In fluid dynamics, the Darcy–Weisbach equation is an empirical equation that relates the head loss, or pressure loss, due to viscous shear forces 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.
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.
Historical background The Darcy-Weisbach equation, combined with the Moody chart for calculating head losses in pipes, is traditionally attributed to Henry Darcy, Julius Weisbach, and Lewis Ferry Moody. However, the development of these formulas and charts also involved other scientists and engineers over its historical development. Generally, the Bernoulli's equation 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.
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. It is likely that Weisbach was aware of Chézy's formula through Prony's publications.
Weisbach's formula was proposed in 1845 in the form we still use today:
where: : head loss. : length of the pipe. : diameter of the pipe. : velocity of the fluid. : acceleration due to gravity. However, the friction factor f was expressed by Weisbach through the following empirical formula:
with and depending on the diameter and the type of pipe wall. Weisbach's work was published in the United States in 1848 and soon became well known there. In contrast, it did not initially gain much traction in France, where Prony equation, 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 dimensional analysis, resulting in a dimensionless friction factor f. The complexity of f, dependent on the mechanics of the boundary layer 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 factor f depended on the flow regime and was independent of the Reynolds number (and thus the velocity) only in the case of rough pipes in a fully turbulent flow regime (Prandtl-von Kármán equation).
Pressure-loss equation In a cylindrical pipe of uniform diameter , flowing full, the pressure loss due to viscous effects is proportional to length and can be characterized by the Darcy–Weisbach equation:
where the pressure loss per unit length (SI units: Pa/m) is a function of: , the density of the fluid (kg/m3); , the hydraulic diameter of the pipe (for a pipe of circular section, this equals ; otherwise for a pipe of cross-sectional area and perimeter ) (m); , the mean flow velocity, experimentally measured as the volumetric flow rate per unit cross-sectional wetted area (m/s); , the Darcy friction factor (also called flow coefficient ). For laminar flow in a circular pipe of diameter , the friction factor is inversely proportional to the Reynolds number alone () 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
where is the dynamic viscosity of the fluid (Pa·s = N·s/m2 = kg/(m·s)); is the volumetric flow rate, used here to measure flow instead of mean velocity according to (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 formula The head loss (or ) 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
where:
= The head loss due to pipe friction over the given length of pipe (SI units: m); = The local acceleration due to gravity (m/s2).
It is useful to present head loss per length of pipe (dimensionless):
where is the pipe length (m).
Therefore, the Darcy–Weisbach equation can also be written in terms of head loss:
In terms of volumetric flow The relationship between mean flow velocity and volumetric flow rate is
where: = The volumetric flow (m3/s), = The cross-sectional wetted area (m2).
In a full-flowing, circular pipe of diameter ,
Then the Darcy–Weisbach equation in terms of is
Shear-stress form
The mean wall shear stress in a pipe or open channel is expressed in terms of the Darcy–Weisbach friction factor as
The wall shear stress has the SI unit of pascals (Pa).
Darcy friction factor
The friction factor is not a constant: it depends on such things as the characteristics of the pipe (diameter and roughness height ), the characteristics of the fluid (its kinematic viscosity [nu]), and the velocity of the fluid flow . 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 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.
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
where is the Reynolds number
and where is the viscosity of the fluid and
is known as the kinematic viscosity. In this expression for Reynolds number, the characteristic length 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 demonstrates laminar flow; the friction factor is well represented by the above equation.
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 : the flow velocity in the neighborhood of the pipe wall is zero.
Critical regime For Reynolds numbers in the range , 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.
Turbulent regime
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 (), the friction factor varies less than one order of magnitude (). 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
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
The numbers 1.930 and 0.537 are phenomenological; these specific values provide a fairly good fit to the data. The product (called the "friction Reynolds number") can be considered, like the Reynolds number, to be a (dimensionless) parameter of the flow: at fixed values of , the friction factor is also fixed.
In the Kármán–Prandtl resistance equation, can be expressed in closed form as an analytic function of through the use of the Lambert function:
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 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
When the pipe surface's roughness height 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 In translation, NACA TM 1292. The data are available in digital form.
where the roughness height is scaled to the pipe diameter .
It is illustrative to plot the roughness function :
Figure 3 shows versus for the rough pipe data of Nikuradse, Shockling, and Langelandsvik.
In this view, the data at different roughness ratio fall together when plotted against , demonstrating scaling in the variable . The following features are present: When , then 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 , the data lie on the line ; flow is in the smooth pipe regime. When , the data asymptotically approach a horizontal line; they are independent of , , and . The intermediate range of constitutes a transition from one behavior to the other. The data depart from the line very slowly, reach a maximum near , 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 that ensures proper behavior for (the transition from the smooth pipe regime to the rough pipe regime):
and
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 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
and
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 along with one further parameter, 26, to govern the transition from smooth to rough flow.