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fluid dynamics In physics, physical chemistry and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids – liquids and gases. It has several subdisciplines, including (the study of air and other gases in motion ...
, the Darcy–Weisbach equation is an
empirical Empirical evidence is evidence obtained through sense experience or experimental procedure. It is of central importance to the sciences and plays a role in various other fields, like epistemology and law. There is no general agreement on how t ...
equation that relates the
head loss Hydraulic head or piezometric head is a measurement related to liquid pressure (normalized by specific weight) and the liquid elevation above a vertical datum., 410 pages. See pp. 43–44., 650 pages. See p. 22, eq.3.2a. It is usually measur ...
, or
pressure Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and eve ...
loss, due to
friction Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. Types of friction include dry, fluid, lubricated, skin, and internal -- an incomplete list. The study of t ...
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 In fluid dynamics, the Darcy friction factor formulae are equations that allow the calculation of the ''Darcy friction factor'', a dimensionless quantity used in the Darcy–Weisbach equation, for the description of friction losses in pipe flow as ...
. The Darcy–Weisbach equation contains a
dimensionless Dimensionless quantities, or quantities of dimension one, are quantities implicitly defined in a manner that prevents their aggregation into units of measurement. ISBN 978-92-822-2272-0. Typically expressed as ratios that align with another sy ...
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 In engineering, the Moody chart or Moody diagram (also Stanton diagram) is a graph in non-dimensional form that relates the Darcy–Weisbach friction factor ''f'D'', Reynolds number Re, and surface roughness for fully developed flow in a circu ...
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 Bernoulli's principle is a key concept in fluid dynamics that relates pressure, speed and height. For example, for a fluid flowing horizontally Bernoulli's principle states that an increase in the speed occurs simultaneously with a decrease i ...
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 The Manning formula or Manning's equation is an empirical formula estimating the average velocity of a liquid in an open channel flow (flowing in a conduit that does not completely enclose the liquid). However, this equation is also used for calc ...
. However, Chézy's formula was lost until 1800, when
Gaspard de Prony Baron Gaspard Clair François Marie Riche de Prony (22 July 1755 – 29 July 1839) was a French mathematician and engineer, who worked on hydraulics. He was born at Chamelet, Beaujolais, France and died in Asnières-sur-Seine, France. Educat ...
(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: : \Delta H = f \cdot where: * \Delta H : head loss. * L : length of the pipe. * D :
diameter In geometry, a diameter of a circle is any straight line segment that passes through the centre of the circle and whose endpoints lie on the circle. It can also be defined as the longest Chord (geometry), chord of the circle. Both definitions a ...
of the pipe. * V :
velocity Velocity is a measurement of speed in a certain direction of motion. It is a fundamental concept in kinematics, the branch of classical mechanics that describes the motion of physical objects. Velocity is a vector (geometry), vector Physical q ...
of the fluid. * g : acceleration due to gravity. However, the friction factor f was expressed by Weisbach through the following empirical formula: : f = \alpha + with \alpha and \beta depending on the diameter and the type of pipe wall. Weisbach's work was published in the
United States of America The United States of America (USA), also known as the United States (U.S.) or America, is a country primarily located in North America. It is a federal republic of 50 states and a federal capital district, Washington, D.C. The 48 contiguo ...
in 1848 and soon became well known there. In contrast, it did not initially gain much traction in
France France, officially the French Republic, is a country located primarily in Western Europe. Overseas France, Its overseas regions and territories include French Guiana in South America, Saint Pierre and Miquelon in the Atlantic Ocean#North Atlan ...
, 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 In physics and fluid mechanics, a boundary layer is the thin layer of fluid in the immediate vicinity of a Boundary (thermodynamic), bounding surface formed by the fluid flowing along the surface. The fluid's interaction with the wall induces ...
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 In fluid dynamics, the Reynolds number () is a dimensionless quantity that helps predict fluid flow patterns in different situations by measuring the ratio between Inertia, inertial and viscous forces. At low Reynolds numbers, flows tend to ...
(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: :\frac =f_\mathrm \cdot \frac \cdot \frac, where the pressure loss per unit length (SI units: Pa/ m) is a function of: : \rho, the density of the fluid (kg/m3); : D_H, the
hydraulic diameter The hydraulic diameter, , is a commonly used term when handling flow in non-circular tubes and channels. Using this term, one can calculate many things in the same way as for a round tube. When the cross-section is uniform along the tube or channe ...
of the pipe (for a pipe of circular section, this equals ; otherwise for a pipe of cross-sectional area and perimeter ) (m); : \langle v \rangle, the mean
flow velocity In continuum mechanics the flow velocity in fluid dynamics, also macroscopic velocity in statistical mechanics, or drift velocity in electromagnetism, is a vector field used to mathematically describe the motion of a continuum. The length of the f ...
, experimentally measured as the
volumetric flow rate In physics and engineering, in particular fluid dynamics, the volumetric flow rate (also known as volume flow rate, or volume velocity) is the volume of fluid which passes per unit time; usually it is represented by the symbol (sometimes \do ...
per unit cross-sectional
wetted area In fluid dynamics, the wetted area is the surface area that interacts with the working fluid or gas. * In maritime use, the wetted area is the area of the watercraft's hull which is immersed in water. This has a direct relationship on the overal ...
(m/s); : f_\mathrm, the
Darcy friction factor Darcy, Darci or Darcey may refer to different people such as: Science * Darcy's law, which describes the flow of a fluid through porous material * Darcy (unit), a unit of permeability of fluids in porous material * Darcy friction factor in the ...
(also called flow coefficient ). For
laminar flow Laminar flow () is the property of fluid particles in fluid dynamics to follow smooth paths in layers, with each layer moving smoothly past the adjacent layers with little or no mixing. At low velocities, the fluid tends to flow without lateral m ...
in a circular pipe of diameter , the friction factor is inversely proportional to the
Reynolds number In fluid dynamics, the Reynolds number () is a dimensionless quantity that helps predict fluid flow patterns in different situations by measuring the ratio between Inertia, inertial and viscous forces. At low Reynolds numbers, flows tend to ...
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 :\frac = \frac \cdot \frac, where : is the
dynamic viscosity Viscosity is a measure of a fluid's rate-dependent resistance to a change in shape or to movement of its neighboring portions relative to one another. For liquids, it corresponds to the informal concept of ''thickness''; for example, syrup h ...
of the
fluid In physics, a fluid is a liquid, gas, or other material that may continuously motion, move and Deformation (physics), deform (''flow'') under an applied shear stress, or external force. They have zero shear modulus, or, in simpler terms, are M ...
(Pa·s = N·s/m2 = kg/(m·s)); : is the
volumetric flow rate In physics and engineering, in particular fluid dynamics, the volumetric flow rate (also known as volume flow rate, or volume velocity) is the volume of fluid which passes per unit time; usually it is represented by the symbol (sometimes \do ...
, 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 In fluid dynamics, the Hagen–Poiseuille equation, also known as the Hagen–Poiseuille law, Poiseuille law or Poiseuille equation, is a physical law that gives the pressure drop in an incompressible and Newtonian fluid in laminar flow flowing t ...
, which is analytically derived from the
Navier–Stokes equations The Navier–Stokes equations ( ) are partial differential equations which describe the motion of viscous fluid substances. They were named after French engineer and physicist Claude-Louis Navier and the Irish physicist and mathematician Georg ...
.


Head-loss formula

The
head loss Hydraulic head or piezometric head is a measurement related to liquid pressure (normalized by specific weight) and the liquid elevation above a vertical datum., 410 pages. See pp. 43–44., 650 pages. See p. 22, eq.3.2a. It is usually measur ...
(or ) expresses the pressure loss due to friction in terms of the equivalent height of a column of the working fluid, so the
pressure Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and eve ...
drop is :\Delta p = \rho g \, \Delta h, where: : = The head loss due to pipe friction over the given length of pipe (SI units: m); : = The local acceleration due to
gravity In physics, gravity (), also known as gravitation or a gravitational interaction, is a fundamental interaction, a mutual attraction between all massive particles. On Earth, gravity takes a slightly different meaning: the observed force b ...
(m/s2). It is useful to present head loss per length of pipe (dimensionless): :S = \frac = \frac \cdot \frac, where is the pipe length (''m''). Therefore, the Darcy–Weisbach equation can also be written in terms of head loss: :S = f_\text \cdot \frac \cdot \frac.


In terms of volumetric flow

The relationship between mean flow velocity and volumetric flow rate is :Q = A \cdot \langle v \rangle, where: : = The volumetric flow (m3/s), : = The cross-sectional wetted area (m2). In a full-flowing, circular pipe of diameter , : Q = \frac D_c^2 \langle v \rangle. Then the Darcy–Weisbach equation in terms of is :S = f_\text \cdot \frac \cdot \frac.


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 :\tau = \frac18 f_\text \rho ^2. The wall
shear stress Shear stress (often denoted by , Greek alphabet, Greek: tau) is the component of stress (physics), stress coplanar with a material cross section. It arises from the shear force, the component of force vector parallel to the material cross secti ...
has the
SI unit The International System of Units, internationally known by the abbreviation SI (from French ), is the modern form of the metric system and the world's most widely used system of units of measurement, system of measurement. It is the only system ...
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 u, 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 :f_ = \frac, where is the
Reynolds number In fluid dynamics, the Reynolds number () is a dimensionless quantity that helps predict fluid flow patterns in different situations by measuring the ratio between Inertia, inertial and viscous forces. At low Reynolds numbers, flows tend to ...
: \mathrm = \frac \rho \mu \langle v \rangle D = \frac \nu, and where is the viscosity of the fluid and :\nu = \frac is known as the
kinematic viscosity Viscosity is a measure of a fluid's rate-dependent drag (physics), resistance to a change in shape or to movement of its neighboring portions relative to one another. For liquids, it corresponds to the informal concept of ''thickness''; for e ...
. In this expression for Reynolds number, the characteristic length is taken to be the
hydraulic diameter The hydraulic diameter, , is a commonly used term when handling flow in non-circular tubes and channels. Using this term, one can calculate many things in the same way as for a round tube. When the cross-section is uniform along the tube or channe ...
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 : \frac = 1.930 \log\left(\mathrm\sqrt\right) - 0.537. 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: : \frac 1 = \frac W\left( 10^\frac \mathrm \right) = 0.838\ W(0.629\ \mathrm) 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 In physics and fluid mechanics, a Blasius boundary layer (named after Paul Richard Heinrich Blasius) describes the steady two-dimensional laminar boundary layer that forms on a semi-infinite plate which is held parallel to a constant unidirectional ...
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 i
digital form
: R_* = \frac 1 \left( \mathrm\sqrt \, \right) \frac \varepsilon D where the roughness height is scaled to the pipe diameter . It is illustrative to plot the roughness function : : B(R_*) = \frac 1 + \log\left( \frac \cdot \frac\right) 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): : \frac = -2 \log\left( \frac \left( 1 + 0.305 R_* \exp\frac \right) \right) , and : \frac = -1.930 \log\left( \frac \left( 1 + 0.34 R_* \exp\frac \right) \right) , 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 : \frac = - 2.0 \, \log_ \left ( \frac \left \ \right) , and : \frac = - 1.93 \, \log_ \left ( \frac \left \ \right) , 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. The Colebrook–White relation fits the friction factor with a function of the form :\frac = -2.00 \log\left( \frac \left(1 + \frac\right) \right). This relation has the correct behavior at extreme values of , as shown by the labeled curve in Figure 3: when 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 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.


Calculating the friction factor from its parametrization

For
turbulent flow In fluid dynamics, turbulence or turbulent flow is fluid motion characterized by Chaos theory, chaotic changes in pressure and flow velocity. It is in contrast to laminar flow, which occurs when a fluid flows in parallel layers with no disrupt ...
, methods for finding the friction factor include using a diagram, such as the
Moody chart In engineering, the Moody chart or Moody diagram (also Stanton diagram) is a graph in non-dimensional form that relates the Darcy–Weisbach friction factor ''f'D'', Reynolds number Re, and surface roughness for fully developed flow in a circu ...
, 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 to be found directly for full flow in a circular pipe. :\frac = 2 \log\left(\mathrm \sqrt\right) - 0.8 \quad \text \mathrm > 3000.


Direct calculation when friction loss ' is known

In typical engineering applications, there will be a set of given or known quantities. The acceleration of gravity and the kinematic viscosity of the fluid are known, as are the diameter of the pipe and its roughness height . If as well the head loss per unit length is a known quantity, then the friction factor can be calculated directly from the chosen fitting function. Solving the Darcy–Weisbach equation for , :\sqrt = \frac we can now express : :\mathrm\sqrt = \frac \sqrt \sqrt \sqrt Expressing the roughness Reynolds number , :\begin R_* &= \frac \varepsilon D \cdot \mathrm\sqrt \cdot \frac 1 \\ &= \frac12 \frac \varepsilon \sqrt \sqrt \end we have the two parameters needed to substitute into the Colebrook–White relation, or any other function, for the friction factor , the flow velocity , and the volumetric flow rate .


Confusion with the Fanning friction factor

The Darcy–Weisbach friction factor is 4 times larger than the
Fanning friction factor The Fanning friction factor (named after American engineer John T. Fanning) is a dimensionless number used as a local parameter in continuum mechanics calculations. It is defined as the ratio between the local shear stress and the local flow ki ...
, 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 is more commonly used by civil and mechanical engineers, and the Fanning factor 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_ \cdot \frac \cdot \frac = f \cdot \frac \cdot 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 , it is the Fanning factor , and if the formula for laminar flow is , it is the Darcy–Weisbach factor . 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: . #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: . 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 Saxony, officially the Free State of Saxony, is a landlocked state of Germany, bordering the states of Brandenburg, Saxony-Anhalt, Thuringia, and Bavaria, as well as the countries of Poland and the Czech Republic. Its capital is Dresden, and ...
in 1845. Initially, data on the variation of 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 The Manning formula or Manning's equation is an empirical formula estimating the average velocity of a liquid in an open channel flow (flowing in a conduit that does not completely enclose the liquid). However, this equation is also used for calcu ...
, most of which were significantly easier to use in calculations. However, since the advent of the
calculator An electronic calculator is typically a portable electronic device used to perform calculations, ranging from basic arithmetic to complex mathematics. The first solid-state electronic calculator was created in the early 1960s. Pocket-si ...
, ease of calculation is no longer a major issue, and so the Darcy–Weisbach equation's generality has made it the preferred one.


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, , and the volumetric flow rate. The flow rate can be converted to a mean flow velocity by dividing by the
wetted area In fluid dynamics, the wetted area is the surface area that interacts with the working fluid or gas. * In maritime use, the wetted area is the area of the watercraft's hull which is immersed in water. This has a direct relationship on the overal ...
of the flow (which equals the
cross-sectional In statistics and econometrics, cross-sectional data is a type of data collected by observing many subjects (such as individuals, firms, countries, or regions) at a single point or period of time. Analysis of cross-sectional data usually consists ...
area Area is the measure of a region's size on a surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an open surface or the boundary of a three-di ...
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 In fluid dynamics, dynamic pressure (denoted by or and sometimes called velocity pressure) is the quantity defined by:Clancy, L.J., ''Aerodynamics'', Section 3.5 :q = \frac\rho\, u^2 where (in SI units): * is the dynamic pressure in pascals ...
q. We also know that pressure must be proportional to the length of the pipe between the two points 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, , which is also constant along the pipe. Therefore, :\Delta p \propto \frac q = \frac \cdot \frac \cdot ^2 The proportionality coefficient is the dimensionless "
Darcy friction factor Darcy, Darci or Darcey may refer to different people such as: Science * Darcy's law, which describes the flow of a fluid through porous material * Darcy (unit), a unit of permeability of fluids in porous material * Darcy friction factor in the ...
" or "flow coefficient". This dimensionless coefficient will be a combination of geometric factors such as , the Reynolds number and (outside the laminar regime) the relative roughness of the pipe (the ratio of the roughness height to the
hydraulic diameter The hydraulic diameter, , is a commonly used term when handling flow in non-circular tubes and channels. Using this term, one can calculate many things in the same way as for a round tube. When the cross-section is uniform along the tube or channe ...
). 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
laminar flow Laminar flow () is the property of fluid particles in fluid dynamics to follow smooth paths in layers, with each layer moving smoothly past the adjacent layers with little or no mixing. At low velocities, the fluid tends to flow without lateral m ...
, 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 In fluid dynamics, turbulence or turbulent flow is fluid motion characterized by Chaos theory, chaotic changes in pressure and flow velocity. It is in contrast to laminar flow, which occurs when a fluid flows in parallel layers with no disrupt ...
, 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 Hydraulic engineering as a sub-discipline of civil engineering is concerned with the flow and conveyance of fluids, principally water and sewage. One feature of these systems is the extensive use of gravity as the motive force to cause the move ...
application, it is typical for the volumetric flow within a pipe (that is, its productivity) and the head loss per unit length (the concomitant power consumption) to be the critical important factors. The practical consequence is that, for a fixed volumetric flow rate , head loss ''decreases'' with the inverse fifth power of the pipe diameter, . 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: * 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.


See also

*
Bernoulli's principle Bernoulli's principle is a key concept in fluid dynamics that relates pressure, speed and height. For example, for a fluid flowing horizontally Bernoulli's principle states that an increase in the speed occurs simultaneously with a decrease i ...
*
Darcy friction factor formulae In fluid dynamics, the Darcy friction factor formulae are equations that allow the calculation of the ''Darcy friction factor'', a dimensionless quantity used in the Darcy–Weisbach equation, for the description of friction losses in pipe flow as ...
*
Euler number Leonhard Euler ( ; ; ; 15 April 170718 September 1783) was a Swiss polymath who was active as a mathematician, physicist, astronomer, logician, geographer, and engineer. He founded the studies of graph theory and topology and made influential ...
*
Friction loss Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. Types of friction include dry, fluid, lubricated, skin, and internal -- an incomplete list. The study of t ...
* Hazen–Williams equation *
Hagen–Poiseuille equation In fluid dynamics, the Hagen–Poiseuille equation, also known as the Hagen–Poiseuille law, Poiseuille law or Poiseuille equation, is a physical law that gives the pressure drop in an incompressible and Newtonian fluid in laminar flow flowing t ...
*
Water pipe Plumbing is any system that conveys fluids for a wide range of applications. Plumbing uses pipes, valves, plumbing fixtures, tanks, and other apparatuses to convey fluids. Heating and cooling (HVAC), waste removal, and potable water deliv ...


Notes


References

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


Further reading

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External links


The History of the Darcy–Weisbach Equation

Darcy–Weisbach equation calculator

Pipe pressure drop calculator
for single phase flows.
Pipe pressure drop calculator for two phase flows.

Open source pipe pressure drop calculator.

Web application with pressure drop calculations for pipes and ductsThermoTurb – A web application for thermal and turbulent flow analysis
{{DEFAULTSORT:Darcy-Weisbach Equation Dimensionless numbers of fluid mechanics Eponymous equations of physics Equations of fluid dynamics Piping