Английская Википедия:Darcy–Weisbach equation

Шаблон:Short description In fluid dynamics, the Darcy–Weisbach equation is an empirical equation that relates the head loss, or pressure loss, due to friction along a given length of pipe to the average velocity of the fluid flow for an incompressible fluid. The equation is named after Henry Darcy and Julius Weisbach. Currently, there is no formula more accurate or universally applicable than the Darcy-Weisbach supplemented by the Moody diagram or Colebrook equation.^[1]

The Darcy–Weisbach equation contains a dimensionless friction factor, known as the Darcy friction factor. This is also variously called the Darcy–Weisbach friction factor, friction factor, resistance coefficient, or flow coefficient.Шаблон:Efn

Pressure-loss equation

In a cylindrical pipe of uniform diameter Шаблон:Mvar, flowing full, the pressure loss due to viscous effects Шаблон:Math is proportional to length Шаблон:Mvar and can be characterized by the Darcy–Weisbach equation:^[2]

<math>\frac{\Delta p}{L} =f_\mathrm{D} \cdot \frac{\rho }{2} \cdot \frac{{\langle v \rangle}^2}{D_H},</math>

where the pressure loss per unit length Шаблон:Math (SI units: Pa/m) is a function of:

<math>\rho</math>, the density of the fluid (kg/m³);

<math>D_H</math>, the hydraulic diameter of the pipe (for a pipe of circular section, this equals Шаблон:Math; otherwise Шаблон:Math for a pipe of cross-sectional area Шаблон:Mvar and perimeter Шаблон:Mvar) (m);

<math>\langle v \rangle</math>, the mean flow velocity, experimentally measured as the volumetric flow rate Шаблон:Mvar per unit cross-sectional wetted area (m/s);

<math>f_\mathrm{D}</math>, the Darcy friction factor (also called flow coefficient Шаблон:Mvar^[3]^[4]).

For laminar flow in a circular pipe of diameter Шаблон:Math, the friction factor is inversely proportional to the Reynolds number alone (Шаблон:Math) 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

<math>\frac{\Delta p}{L} = \frac{128}{\pi} \cdot \frac{\mu Q}{D_c^4},</math>

where

Шаблон:Math is the dynamic viscosity of the fluid (Pa·s = N·s/m² = kg/(m·s));

Шаблон:Math is the volumetric flow rate, used here to measure flow instead of mean velocity according to Шаблон:Math (m³/s).

Note that this laminar form of Darcy–Weisbach is equivalent to the Hagen–Poiseuille equation, which is analytically derived from the Navier–Stokes equations.

Head-loss form

The head loss Шаблон:Math (or Шаблон:Math) expresses the pressure loss due to friction in terms of the equivalent height of a column of the working fluid, so the pressure drop is

<math>\Delta p = \rho g \, \Delta h,</math>

where:

Шаблон:Math = The head loss due to pipe friction over the given length of pipe (SI units: m);Шаблон:Efn

Шаблон:Mvar = The local acceleration due to gravity (m/s²).

It is useful to present head loss per length of pipe (dimensionless):

<math>S = \frac{\Delta h}{L} = \frac{1}{\rho g} \cdot \frac{\Delta p}{L},</math>

where Шаблон:Mvar is the pipe length (m).

Therefore, the Darcy–Weisbach equation can also be written in terms of head loss:^[5]

<math>S = f_\text{D} \cdot \frac{1}{2g} \cdot \frac{{\langle v \rangle}^2}{D}.</math>

In terms of volumetric flow

The relationship between mean flow velocity Шаблон:Math and volumetric flow rate Шаблон:Mvar is

<math>Q = A \cdot \langle v \rangle,</math>

where:

Шаблон:Mvar = The volumetric flow (m³/s),

Шаблон:Mvar = The cross-sectional wetted area (m²).

In a full-flowing, circular pipe of diameter Шаблон:Mvar,

<math> Q = \frac{\pi}{4} D_c^2 \langle v \rangle.</math>

Then the Darcy–Weisbach equation in terms of Шаблон:Mvar is

Shear-stress form

The mean wall shear stress Шаблон:Mvar in a pipe or open channel is expressed in terms of the Darcy–Weisbach friction factor as^[6]

<math>\tau = \frac18 f_\text{D} \rho {\langle v \rangle}^2.</math>

The wall shear stress has the SI unit of pascals (Pa).

Darcy friction factor

Файл:Darcy Friction factor for Re between 10 and 10E8 for values of relative roughness.svg

Figure 1. The Darcy friction factor versus Reynolds number for Шаблон:Math for smooth pipe and a range of values of relative roughness Шаблон:Mvar. Data are from Nikuradse (1932, 1933), Colebrook (1939), and McKeon (2004).

The friction factor Шаблон:Math is not a constant: it depends on such things as the characteristics of the pipe (diameter Шаблон:Mvar and roughness height Шаблон:Mvar), the characteristics of the fluid (its kinematic viscosity Шаблон:Mvar [nu]), and the velocity of the fluid flow Шаблон:Math. It has been measured to high accuracy within certain flow regimes 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 Moody diagrams, after L. F. Moody, and hence the factor itself is sometimes erroneously called the Moody friction factor. It is also sometimes called the Blasius friction factor, after the approximate formula he proposed.

Figure 1 shows the value of Шаблон:Math 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. Шаблон:Clear left

Laminar regime

For laminar (smooth) flows, it is a consequence of Poiseuille's law (which stems from an exact classical solution for the fluid flow) that

<math>f_{\mathrm D} = \frac{64}{\mathrm{Re}},</math>

where Шаблон:Math is the Reynolds number

<math>\mathrm{Re} = \frac \rho \mu \langle v \rangle D = \frac{\langle v \rangle D} \nu, </math>

and where Шаблон:Mvar is the viscosity of the fluid and

is known as the kinematic viscosity. In this expression for Reynolds number, the characteristic length Шаблон:Mvar is taken to be the hydraulic diameter 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 Шаблон:Math demonstrates laminar flow; the friction factor is well represented by the above equation.Шаблон:Efn

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.

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 Шаблон:Mvar: the flow velocity in the neighborhood of the pipe wall is zero.

Critical regime

For Reynolds numbers in the range Шаблон:Math, 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. Шаблон:Clear left

Turbulent regime

Файл:Darcy Friction factor for Re between 10E3 and 10E8 for values of relative roughness.svg

Figure 2. The Darcy friction factor versus Reynolds number for Шаблон:Math for smooth pipe and a range of values of relative roughness Шаблон:Mvar. Data are from Nikuradse (1932, 1933), Colebrook (1939), and McKeon (2004).

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 Шаблон:Math (Шаблон:Math), the friction factor varies less than one order of magnitude (Шаблон:Math). 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.

Smooth-pipe regime

Шаблон:Clear left

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^[3] with the parameters suitably adjusted

<math>\frac{1}{\sqrt{f_{\mathrm D}}} = 1.930 \log\left(\mathrm{Re}\sqrt{f_{\mathrm D}}\right) - 0.537.</math>

The numbers 1.930 and 0.537 are phenomenological; these specific values provide a fairly good fit to the data.^[7] The product Шаблон:Math (called the "friction Reynolds number") can be considered, like the Reynolds number, to be a (dimensionless) parameter of the flow: at fixed values of Шаблон:Math, the friction factor is also fixed.

In the Kármán–Prandtl resistance equation, Шаблон:Math can be expressed in closed form as an analytic function of Шаблон:Math through the use of the [[Lambert W function|Lambert Шаблон:Mvar function]]:

<math>\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})</math>

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 Шаблон:Math increases, the profile of the fluid velocity approaches the wall asymptotically, thereby transferring more momentum to the pipe wall, as modeled in Blasius boundary layer theory.

Rough-pipe regime

Шаблон:Clear left

When the pipe surface's roughness height Шаблон:Mvar 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 number. Here, it is useful to employ yet another dimensionless parameter of the flow, the roughness Reynolds number^[8]

<math>R_* = \frac 1 {\sqrt 8} \left( \mathrm{Re}\sqrt{f_{\mathrm D}} \, \right) \frac \varepsilon D </math>

where the roughness height Шаблон:Mvar is scaled to the pipe diameter Шаблон:Mvar.

Roughness function Шаблон:Mvar vs. friction Reynolds number Шаблон:Math

Figure 3. Roughness function B vs. friction Reynolds number Шаблон:Math. The data fall on a single trajectory when plotted in this way. The regime Шаблон:Math is effectively that of smooth pipe flow. For large Шаблон:Math, the roughness function Шаблон:Mvar approaches a constant value. Phenomenological functions attempting to fit these data, including the Afzal^[9] and Colebrook–White^[10] are shown.

It is illustrative to plot the roughness function Шаблон:Mvar:^[11]

<math>B(R_*) = \frac 1 {1.930 \sqrt{f_{\mathrm D}}} + \log\left( \frac{1.90}{\sqrt{8}} \cdot \frac{\varepsilon}{D}\right) </math>

Figure 3 shows Шаблон:Mvar versus Шаблон:Math for the rough pipe data of Nikuradse,^[8] Shockling,^[12] and Langelandsvik.^[13]

In this view, the data at different roughness ratio Шаблон:Mvar fall together when plotted against Шаблон:Math, demonstrating scaling in the variable Шаблон:Math. The following features are present:

When Шаблон:Math, then Шаблон:Math 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.
When Шаблон:Math, the data lie on the line Шаблон:Math; flow is in the smooth pipe regime.
When Шаблон:Math, the data asymptotically approach a horizontal line; they are independent of Шаблон:Math, Шаблон:Math, and Шаблон:Mvar.
The intermediate range of Шаблон:Math constitutes a transition from one behavior to the other. The data depart from the line Шаблон:Math very slowly, reach a maximum near Шаблон:Math, then fall to a constant value.

Afzal's fit to these data in the transition from smooth pipe flow to rough pipe flow employs an exponential expression in Шаблон:Math that ensures proper behavior for Шаблон:Math (the transition from the smooth pipe regime to the rough pipe regime):^[9]^[14]^[15]

<math> \frac{1}{ \sqrt{ f_{\mathrm D}}} = -2 \log\left( \frac {2.51} {\mathrm{Re}\sqrt{f_{\mathrm D}}} \left( 1 + 0.305 R_* \exp\frac{-11}{R_*} \right) \right) ,</math>

and

<math> \frac{1}{ \sqrt{ f_{\mathrm D}}} = -1.930 \log\left( \frac {1.90} {\mathrm{Re}\sqrt{f_{\mathrm D}}} \left( 1 + 0.34 R_* \exp\frac{-11}{R_*} \right) \right) ,</math>

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 Шаблон:Math along with one further parameter, 11, to govern the transition from smooth to rough flow. It is exhibited in Figure 3.

The friction factor for another analogous roughness becomes

 : <math> \frac{1}{ \sqrt{ f_{\mathrm D}}}   = - 2.0 \, \log_{10}   \left ( \frac{2.51 } {\mathrm{Re}\sqrt{f_{\mathrm D}}}

\left \{ 1 + 0.305 R_* \; \left ( 1 - \exp \frac{- R_*}{26} \right) \right\} \right) ,</math>

and

: <math> \frac{1}{ \sqrt{ f_{\mathrm D}}} = - 1.93 \, \log_{10}   \left (
\frac{1.91} {\mathrm{Re}\sqrt{f_{\mathrm D}}}

\left \{1 + 0.34 R_* \; \left ( 1 - \exp \frac{- R_*}{26} \right)\right\} \right) ,</math>

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 Шаблон:Math along with one further parameter, 26, to govern the transition from smooth to rough flow.

The Colebrook–White relation^[10] fits the friction factor with a function of the form

<math>\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). </math>Шаблон:Efn} = -2.00 \log\left(2.51\frac{1}{\mathrm{Re} \sqrt{f_{\mathrm D}}} + \frac{1}{3.7}\frac{\varepsilon}{D}\right)</math>}}

This relation has the correct behavior at extreme values of Шаблон:Math, as shown by the labeled curve in Figure 3: when Шаблон:Math 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.^[12] Colebrook acknowledges the discrepancy with Nikuradze's data but argues that his relation 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.

Шаблон:Clear left

Calculating the friction factor from its parametrization

Шаблон:Main

For turbulent flow, methods for finding the friction factor Шаблон:Math include using a diagram, such as the Moody chart, or solving equations such as the Colebrook–White equation (upon which the Moody chart is based), or the Swamee–Jain equation. While the Colebrook–White relation is, in the general case, an iterative method, the Swamee–Jain equation allows Шаблон:Math to be found directly for full flow in a circular pipe.^[5]

Direct calculation when friction loss Шаблон:Mvar is known

In typical engineering applications, there will be a set of given or known quantities. The acceleration of gravity Шаблон:Mvar and the kinematic viscosity of the fluid Шаблон:Mvar are known, as are the diameter of the pipe Шаблон:Mvar and its roughness height Шаблон:Mvar. If as well the head loss per unit length Шаблон:Mvar is a known quantity, then the friction factor Шаблон:Math can be calculated directly from the chosen fitting function. Solving the Darcy–Weisbach equation for Шаблон:Math,

<math>\sqrt{f_{\mathrm D}} = \frac{ \sqrt{2gSD} }{ \langle v \rangle }</math>

we can now express Шаблон:Math:

<math>\mathrm{Re}\sqrt{f_{\mathrm D}} = \frac{ 1 }{ \nu } \sqrt{2g} \sqrt{ S } \sqrt{ D^3 }</math>

Expressing the roughness Reynolds number Шаблон:Math,

<math>\begin{align}

R_* &= \frac \varepsilon D \cdot \mathrm{Re}\sqrt{f_{\mathrm D}} \cdot \frac 1 {\sqrt 8} \\ &= \frac12 \frac{\sqrt g}{\nu} \varepsilon \sqrt{S} \sqrt{D} \end{align}</math>

we have the two parameters needed to substitute into the Colebrook–White relation, or any other function, for the friction factor Шаблон:Math, the flow velocity Шаблон:Math, and the volumetric flow rate Шаблон:Mvar.

Confusion with the Fanning friction factor

The Darcy–Weisbach friction factor Шаблон:Math is 4 times larger than the Fanning friction factor Шаблон:Mvar, 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 Шаблон:Math is more commonly used by civil and mechanical engineers, and the Fanning factor Шаблон:Mvar by chemical engineers, but care should be taken to identify the correct factor regardless of the source of the chart or formula.

Note that

<math>\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}</math>

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 Шаблон:Math, it is the Fanning factor Шаблон:Mvar, and if the formula for laminar flow is Шаблон:Math, it is the Darcy–Weisbach factor Шаблон:Math.

Which friction factor is plotted in a Moody diagram may be determined by inspection if the publisher did not include the formula described above:

Observe the value of the friction factor for laminar flow at a Reynolds number of 1000.
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: Шаблон:Math.
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: Шаблон:Math.

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.

History

Historically this equation arose as a variant on the Prony equation; this variant was developed by Henry Darcy of France, and further refined into the form used today by Julius Weisbach of Saxony in 1845. Initially, data on the variation of Шаблон:Math 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 empirical equations valid only for certain flow regimes, notably the Hazen–Williams equation or the Manning equation, most of which were significantly easier to use in calculations. However, since the advent of the calculator, ease of calculation is no longer a major issue, and so the Darcy–Weisbach equation's generality has made it the preferred one.^[16]

Derivation by dimensional analysis

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, Шаблон:Math, and the volumetric flow rate. The flow rate can be converted to a mean flow velocity Шаблон:Mvar by dividing by the wetted area of the flow (which equals the cross-sectional area of the pipe if the pipe is full of fluid).

Pressure has dimensions of energy per unit volume, therefore the pressure drop between two points must be proportional to the dynamic pressure q. We also know that pressure must be proportional to the length of the pipe between the two points Шаблон:Mvar 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, Шаблон:Mvar, which is also constant along the pipe. Therefore,

<math>\Delta p \propto \frac{L}{D} q = \frac{L}{D} \cdot \frac{\rho }{2} \cdot {\langle v \rangle}^2 </math>

The proportionality coefficient is the dimensionless "Darcy friction factor" or "flow coefficient". This dimensionless coefficient will be a combination of geometric factors such as Шаблон:Mvar, the Reynolds number and (outside the laminar regime) the relative roughness of the pipe (the ratio of the roughness height to the hydraulic diameter).

Note that the dynamic pressure is not the kinetic energy of the fluid per unit volume,Шаблон:Citation needed for the following reasons. Even in the case of laminar flow, where all the flow lines 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 energy then involves the root mean-square velocity, which always exceeds the mean velocity. In the case of turbulent flow, 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.

Practical application

In a hydraulic engineering application, it is typical for the volumetric flow Шаблон:Mvar within a pipe (that is, its productivity) and the head loss per unit length Шаблон:Mvar (the concomitant power consumption) to be the critical important factors. The practical consequence is that, for a fixed volumetric flow rate Шаблон:Mvar, head loss Шаблон:Mvar decreases with the inverse fifth power of the pipe diameter, Шаблон:Mvar. 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.

Advantages

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:^[1]

It is based on fundamentals.
It is dimensionally consistent.
It is useful for any fluid, including oil, gas, brine, and sludges.
It can be derived analytically in the laminar flow region.
It is useful in the transition region between laminar flow and fully developed turbulent flow.
The friction factor variation is well documented.

Notes

Шаблон:Notelist

References

Шаблон:Reflist 18. Afzal, Noor (2013) "Roughness effects of commercial steel pipe in turbulent flow: Universal scaling". Canadian Journal of Civil Engineering 40, 188-193.

External links

Шаблон:Hydraulics

↑ ^1,0 ^1,1 Шаблон:Cite book
↑ Шаблон:Cite book
↑ ^3,0 ^3,1 Шаблон:Cite book
↑ Шаблон:Cite book
↑ ^5,0 ^5,1 Шаблон:Cite book
<math>\frac{1}{\sqrt{f_\mathrm{D}}} = 2 \log\left(\mathrm{Re} \sqrt{f_\mathrm{D}}\right) - 0.8 \quad \text{for } \mathrm{Re} > 3000.</math>
↑ Шаблон:Cite book
↑ Шаблон:Cite journal
↑ ^8,0 ^8,1 Шаблон:Cite journal In translation, NACA TM 1292. The data are available in digital form Шаблон:Dead link.
↑ ^9,0 ^9,1 Шаблон:Cite journal
↑ ^10,0 ^10,1 Шаблон:Cite journal
↑ Шаблон:Cite book
↑ ^12,0 ^12,1 Шаблон:Cite journal
↑ Шаблон:Cite journal
↑ Шаблон:Cite journal
↑ Шаблон:Cite journal
↑ Шаблон:Cite book

[:0-1] 1,0 ^1,1 Шаблон:Cite book

[Brown-2] Шаблон:Cite book

[Rouse1946-3] 3,0 ^3,1 Шаблон:Cite book

[Incopera2002-4] Шаблон:Cite book

[Crowe2005-5] 5,0 ^5,1 Шаблон:Cite book
<math>\frac{1}{\sqrt{f_\mathrm{D}}} = 2 \log\left(\mathrm{Re} \sqrt{f_\mathrm{D}}\right) - 0.8 \quad \text{for } \mathrm{Re} > 3000.</math>

[6] Шаблон:Cite book

[McKeon2005-7] Шаблон:Cite journal

[Nikuradse1933-8] 8,0 ^8,1 Шаблон:Cite journal In translation, NACA TM 1292. The data are available in digital form Шаблон:Dead link.

[Afzal2007-9] 9,0 ^9,1 Шаблон:Cite journal

[Colebrook1939-10] 10,0 ^10,1 Шаблон:Cite journal

[Schlichting1955-11] Шаблон:Cite book

[Shockling2006-12] 12,0 ^12,1 Шаблон:Cite journal

[Langelandsvik2008-13] Шаблон:Cite journal

[Afzal2011-14] Шаблон:Cite journal

[Afzal2013-15] Шаблон:Cite journal

[16] Шаблон:Cite book

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Английская Википедия:Darcy–Weisbach equation

Содержание

Pressure-loss equation

Head-loss form

In terms of volumetric flow

Shear-stress form

Darcy friction factor

Laminar regime

Critical regime

Turbulent regime

Smooth-pipe regime

Rough-pipe regime

Calculating the friction factor from its parametrization

Direct calculation when friction loss Шаблон:Mvar is known

Confusion with the Fanning friction factor

History

Derivation by dimensional analysis

Practical application

Advantages

See also

Notes

References

Further reading

External links

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