Darcy–Weisbach equation

Therefore, empirical relationships were sought to correlate the head loss with quantities like pipe diameter and fluid velocity.

[3] 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.

Antoine Chézy (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 hydraulic radius.

However, Chézy's formula was lost until 1800, when Gaspard de Prony (a former student of his) published an account describing his results.

[7] In a cylindrical pipe of uniform diameter D, flowing full, the pressure loss due to viscous effects Δp is proportional to length L and can be characterized by the Darcy–Weisbach equation:[8] where the pressure loss per unit length ⁠Δp/L⁠ (SI units: Pa/m) is a function of: For laminar flow in a circular pipe of diameter

, the friction factor is inversely proportional to the Reynolds number alone (fD = ⁠64/Re⁠) which itself can be expressed in terms of easily measured or published physical quantities (see section below).

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.

Figure 1 shows the value of fD as measured by experimenters for many different fluids, over a wide range of Reynolds numbers, and for pipes of various roughness heights.

The Colebrook–White relation[16] fits the friction factor with a function of the form This relation has the correct behavior at extreme values of R∗, as shown by the labeled curve in Figure 3: when R∗ is small, it is consistent with smooth pipe flow, when large, it is consistent with rough pipe flow.

[18] Colebrook acknowledges the discrepancy with Nikuradze's data but argues that his relation is consistent with the measurements on commercial pipes.

While the Colebrook–White relation is, in the general case, an iterative method, the Swamee–Jain equation allows fD to be found directly for full flow in a circular pipe.

The acceleration of gravity g and the kinematic viscosity of the fluid ν are known, as are the diameter of the pipe D and its roughness height ε.

If as well the head loss per unit length S is a known quantity, then the friction factor fD can be calculated directly from the chosen fitting function.

Which friction factor is plotted in a Moody diagram may be determined by inspection if the publisher did not include the formula described above: The procedure above is similar for any available Reynolds number that is an integer power of ten.

The key quantities are then the pressure drop along the pipe per unit length, ⁠Δp/L⁠, and the volumetric flow rate.

Note that the dynamic pressure is not the kinetic energy of the fluid per unit volume,[citation needed] for the following reasons.

In a hydraulic engineering application, it is typical for the volumetric flow Q within a pipe (that is, its productivity) and the head loss per unit length S (the concomitant power consumption) to be the critical important factors.

The practical consequence is that, for a fixed volumetric flow rate Q, head loss S decreases with the inverse fifth power of the pipe diameter, D. 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.

Thus the energy consumed in moving a given volumetric flow of the fluid is cut down dramatically for a modest increase in capital cost.

Afzal, Noor (2013) "Roughness effects of commercial steel pipe in turbulent flow: Universal scaling".

Figure 1. The Darcy friction factor versus Reynolds number for 10 < Re < 10 8 for smooth pipe and a range of values of relative roughness ⁠ ε / D ⁠ . Data are from Nikuradse (1932, 1933), Colebrook (1939), and McKeon (2004).
Figure 2. The Darcy friction factor versus Reynolds number for 1000 < Re < 10 8 for smooth pipe and a range of values of relative roughness ⁠ ε / D ⁠ . Data are from Nikuradse (1932, 1933), Colebrook (1939), and McKeon (2004).
Roughness function B vs. friction Reynolds number R∗
Figure 3. Roughness function B vs. friction Reynolds number R ∗ . The data fall on a single trajectory when plotted in this way. The regime R ∗ < 1 is effectively that of smooth pipe flow. For large R ∗ , the roughness function B approaches a constant value. Phenomenological functions attempting to fit these data, including the Afzal [ 15 ] and Colebrook–White [ 16 ] are shown.