397:
3390:
3386:
3382:
3379:
3376:
3372:
3360:
3355:
3348:
3345:
3340:
3334:
3330:
3325:
3321:
3316:
3313:
3310:
3301:
3296:
3291:
3279:
3266:
3259:
3258:
3247:
3243:
3238:
3230:
3226:
3221:
3218:
3212:
3209:
3204:
3200:
3196:
3193:
3190:
3186:
3174:
3169:
3162:
3159:
3154:
3148:
3144:
3141:
3138:
3135:
3132:
3123:
3118:
3113:
3097:
3096:
3085:
3081:
3076:
3068:
3064:
3059:
3056:
3050:
3047:
3042:
3038:
3034:
3031:
3028:
3024:
3012:
3007:
3000:
2997:
2992:
2986:
2982:
2979:
2976:
2973:
2970:
2961:
2956:
2951:
2932:
2922:
2915:
2914:
2908:
2899:
2892:
2878:
2870:
2847:
2833:
2826:
2821:
2814:
2800:
2793:
2788:
2768:
2759:
2729:
2718:
2717:
2705:
2699:
2696:
2691:
2685:
2681:
2675:
2671:
2668:
2665:
2654:
2649:
2643:
2639:
2634:
2631:
2626:
2622:
2618:
2615:
2587:
2577:
2568:
2543:
2542:
2529:
2526:
2520:
2510:
2505:
2498:
2495:
2490:
2483:
2479:
2474:
2469:
2465:
2441:
2438:
2426:
2415:
2414:
2403:
2399:
2396:
2389:
2386:
2383:
2377:
2374:
2370:
2365:
2362:
2356:
2352:
2349:
2346:
2343:
2340:
2331:
2327:
2324:
2318:
2313:
2309:
2303:
2300:
2297:
2294:
2291:
2287:
2282:
2273:
2268:
2263:
2233:
2219:
2204:
2193:
2192:
2181:
2178:
2174:
2165:
2160:
2153:
2150:
2145:
2141:
2138:
2135:
2132:
2123:
2118:
2113:
2095:
2092:
2085:
2030:
2027:
2018:
2015:
1983:
1982:
1969:
1966:
1961:
1958:
1940:
1939:
1928:
1923:
1919:
1916:
1913:
1910:
1904:
1901:
1898:
1895:
1892:
1887:
1884:
1879:
1875:
1872:
1850:
1849:
1838:
1832:
1829:
1825:
1820:
1814:
1809:
1785:
1784:Laminar regime
1782:
1775:
1753:Moody diagrams
1723:
1674:
1671:
1655:
1654:
1643:
1638:
1633:
1630:
1627:
1622:
1613:
1607:
1604:
1599:
1596:
1570:
1567:
1566:
1565:
1554:
1547:
1542:
1538:
1532:
1528:
1522:
1516:
1511:
1507:
1502:
1497:
1488:
1484:
1481:
1463:
1462:
1451:
1448:
1445:
1442:
1437:
1432:
1428:
1422:
1419:
1414:
1411:
1385:
1381:
1368:
1367:
1359:
1343:
1342:
1331:
1328:
1325:
1322:
1319:
1316:
1313:
1310:
1283:
1280:
1279:
1278:
1267:
1262:
1257:
1252:
1249:
1246:
1239:
1233:
1230:
1226:
1221:
1212:
1208:
1205:
1180:
1179:
1168:
1163:
1159:
1156:
1150:
1144:
1141:
1137:
1132:
1127:
1123:
1120:
1114:
1111:
1097:
1096:
1084:
1065:
1064:
1053:
1050:
1047:
1043:
1040:
1037:
1034:
1031:
1011:
993:
990:
978:
977:
967:
932:
912:
911:
900:
893:
888:
884:
879:
876:
870:
865:
862:
857:
852:
848:
845:
810:
787:
783:
766:
765:
742:
737:
726:
699:
696:
693:
683:
678:and perimeter
665:
637:
633:
622:
610:
565:
564:
553:
546:
542:
536:
531:
528:
525:
518:
513:
510:
505:
499:
494:
490:
485:
481:
478:
445:
442:
433:boundary layer
425:Prony equation
404:
384:
373:
372:
357:
352:
347:
344:
341:
338:
324:
323:
307:
297:
281:
271:
255:
245:
233:
223:
211:
208:
194:
193:
179:
176:
173:
166:
162:
158:
152:
149:
146:
143:
140:
86:
83:
32:fluid dynamics
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
6240:
6229:
6226:
6224:
6221:
6219:
6216:
6214:
6211:
6210:
6208:
6193:
6190:
6188:
6185:
6183:
6180:
6179:
6177:
6173:
6167:
6164:
6162:
6159:
6157:
6154:
6152:
6149:
6147:
6144:
6142:
6141:Power network
6139:
6137:
6134:
6132:
6129:
6127:
6124:
6122:
6119:
6117:
6114:
6112:
6109:
6107:
6104:
6102:
6099:
6098:
6096:
6092:
6086:
6083:
6080:
6076:
6073:
6071:
6068:
6066:
6063:
6061:
6058:
6056:
6053:
6051:
6048:
6047:
6045:
6041:
6035:
6032:
6030:
6027:
6025:
6022:
6020:
6017:
6016:
6014:
6010:
6006:
5999:
5994:
5992:
5987:
5985:
5980:
5979:
5976:
5970:
5967:
5965:
5962:
5960:
5957:
5955:
5951:
5948:
5945:
5942:
5938:
5935:
5932:
5930:
5927:
5925:
5921:
5918:
5915:
5914:
5910:
5904:
5900:
5895:
5891:
5889:0-07-053554-X
5885:
5881:
5876:
5872:
5867:
5863:
5861:0-201-01497-1
5857:
5853:
5848:
5847:
5843:
5841:
5832:
5826:
5822:
5818:
5814:
5813:
5808:
5801:
5798:
5793:
5789:
5785:
5781:
5777:
5770:
5767:
5762:
5758:
5754:
5750:
5746:
5739:
5736:
5720:
5716:
5712:
5708:
5704:
5700:
5696:
5692:
5688:
5681:
5674:
5671:
5666:
5662:
5658:
5654:
5650:
5646:
5642:
5638:
5631:
5629:
5625:
5620:
5613:
5610:
5605:
5601:
5597:
5590:
5588:
5584:
5579:
5575:
5571:
5567:
5563:
5556:
5554:
5550:
5546:
5540:
5536:
5529:
5522:
5520:
5516:
5504:
5500:
5496:
5492:
5488:
5484:
5480:
5476:
5469:
5465:
5464:McKeon, B. J.
5459:
5456:
5451:
5445:
5441:
5434:
5431:
5414:
5411:
5394:
5391:
5387:
5373:
5358:
5354:
5351:
5348:
5345:
5331:
5326:
5317:
5316:
5313:
5306:
5304:
5300:
5295:
5288:
5285:
5280:
5279:
5271:
5269:
5265:
5260:
5256:
5255:
5247:
5244:
5240:
5235:
5232:
5228:
5223:
5220:
5216:
5211:
5208:
5204:
5199:
5196:
5192:
5187:
5184:
5179:
5177:0-87814-343-2
5173:
5169:
5162:
5159:
5154:
5150:
5146:
5140:
5136:
5129:
5127:
5123:
5117:
5095:
5089:
5086:
5079:
5076:
5071:
5055:
5040:
5035:
5031:
5027:
5024:
5021:
5018:
5015:
5001:
4996:
4987:
4986:
4982:
4979:
4969:
4966:
4962:
4956:
4953:
4949:
4943:
4940:
4933:
4929:
4926:
4924:
4921:
4919:
4916:
4914:
4913:Friction loss
4911:
4909:
4906:
4904:
4901:
4899:
4896:
4895:
4891:
4886:
4883:
4880:
4877:
4874:
4871:
4870:
4869:
4863:
4861:
4855:
4836:
4828:
4826:
4824:
4820:
4815:
4811:
4806:
4804:
4800:
4792:
4771:
4763:
4755:
4750:
4747:
4742:
4737:
4734:
4729:
4726:
4721:
4718:
4713:
4710:
4700:
4699:
4698:
4688:
4683:
4681:
4678:
4674:
4661:
4653:
4639:
4637:
4635:
4631:
4627:
4623:
4614:
4609:
4605:
4601:
4597:
4589:
4587:
4563:
4557:
4534:
4528:
4525:
4524:
4523:
4520:
4514:
4489:
4463:
4440:
4432:
4424:
4421:
4417:
4412:
4409:
4404:
4401:
4398:
4393:
4387:
4379:
4371:
4365:
4360:
4357:
4352:
4341:
4337:
4334:
4324:
4323:
4322:
4319:
4309:
4301:
4293:
4284:
4282:
4274:
4264:
4238:
4231:
4226:
4221:
4217:
4209:
4206:
4201:
4199:
4188:
4184:
4179:
4166:
4152:
4147:
4144:
4139:
4137:
4130:
4126:
4114:
4113:
4112:
4106:
4082:
4078:
4070:
4063:
4060:
4053:
4050:
4045:
4032:
4014:
4013:
4012:
4004:
3974:
3966:
3963:
3960:
3957:
3951:
3938:
3928:
3927:
3926:
3918:
3906:
3877:
3871:
3869:
3863:
3858:
3854:
3850:
3842:
3837:
3831:
3823:
3821:
3814:
3805:
3785:
3781:
3776:
3770:
3765:
3761:
3755:
3752:
3748:
3731:
3716:
3710:
3706:
3703:
3700:
3697:
3694:
3680:
3675:
3666:
3665:
3664:
3660:
3653:
3634:
3630:
3625:
3620:
3614:
3608:
3604:
3600:
3594:
3591:
3588:
3585:
3581:
3574:
3570:
3566:
3563:
3560:
3556:
3539:
3524:
3518:
3514:
3509:
3505:
3500:
3497:
3494:
3480:
3475:
3464:
3448:
3444:
3439:
3434:
3428:
3422:
3418:
3414:
3408:
3405:
3402:
3399:
3395:
3388:
3384:
3380:
3377:
3374:
3370:
3353:
3338:
3332:
3328:
3323:
3319:
3314:
3311:
3308:
3294:
3289:
3278:
3275:
3272:
3265:
3245:
3241:
3236:
3228:
3224:
3219:
3216:
3210:
3207:
3202:
3198:
3194:
3191:
3188:
3184:
3167:
3152:
3146:
3142:
3139:
3136:
3133:
3130:
3116:
3111:
3102:
3101:
3100:
3083:
3079:
3074:
3066:
3062:
3057:
3054:
3048:
3045:
3040:
3036:
3032:
3029:
3026:
3022:
3005:
2990:
2984:
2980:
2977:
2974:
2971:
2968:
2954:
2949:
2940:
2939:
2938:
2931:
2921:
2907:
2898:
2891:
2887:
2877:
2871:
2846:
2832:
2827:
2820:
2813:
2809:
2799:
2794:
2787:
2780:
2775:
2774:
2773:
2767:
2758:
2734:
2728:
2703:
2697:
2694:
2689:
2683:
2679:
2673:
2669:
2666:
2663:
2647:
2641:
2637:
2632:
2624:
2620:
2613:
2606:
2605:
2604:
2586:
2576:
2567:
2562:
2558:
2554:
2527:
2524:
2518:
2503:
2488:
2481:
2477:
2472:
2467:
2463:
2455:
2454:
2453:
2452:
2439:
2437:
2435:
2425:
2387:
2381:
2375:
2372:
2368:
2354:
2347:
2341:
2338:
2329:
2325:
2322:
2316:
2311:
2307:
2298:
2292:
2289:
2285:
2280:
2266:
2261:
2252:
2251:
2250:
2248:
2232:
2226:
2218:
2203:
2179:
2176:
2172:
2158:
2143:
2139:
2136:
2133:
2130:
2116:
2111:
2102:
2101:
2100:
2093:
2091:
2084:
2043:
2039:
2035:
2028:
2026:
2016:
2014:
2006:
2002:
1996:
1988:
1967:
1964:
1959:
1956:
1949:
1948:
1947:
1926:
1921:
1917:
1911:
1902:
1899:
1893:
1885:
1882:
1877:
1862:
1861:
1860:
1859:
1836:
1823:
1818:
1807:
1799:
1798:
1797:
1795:
1791:
1783:
1781:
1774:
1768:
1766:
1762:
1758:
1754:
1750:
1744:
1722:
1691:
1687:
1683:
1679:
1672:
1670:
1668:
1664:
1660:
1641:
1636:
1628:
1620:
1611:
1605:
1602:
1597:
1594:
1587:
1586:
1585:
1583:
1580:in a pipe or
1576:
1568:
1552:
1545:
1540:
1536:
1530:
1526:
1520:
1514:
1509:
1505:
1500:
1495:
1486:
1482:
1479:
1472:
1471:
1470:
1449:
1443:
1435:
1430:
1426:
1420:
1417:
1412:
1409:
1402:
1401:
1400:
1383:
1379:
1365:
1360:
1357:
1352:
1351:
1350:
1349:
1348:
1329:
1323:
1317:
1314:
1311:
1308:
1301:
1300:
1299:
1291:
1281:
1265:
1260:
1255:
1247:
1237:
1231:
1228:
1224:
1219:
1210:
1206:
1203:
1196:
1195:
1194:
1191:
1189:
1166:
1161:
1157:
1148:
1142:
1139:
1135:
1130:
1125:
1121:
1112:
1109:
1102:
1101:
1100:
1094:
1090:
1085:
1082:
1080:
1074:
1073:
1072:
1071:
1070:
1051:
1048:
1041:
1038:
1035:
1032:
1022:
1021:
1020:
1018:
1010:
1004:
999:
991:
989:
987:
983:
973:
966:
947:
942:
937:
933:
930:
926:
921:
917:
916:
915:
898:
891:
886:
882:
877:
874:
868:
863:
860:
855:
850:
846:
833:
832:
831:
813: =
809:
804:
785:
781:
771:
759:
735:
727:
724:
717:
713:
712:flow velocity
694:
684:
672:
668:
664:
658:
653:
635:
631:
623:
608:
601:
600:
599:
597:
593:
584:
576:
551:
544:
540:
534:
526:
516:
511:
508:
503:
492:
488:
483:
479:
466:
465:
464:
457:
443:
441:
439:
434:
430:
426:
422:
418:
402:
382:
355:
350:
345:
342:
339:
336:
329:
328:
327:
321:
305:
298:
296:of the fluid.
295:
279:
272:
269:
253:
246:
231:
224:
209:
199:
198:
197:
177:
174:
171:
164:
160:
156:
150:
147:
144:
141:
131:
130:
129:
126:
124:
120:
116:
115:Antoine Chézy
111:
108:
104:
100:
96:
92:
84:
82:
80:
76:
75:dimensionless
71:
69:
65:
64:Moody diagram
61:
57:
53:
50:loss, due to
49:
45:
41:
37:
33:
19:
6161:Rescue tools
6126:Drive system
6094:Technologies
6054:
5902:
5879:
5870:
5851:
5839:
5811:
5800:
5783:
5779:
5769:
5752:
5748:
5738:
5726:. Retrieved
5719:the original
5690:
5686:
5673:
5640:
5636:
5618:
5612:
5595:
5569:
5565:
5545:digital form
5538:
5534:
5506:. Retrieved
5478:
5474:
5458:
5439:
5433:
5311:
5293:
5287:
5277:
5253:
5246:
5241:, p. 39
5234:
5229:, p. 37
5222:
5210:
5198:
5186:
5167:
5161:
5134:
4981:
4968:
4955:
4942:
4908:Euler number
4867:
4853:
4849:, head loss
4832:
4810:laminar flow
4807:
4788:
4684:
4659:
4651:
4643:
4612:
4593:
4584:
4561:
4532:
4521:
4512:
4487:
4461:
4458:
4320:
4307:
4291:
4288:
4272:
4262:
4259:
4104:
4101:
4002:
3995:
3916:
3904:
3881:
3873:
3861:
3840:
3833:
3812:
3803:
3800:
3661:
3651:
3648:
3462:
3276:
3273:
3263:
3260:
3098:
2929:
2919:
2916:
2905:
2896:
2889:
2885:
2875:
2844:
2830:
2818:
2811:
2807:
2797:
2785:
2778:
2765:
2756:
2735:
2726:
2719:
2598:
2584:
2574:
2565:
2560:
2544:
2450:
2443:
2423:
2416:
2230:
2227:
2216:
2201:
2194:
2097:
2082:
2069:
2037:
2020:
2007:
2003:
1999:Re < 2000
1984:
1941:
1851:
1787:
1772:
1769:
1760:
1742:
1720:
1717:
1681:
1659:shear stress
1656:
1582:open channel
1572:
1464:
1369:
1361:
1353:
1346:
1345:
1344:
1289:
1285:
1192:
1187:
1181:
1098:
1086:
1078:
1075:
1068:
1067:
1066:
1008:
1002:
995:
979:
971:
964:
945:
935:
919:
913:
807:
770:laminar flow
767:
670:
666:
662:
660:; otherwise
656:
582:
574:
566:
455:
447:
374:
325:
270:of the pipe.
222:: head loss.
195:
127:
112:
88:
78:
72:
35:
29:
6106:Accumulator
6029:Fluid power
5643:: 267–285.
4974:Re > 500
4673:wetted area
4600:Henry Darcy
3849:Moody chart
2081:0.006 <
1757:L. F. Moody
723:wetted area
710:, the mean
590:(SI units:
95:Henry Darcy
91:Moody chart
56:Henry Darcy
6207:Categories
6192:Manchester
6019:Hydraulics
6005:Hydraulics
5598:. London.
5239:Brown 2002
5227:Brown 2002
5215:Brown 2002
5203:Brown 2002
5191:Brown 2002
5118:References
4928:Water pipe
4864:Advantages
4814:flow lines
4634:calculator
4321:Note that
1942:and where
6182:Liverpool
6101:Machinery
5665:120958504
5400:for
5392:−
5355:
5153:144609617
5087:ε
5028:
5019:−
4854:decreases
4767:⟩
4761:⟨
4756:⋅
4748:ρ
4743:⋅
4714:∝
4708:Δ
4436:⟩
4430:⟨
4425:ρ
4418:⋅
4405:⋅
4383:⟩
4377:⟨
4372:ρ
4366:⋅
4353:⋅
4332:Δ
4227:ε
4222:ν
4180:⋅
4153:⋅
4145:ε
4131:∗
4054:ν
3978:⟩
3972:⟨
3766:∗
3707:
3698:−
3609:∗
3601:−
3595:
3589:−
3575:∗
3515:
3498:−
3423:∗
3415:−
3409:
3403:−
3389:∗
3329:
3312:−
3229:∗
3217:−
3211:
3203:∗
3143:
3134:−
3067:∗
3055:−
3049:
3041:∗
2981:
2972:−
2695:ε
2690:⋅
2670:
2625:∗
2561:Figure 3.
2525:ε
2468:∗
2342:
2323:−
2293:
2177:−
2140:
2088:< 0.06
2038:Figure 2.
1968:ρ
1965:μ
1957:ν
1922:ν
1915:⟩
1909:⟨
1897:⟩
1891:⟨
1886:μ
1883:ρ
1682:Figure 1.
1657:The wall
1632:⟩
1626:⟨
1621:ρ
1595:τ
1573:The mean
1521:⋅
1506:π
1496:⋅
1447:⟩
1441:⟨
1418:π
1327:⟩
1321:⟨
1318:⋅
1251:⟩
1245:⟨
1238:⋅
1220:⋅
1155:Δ
1149:⋅
1140:ρ
1119:Δ
1046:Δ
1039:ρ
1030:Δ
998:head loss
875:μ
869:⋅
864:π
844:Δ
698:⟩
692:⟨
609:ρ
530:⟩
524:⟨
517:⋅
509:ρ
504:⋅
477:Δ
403:β
383:α
351:β
343:α
207:Δ
151:⋅
139:Δ
44:head loss
40:empirical
6131:Manifold
6121:Cylinder
6043:Modeling
6012:Concepts
5950:Archived
5937:Archived
5920:Archived
5715:59433444
5503:15642454
5259:Archived
4892:See also
3878:is known
3280: :
2881:< 100
2836:> 100
2436:theory.
2247:function
2243:Lambert
1755:, after
1661:has the
1019:drop is
1017:pressure
423:, where
294:velocity
268:diameter
52:friction
48:pressure
6116:Circuit
5728:25 June
5695:Bibcode
5645:Bibcode
5508:25 June
5483:Bibcode
4801:to the
4664:
4647:
4628:or the
4590:History
4578:
4566:
4552:
4540:
4507:
4495:
4478:
4466:
4000:√
3914:√
2935:< 50
2928:1 <
2874:5 <
2866:
2854:
2783:, then
2751:
2739:
2724:versus
2421:√
2214:√
2199:√
2063:
2051:
1856:is the
1765:Blasius
1711:
1699:
1688:versus
1667:pascals
1663:SI unit
1093:gravity
962:
950:
939:is the
927:of the
923:is the
827:
815:
805:alone (
587:
570:
196:where:
6228:Piping
6187:London
5886:
5858:
5827:
5713:
5663:
5501:
5446:
5174:
5151:
5141:
4608:Saxony
2851:, and
2803:< 5
2580:< 1
2391:
2379:
2180:0.537.
1852:where
1669:(Pa).
1347:where:
1182:where
1095:(m/s).
1069:where:
976:(m/s).
914:where
756:, the
725:(m/s);
682:) (m);
671:= 4A/P
650:, the
421:France
101:, and
38:is an
34:, the
6146:Press
6136:Motor
6111:Brake
5722:(PDF)
5711:S2CID
5683:(PDF)
5661:S2CID
5531:(PDF)
5499:S2CID
5471:(PDF)
5415:3000.
4934:Notes
4833:In a
3381:0.305
3137:1.930
3099:and
3033:0.305
2828:When
2795:When
2776:When
2642:1.930
2388:0.629
2376:0.838
2355:1.930
2330:1.930
2326:0.537
2286:1.930
2134:1.930
929:fluid
375:with
46:, or
6166:Seal
6151:Pump
5884:ISBN
5856:ISBN
5825:ISBN
5730:2016
5510:2016
5444:ISBN
5412:>
5172:ISBN
5149:OCLC
5139:ISBN
5036:2.51
5022:2.00
4680:area
3834:For
3717:2.51
3701:2.00
3567:0.34
3525:1.91
3501:1.93
3463:and
3339:2.51
3195:0.34
3153:1.90
2991:2.51
2911:= 10
2895:) =
2817:) =
2680:1.90
2040:The
1788:For
1692:for
1684:The
1292:>
1288:<
1006:(or
996:The
974:>
970:<
768:For
395:and
58:and
6156:Ram
5817:doi
5788:doi
5757:doi
5753:133
5703:doi
5691:595
5653:doi
5641:564
5600:doi
5574:doi
5570:129
5539:361
5491:doi
5479:538
5395:0.8
5352:log
5080:3.7
5025:log
4805:).
4606:of
3771:3.3
3704:log
3657:→ ∞
3592:exp
3506:log
3406:exp
3320:log
3315:2.0
3269:→ ∞
3208:exp
3140:log
3046:exp
2978:log
2781:= 0
2667:log
2137:log
1665:of
1469:is
1298:is
1190:).
861:128
66:or
30:In
6209::
5901:.
5823:.
5782:.
5778:.
5751:.
5747:.
5709:.
5701:.
5689:.
5685:.
5659:.
5651:.
5639:.
5627:^
5586:^
5568:.
5564:.
5552:^
5537:.
5533:.
5518:^
5497:.
5489:.
5477:.
5473:.
5302:^
5267:^
5147:.
5125:^
4575:Re
4569:16
4564:=
4549:Re
4543:64
4538:=
4519:.
4504:Re
4498:64
4493:=
4475:Re
4469:16
4464:=
4281:.
4111:,
4011::
3998:Re
3925:,
3615:26
3510:10
3466::
3429:26
3324:10
3220:11
3058:11
2842:,
2840:Re
2603::
2553:.
2419:Re
2348:10
2339:ln
2317:10
2299:10
2290:ln
2249::
2239:Re
2212:Re
2197:Re
2073:Re
1854:Re
1824:64
1399:,
988:.
948:=
824:Re
818:64
764:).
592:Pa
318::
292::
266::
97:,
70:.
6081:)
6077:(
5997:e
5990:t
5983:v
5905:.
5892:.
5864:.
5833:.
5819::
5794:.
5790::
5784:7
5763:.
5759::
5732:.
5705::
5697::
5667:.
5655::
5647::
5606:.
5602::
5580:.
5576::
5547:.
5512:.
5493::
5485::
5452:.
5408:e
5405:R
5388:)
5379:D
5374:f
5367:e
5364:R
5359:(
5349:2
5346:=
5337:D
5332:f
5327:1
5180:.
5155:.
5096:)
5090:D
5077:1
5072:+
5061:D
5056:f
5049:e
5046:R
5041:1
5032:(
5016:=
5007:D
5002:f
4997:1
4858:D
4851:S
4847:Q
4843:S
4839:Q
4795:π
4772:2
4764:v
4751:2
4738:D
4735:L
4730:=
4727:q
4722:D
4719:L
4711:p
4695:D
4691:L
4669:V
4660:L
4656:/
4652:p
4650:Δ
4616:D
4613:f
4581:.
4572:/
4562:f
4555:.
4546:/
4536:D
4533:f
4516:D
4513:f
4501:/
4491:D
4488:f
4483:f
4472:/
4462:f
4441:2
4433:v
4422:2
4413:D
4410:L
4402:f
4399:=
4394:2
4388:2
4380:v
4361:D
4358:L
4347:D
4342:f
4338:=
4335:p
4316:f
4311:D
4308:f
4303:f
4295:D
4292:f
4279:Q
4275:⟩
4273:v
4271:⟨
4266:D
4263:f
4239:D
4232:S
4218:g
4210:2
4207:1
4202:=
4189:8
4185:1
4172:D
4167:f
4160:e
4157:R
4148:D
4140:=
4127:R
4108:∗
4105:R
4083:3
4079:D
4071:S
4064:g
4061:2
4051:1
4046:=
4038:D
4033:f
4026:e
4023:R
4006:D
4003:f
3975:v
3967:D
3964:S
3961:g
3958:2
3952:=
3944:D
3939:f
3920:D
3917:f
3908:D
3905:f
3900:S
3896:ε
3892:D
3888:ν
3884:g
3875:S
3865:D
3862:f
3844:D
3841:f
3816:∗
3813:R
3807:∗
3804:R
3786:.
3782:)
3777:)
3762:R
3756:+
3753:1
3749:(
3737:D
3732:f
3725:e
3722:R
3711:(
3695:=
3686:D
3681:f
3676:1
3655:∗
3652:R
3635:,
3631:)
3626:}
3621:)
3605:R
3586:1
3582:(
3571:R
3564:+
3561:1
3557:{
3545:D
3540:f
3533:e
3530:R
3519:(
3495:=
3486:D
3481:f
3476:1
3449:,
3445:)
3440:}
3435:)
3419:R
3400:1
3396:(
3385:R
3378:+
3375:1
3371:{
3359:D
3354:f
3347:e
3344:R
3333:(
3309:=
3300:D
3295:f
3290:1
3267:∗
3264:R
3246:,
3242:)
3237:)
3225:R
3199:R
3192:+
3189:1
3185:(
3173:D
3168:f
3161:e
3158:R
3147:(
3131:=
3122:D
3117:f
3112:1
3084:,
3080:)
3075:)
3063:R
3037:R
3030:+
3027:1
3023:(
3011:D
3006:f
2999:e
2996:R
2985:(
2975:2
2969:=
2960:D
2955:f
2950:1
2933:∗
2930:R
2923:∗
2920:R
2909:∗
2906:R
2900:∗
2897:R
2893:∗
2890:R
2888:(
2886:B
2879:∗
2876:R
2869:.
2863:D
2860:/
2857:ε
2848:D
2845:f
2834:∗
2831:R
2822:∗
2819:R
2815:∗
2812:R
2810:(
2808:B
2801:∗
2798:R
2789:∗
2786:R
2779:ε
2769:∗
2766:R
2760:∗
2757:R
2748:D
2745:/
2742:ε
2730:∗
2727:R
2722:B
2704:)
2698:D
2684:8
2674:(
2664:+
2653:D
2648:f
2638:1
2633:=
2630:)
2621:R
2617:(
2614:B
2601:B
2593:B
2588:∗
2585:R
2578:∗
2575:R
2569:∗
2566:R
2551:D
2547:ε
2528:D
2519:)
2509:D
2504:f
2497:e
2494:R
2489:(
2482:8
2478:1
2473:=
2464:R
2446:ε
2427:D
2424:f
2402:)
2398:e
2395:R
2385:(
2382:W
2373:=
2369:)
2364:e
2361:R
2351:)
2345:(
2312:(
2308:W
2302:)
2296:(
2281:=
2272:D
2267:f
2262:1
2245:W
2234:D
2231:f
2220:D
2217:f
2205:D
2202:f
2173:)
2164:D
2159:f
2152:e
2149:R
2144:(
2131:=
2122:D
2117:f
2112:1
2086:D
2083:f
2075:(
2060:D
2057:/
2054:ε
2011:ε
1991:D
1960:=
1944:μ
1927:,
1918:D
1912:v
1903:=
1900:D
1894:v
1878:=
1874:e
1871:R
1837:,
1831:e
1828:R
1819:=
1813:D
1808:f
1776:D
1773:f
1745:⟩
1743:v
1741:⟨
1737:ν
1733:ε
1729:D
1724:D
1721:f
1708:D
1705:/
1702:ε
1642:.
1637:2
1629:v
1616:D
1612:f
1606:8
1603:1
1598:=
1578:τ
1553:.
1546:5
1541:c
1537:D
1531:2
1527:Q
1515:g
1510:2
1501:8
1491:D
1487:f
1483:=
1480:S
1467:Q
1450:.
1444:v
1436:2
1431:c
1427:D
1421:4
1413:=
1410:Q
1384:c
1380:D
1363:A
1355:Q
1330:,
1324:v
1315:A
1312:=
1309:Q
1296:Q
1290:v
1266:.
1261:D
1256:2
1248:v
1232:g
1229:2
1225:1
1215:D
1211:f
1207:=
1204:S
1188:m
1184:L
1167:,
1162:L
1158:p
1143:g
1136:1
1131:=
1126:L
1122:h
1113:=
1110:S
1088:g
1079:h
1077:Δ
1052:,
1049:h
1042:g
1036:=
1033:p
1012:f
1009:h
1003:h
1001:Δ
972:v
968:c
965:D
959:4
956:/
953:π
946:Q
936:Q
920:μ
899:,
892:4
887:c
883:D
878:Q
856:=
851:L
847:p
821:/
811:D
808:f
786:c
782:D
762:λ
741:D
736:f
719:Q
695:v
680:P
676:A
667:H
663:D
657:D
636:H
632:D
596:m
594:/
583:L
579:/
575:p
573:Δ
552:,
545:H
541:D
535:2
527:v
512:2
498:D
493:f
489:=
484:L
480:p
461:L
456:p
454:Δ
450:D
356:V
346:+
340:=
337:f
322:.
306:g
280:V
254:D
232:L
210:H
178:D
175:g
172:2
165:2
161:V
157:L
148:f
145:=
142:H
20:)
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