Darcy–Weisbach equation

In fluid dynamics, the Darcy–Weisbach equation is a phenomenological equation, which 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.

The Darcy–Weisbach equation contains a dimensionless friction factor, known as the Darcy friction factor. This is also called the Darcy–Weisbach friction factor, resistance coefficient, or simply friction factor.^{[lower-alpha 1]}

Pressure loss form

In a cylindrical pipe of uniform diameter D, the pressure loss due to viscous effects Δp is proportional to length L and can be characterized by the Darcy Weisbach equation:^[2]

\frac{\Delta p}{L} = f_{\mathrm D} \cdot \frac{\rho }{2} \cdot \frac{V^2}{D}

where the loss per unit length Δp/L (SI units Pa/m) is a function of:

the density of the fluid ρ (kg/m³);
the length of the pipe L (m);
the hydraulic diameter of the pipe D (for a pipe of circular section, this equals the internal diameter of the pipe) (m);
the mean flow velocity V, experimentally measured as the volumetric flow rate Q per unit cross-sectional wetted area (m/s);
the Darcy Friction Factor, f_D.^[3]

Head loss form

The head loss h_f expresses the height of a column of fluid, so the pressure loss Δp is related by,

\Delta p = \rho \cdot g \cdot \Delta h

where ρ is the density of the fluid. Therefore the Darcy–Weisbach equation can also be written in terms of head loss:^[4]

\frac{\Delta h}{L} = f_{\mathrm D} \cdot \frac{1}{2g} \cdot \frac{V^2}{D}

where;

Δh is the head loss due to friction (SI units: m); Note: This is also proportional to the piezometric head along the pipe;
g is the local acceleration due to gravity (m/s²);

Darcy friction factor

The Darcy friction factor versus Reynolds Number for 10 < Re < 10⁸ for smooth pipe and a range of values of relative roughness ε/D. Data are from Nikuradse (1932, 1933), Colebrook (1939), and McKeon (2004).

See also Darcy friction factor formulae

The friction factor f_D (or flow coefficient λ) is not a constant: it depends on such things as the characteristics of the pipe (diameter D and roughness height ε), the characteristics of the fluid (its kinematic viscosity ν), and the velocity of the fluid flow V. 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.

Laminar Regime

For laminar (slow) flows, it is a consequence of Poiseuille's law that

f_{\mathrm D} = \frac{64}{\mathrm{Re}},

where Re is the Reynolds number

\mathrm{Re} = \frac{\rho}{\mu} V D = \frac{VD}{\nu},

and where μ is the viscosity of the fluid and

\nu = \frac{\mu}{\rho}

is known as the kinematic viscosity. In this expression for Reynolds number, the characteristic length D is taken to be the hydraulic diameter of the pipe, which, for circular pipe geometries, equals the inside diameter. In the accompanying plot of friction factor versus Reynolds number, the regime Re < 2000 demonstrates laminar flow; the friction factor is well represented by the above equation.

Critical Regime

For Reynolds numbers in the range 2000 < Re < 4000, the flow is unsteady, characterized by the formation of vortices, and not well understood.

Turbulent Regime

The Darcy friction factor versus Reynolds Number for 10³ < Re < 10⁸ for smooth pipe and a range of values of relative roughness ε/D. 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. When the pipe surface is smooth (the "smooth pipe" regime), the friction factor varies according to the relation^[5]

\frac{1}{\sqrt{f_{\mathrm D}}} = -2.00 \log\left(\frac{2.51}{\mathrm{Re}\sqrt{f_{\mathrm D}}}\right)

The factors 2.00 and 2.51 are phenomenological; these specific values provide a fairly good fit to the data. The product Re√f_D can be considered, like the Reynolds number, to be a (dimensionless) parameter of the flow: at fixed values of Re√f_D, the friction factor is also fixed.

A plot of resistance factor f_D with the smooth pipe dependency factored out, versus Re √f_D. The smooth pipe data lie on a horizontal line with ordinate = 1; the rough pipe data coincide with the smooth for low values of the abscissa, then branch away on parallel trajectories. Data with a given value of ε/D are from Nikuradse,^[6]^[7] smooth pipe data from Princeton^[8] are also included.

It is useful to consider the data with the smooth pipe dependency factored out. The accompanying plot shows the value

10^{\frac{1}{2\sqrt{f_{\mathrm D}}}} \cdot \frac{2.7}{\mathrm{Re}\sqrt{f_{\mathrm D}}}

plotted against Re√f_D: the smooth pipe data lie on a horizontal line with ordinate = 1.

When the pipe surface's roughness ε 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 friction Reynolds number

\mathrm{R}_{*} = \frac{1}{\sqrt{8}} \cdot (\mathrm{Re}\sqrt{f_{\mathrm D}}) \cdot (\epsilon / D)

where the roughness ε is scaled to the pipe diameter D.

If ε = 0, then R_* is identically zero: flow is always in the smooth pipe regime.
When R_* < 5, flow is in the smooth pipe regime.
When R_* > 100, the friction factor reaches its asymptotic value: it is independent of Re.
For intermediate values of R_*, termed the "transition regime", the friction factor varies with both Re√f_D and R_*.

A plot of resistance factor f_D with the smooth pipe dependency factored out, versus R*, the friction Reynolds number. The smooth pipe data correspond to R* = 0, so are out of frame. The rough pipe data fall on a single trajectory when plotted in this way. For a given roughness ratio, ε/D, and for R* large, the Darcy friction factor f_D approaches a constant, independent of Re, the Reynolds number. Phenomenological functions attempting to fit these data, including the Colebrook–White relation are shown.

Interestingly, the data show scaling in the variable R_*: the accompanying plot of the same quantity as in the previous one shows that all of the data at different roughness ratio ε/D fall together when plotted against R_*.

Colebrook^[5] effectively fits this dependence on R_* with a function of the form

-2.00 \log\left(1 + \frac{1}{3.3}\mathrm{R}_{*}\right)

Thus, the Colebrook–White relation

\frac{1}{\sqrt{f_{\mathrm D}}} = -2.00 \log\left( \frac{2.51}{\mathrm{Re}\sqrt{f_{\mathrm D}}} \left(1 + \frac{1}{3.3}\mathrm{R}_{*}\right) \right)

Or, in its originally published form,

\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{\epsilon}{D}\right)

This relation has the correct behavior at extreme values of R_*, as shown in the plot: when R_* is small, the relation is consistent with smooth pipe flow, when large, it is consistent with rough pipe flow. However, as may be understandable from a one-parameter fitting function, its performance in the transitional domain overestimates the friction factor by a substantial margin.^[9] The plot shows alternative fitting functions,^[10] labeled "exp1":

\frac{1}{\sqrt{f_{\mathrm D}}} = 2.00 \log\left[\left( \frac {\mathrm{Re}\sqrt{f_{\mathrm D}}} {2.7} \right) \cdot \left(1 - \exp(-3.4/\mathrm{R}_{*}\right)\right],

The curve labeled "exp2":

\frac{1}{\sqrt{f_{\mathrm D}}} = 2.00 \log\left[ \left( \frac {\mathrm{Re}\sqrt{f_{\mathrm D}}} {2.7} \right) \cdot \left(1 - \exp(-3.4/\mathrm{R}_{*}\left(1+10/\mathrm{R}_{*}\right)\right)\right],

and, labeled "exp3":

\frac{1}{\sqrt{f_{\mathrm D}}} = 2.00 \log\left[ \left( \frac {\mathrm{Re}\sqrt{f_{\mathrm D}}} {2.7} \right) \cdot \left(1 - \exp(-3.4/\mathrm{R}_{*}\left(1 + 10/\mathrm{R}_{*}\left(1 + 4/\mathrm{R}_{*}\right)\right)\right)\right].

These constitute, in essence, a polynomial expansion around R_* = ∞, achieving a better fit to the data by employing five arbitrary parameters (two more than Colebrook–White).

For turbulent flow, methods for finding the friction factor f_D 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 an iterative method, the Swamee–Jain equation allows f_D to be found directly for full flow in a circular pipe.^[4]

Direct calculation

In cases where the pipe roughness ε and diameter D are known, and the head loss h_f is given, the friction factor f_D can be calculated directly. Solving the Darcy–Weisbach equation for √f_D,

\sqrt{f_{\mathrm D}} = \frac{ \sqrt{2gSD} }{ V }

Where S is the head loss per unit length of pipe:

S = \frac{ h_f }{ L }

Then, express Re√f_D:

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

and, with the friction Reynolds number R_*,

R_{*} = \frac{ \epsilon }{ D } \cdot \mathrm{Re}\sqrt{ f_{\mathrm D} } \cdot \frac{ 1 }{ \sqrt{8} }

= \frac{ 1 }{ 2 } \frac{ \sqrt{g} }{ \nu } \epsilon \sqrt{ S } \sqrt{ D }

we have the two parameters needed to substitute into the Colebrook-White relation, or any other function for the friction factor. The givens in this case are, in addition to the usual g and ν, the friction loss S, the pipe roughness ε, and the pipe diameter D.

Confusion with the Fanning friction factor

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

Note that

\Delta p = f_{\mathrm D} \cdot \frac{L}{D} \cdot \frac{\rho V^2}{2} = f \cdot \frac{L}{D} \cdot {2\rho V^2}

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 f = 16/Re, it is the Fanning factor, f, and if the formula for laminar flow is f_D = 64/Re, it is the Darcy–Weisbach factor, f_D.

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: f_D = 64/Re.
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: f = 16/Re.

The procedure above is similar for any available Reynolds number that is an integral power of ten. It is not necessary to remember the value 1000 for this procedure—only that an integral 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 f_D 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.^[11]

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, Δp / L, and the volumetric flow rate. The flow rate can be converted to a mean flow velocity V 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 (1/2)ρV², which has the same dimensions as it resembles (see below) the expression for the kinetic energy per unit volume. We also know that pressure must be proportional to the length of the pipe between the two points L 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, D, which is also constant along the pipe. Therefore,

\Delta p \propto \frac{L}{D} \cdot \frac{1}{2}\rho V^2.

The proportionality coefficient is the dimensionless "Darcy friction factor" or "flow coefficient". This dimensionless coefficient will be a combination of geometric factors such as π, 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 (1/2)ρV² is not the kinetic energy of the fluid per unit volume, 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 mean-square velocity, which always exceeds the square of 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 applications

In hydraulic engineering applications, it is often desirable to express the head loss in terms of volumetric flow rate in the pipe. For this, it is necessary to substitute the following into the original head loss form of the Darcy–Weisbach equation

V^2 = \frac{Q^2}{A_w^2}

where

V is, as above, the average flow velocity (m/s);
Q is the volumetric flow rate (m³/s);
A_w is the cross-sectional wetted area (m²).

For the general case of an arbitrarily-full pipe, the value of A_w will not be immediately known, being an implicit function of pipe slope, cross-sectional shape, flow rate and other variables. If, however, the pipe is assumed to be full flowing and of circular cross-section, as is common in practical scenarios, then

A_w^2 = \left(\frac{\pi D^2}{4}\right)^2 = \frac{\pi^2 D^4}{16}

where D is the diameter of the pipe

Substituting these results into the original formulation yields the final equation for head loss in terms of volumetric flow rate in a full-flowing circular pipe

\Delta p = \frac{8 f_{\mathrm D} \rho L Q^2}{ \pi^2 D^5}

where all symbols are defined as above.

The implication of this equation is that for a fixed volumetric flow rate, head loss increases linearly with the length of the pipe, L, but decreases with the inverse fifth power of the diameter, D. Thus, by merely doubling the diameter of a pipe of a given wall thickness, even though the amount of material required per unit length doubles, the head loss decreases to 1/32, about 3%, of the smaller diameter pipe.

Note

↑ The value of the Darcy friction factor is four times that of the Fanning friction factor, with which it should not be confused.^[1]

References

↑ Manning, Francis S.; Thompson, Richard E. (1991), Oilfield Processing of Petroleum. Vol. 1: Natural Gas, PennWell Books, p. 420, ISBN 0-87814-343-2 See page 293.
↑ Brown, Glenn. "The Darcy-Weisbach Equation". Oklahoma State University–Stillwater.
↑ Incopera, Frank P.; Dewitt, David P. (2002). Fundamentals of Heat and Mass Transfer (5 ed.). John Wiley & Sons, Inc. p. 470. See paragraph 3
1 2 Crowe, Clayton T.; Elger, Donald F.; Robertson, John A. (2005). Engineering Fluid Mechanics (8 ed.). John Wiley & Sons, Inc. p. 379. See Equations 10:23, 10:24, paragraph 4: 1/sqrt(f_D) = 2 log( Re * sqrt(f_D)) – .8 for Re > 3000
1 2 Colebrook, C. F. (February 1939). "Turbulent flow in pipes, with particular reference to the transition region between smooth and rough pipe laws". Journal of the Institution of Civil Engineers (London).
↑ Nikuradse, J. (1932). "Gesetzmassigkeiten der Turbulenten Stromung in Glatten Rohren" (PDF). VDI Forschungsheft Arb. Ing.-Wes. 356: 1–36. In translation, NACA TT F-10 359. The data are available in digital form.
↑ Nikuradse, J. (1933). "Strömungsgesetze in Rauen Rohren". V. D. I. Forschungsheft (Berlin) 361: 1–22.
↑ McKeon, B. J.; Swanson, C. J.; Zagarola, M. V; Donnelly, R. J.; Smits, A. J. (2004). "Friction factors for smooth pipe flow" (PDF). J. Fluid Mech. (Cambridge University Press) 511: 41–44. doi:10.1017/S0022112004009796. Retrieved 20 October 2015.
↑ Shockling, M.A.; Allen, J.J.; Smits, A.J. (2006). "Roughness effects in turbulent pipe flow". J. Fluid Mech 564: 267–285.
↑ Rao, A. R.; Kumar, B. (2006). "Friction Factor for Turbulent Pipe Flow" (PDF). Bangalore, India: The Indian Institute of Science: 16. Retrieved 6 November 2015. The authors offer a phenomenological fitting function, a function of the variable R_*, the roughness Reynolds number. It requires three parameters in addition to the three in the Colebrook–White formula.
↑ Brown, G. O. (2003). "The History of the Darcy-Weisbach Equation for Pipe Flow Resistance". In Rogers, J. R.; Fredrich, A. J. Environmental and Water Resources History. American Society of Civil Engineers. pp. 34–43. ISBN 978-0-7844-0650-2. Text of the article, published on a blog

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