{"history":[],"article":{"id":"ff741ada-f762-47d2-b731-d1a9f1bda4ed","title":"Darcy–Weisbach equation","slug":"darcy-weisbach-equation","content":":::summary\n:::\n\n# \nIn [fluid dynamics](https://bliptext.com/articles/fluid-dynamics), the **Darcy–Weisbach equation** is an [empirical](https://bliptext.com/articles/empirical-research) equation that relates the [head loss](https://bliptext.com/articles/head-loss), or [pressure](https://bliptext.com/articles/pressure) loss, due to [friction](https://bliptext.com/articles/friction) along a given length of pipe to the average velocity of the fluid flow for an incompressible fluid.\nThe equation is named after [Henry Darcy](https://bliptext.com/articles/henry-darcy) and [Julius Weisbach](https://bliptext.com/articles/julius-weisbach).\nCurrently, there is no formula more accurate or universally applicable than the Darcy-Weisbach supplemented by the [Moody diagram](https://bliptext.com/articles/moody-diagram) or [Colebrook equation](https://bliptext.com/articles/colebrook-equation).\nThe Darcy–Weisbach equation contains a [dimensionless](https://bliptext.com/articles/dimension-analysis) friction factor, known as the **Darcy friction factor**.\nThis is also variously called the Darcy–Weisbach friction factor, friction factor, resistance coefficient, or flow coefficient.\n\n# Historical background\nThe Darcy-Weisbach equation, combined with the [Moody chart](https://bliptext.com/articles/moody-chart) for calculating head losses in pipes, is traditionally attributed to [Henry Darcy](https://bliptext.com/articles/henry-darcy), [Julius Weisbach](https://bliptext.com/articles/julius-weisbach), and [Lewis Ferry Moody](https://bliptext.com/articles/lewis-ferry-moody).\nHowever, the development of these formulas and charts also involved other scientists and engineers over its historical development.\nGenerally, the [Bernoulli's equation](https://bliptext.com/articles/bernoulli-s-equation) would provide the head losses but in terms of quantities not known a priori, such as pressure.\nTherefore, empirical relationships were sought to correlate the head loss with quantities like pipe diameter and fluid velocity.\nJulius 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.\n[Antoine Chézy](https://bliptext.com/articles/antoine-ch-zy) (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](https://bliptext.com/articles/hydraulic-radius).\nHowever, Chézy's formula was lost until 1800, when [Gaspard de Prony](https://bliptext.com/articles/gaspard-de-prony) (a former student of his) published an account describing his results.\nIt is likely that Weisbach was aware of Chézy's formula through Prony's publications.\nWeisbach's formula was proposed in 1845 in the form we still use today:\nwhere:\nHowever, the friction factor f was expressed by Weisbach through the following empirical formula:\nwith \\alpha and \\beta depending on the diameter and the type of pipe wall.\nWeisbach's work was published in the [United States of America](https://bliptext.com/articles/united-states-of-america) in 1848 and soon became well known there.\nIn contrast, it did not initially gain much traction in [France](https://bliptext.com/articles/france), where [Prony equation](https://bliptext.com/articles/prony-equation), which had a polynomial form in terms of velocity (often approximated by the square of the velocity), continued to be used.\nBeyond the historical developments, Weisbach's formula had the objective merit of adhering to [dimensional analysis](https://bliptext.com/articles/dimensional-analysis), resulting in a dimensionless friction factor f. The complexity of f, dependent on the mechanics of the [boundary layer](https://bliptext.com/articles/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.\nIt 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](https://bliptext.com/articles/reynolds-number) (and thus the velocity) only in the case of rough pipes in a turbulent flow regime (Prandtl-von Kármán equation).\n\n# Pressure-loss equation\nIn a cylindrical pipe of uniform diameter D, flowing full, the pressure loss due to viscous effects\nΔp\nis proportional to length L and can be characterized by the Darcy–Weisbach equation:\nwhere the pressure loss per unit length\nΔp⁄L\n(SI units: [Pa](https://bliptext.co[m](https://bliptext.com/articles/metre)/articles/pascal-unit)/[m](https://bliptext.com/articles/metre)) is a function of:\nD\nDH = 4A/P\nfor a pipe of cross-sectional area A and perimeter P) (m); For [laminar flow](https://bliptext.com/articles/laminar-flow) in a circular pipe of diameter\nD_c\n, the friction factor is inversely proportional to the [Reynolds number](https://bliptext.com/articles/reynolds-number) alone (\n) 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\nμ\nis the [dynamic viscosity](https://bliptext.com/articles/dynamic-viscosity) of the [fluid](https://bliptext.com/articles/fluid) (Pa·s = N·s/m2 = kg/(m·s));\nQ\nis the [volumetric flow rate](https://bliptext.com/articles/volumetric-flow-rate), used here to measure flow instead of mean velocity according to\n(m3/s).\nNote that this laminar form of Darcy–Weisbach is equivalent to the [Hagen–Poiseuille equation](https://bliptext.com/articles/hagen-poiseuille-equation), which is analytically derived from the [Navier–Stokes equations](https://bliptext.com/articles/navier-stokes-equations).\n\n# Head-loss formula\nThe [head loss](https://bliptext.com/articles/hydraulic-head)\nΔh\n(or\nhf\n) expresses the [pressure](https://bliptext.com/articles/pressure) loss due to friction in terms of the equivalent height of a column of the working fluid, so the [pressure](https://bliptext.com/articles/pressure) drop is\n** where: **\nΔh\n''' = The head loss due to pipe friction over the given length of pipe (SI units: m);\nIt is useful to present head loss per length of pipe (dimensionless):\nwhere L is the pipe length (m).\nTherefore, the Darcy–Weisbach equation can also be written in terms of head loss:\n\n## In terms of volumetric flow\nThe relationship between mean flow velocity\n\nand volumetric flow rate Q is\n**where:**\nIn a full-flowing, circular pipe of diameter D_c,\nThen the Darcy–Weisbach equation in terms of Q is\n\n# Shear-stress form\nThe mean [wall shear stress](https://bliptext.com/articles/wall-shear-stress) τ in a pipe or [open channel](https://bliptext.com/articles/open-channel) is expressed in terms of the Darcy–Weisbach friction factor as\nThe wall [shear stress](https://bliptext.com/articles/shear-stress) has the [SI unit](https://bliptext.com/articles/si-unit) of [pascals](https://bliptext.com/articles/pascal-unit) (Pa).\n\n# Darcy friction factor\n[[File:Darcy Friction factor for Re between 10 and 10E8 for values of relative roughness.svg|thumb|**Figure 1.** The [[Darcy friction factor]] versus [Reynolds number](https://bliptext.com/articles/reynolds-number) for\n10 < Re < 108\nfor smooth pipe and a range of values of relative roughness ε⁄D.\nData are from Nikuradse (1932, 1933), Colebrook (1939), and McKeon (2004).]]\nThe friction factor\nfD\nis 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 [nu]), and the velocity of the fluid flow\n⟨v⟩\n. It has been measured to [high accuracy within certain flow regimes](https://bliptext.com/articles/friction-loss) and may be evaluated by the use of various empirical relations, or it may be read from published charts.\nThese charts are often referred to as [Moody diagrams](https://bliptext.com/articles/moody-diagrams), after [L. F. Moody](https://bliptext.com/articles/lewis-ferry-moody), and hence the factor itself is sometimes erroneously called the Moody friction factor.\nIt is also sometimes called the [Blasius](https://bliptext.com/articles/paul-richard-heinrich-blasius) friction factor, after the approximate formula he proposed.\nFigure 1 shows the value of\nfD\nas measured by experimenters for many different fluids, over a wide range of Reynolds numbers, and for pipes of various roughness heights.\nThere are three broad regimes of fluid flow encountered in these data: laminar, critical, and turbulent.\n\n## Laminar regime\nFor [laminar (smooth) flows](https://bliptext.com/articles/laminar-flow), it is a consequence of [Poiseuille's law](https://bliptext.com/articles/hagen-poiseuille-equation) (which stems from an exact classical solution for the fluid flow) that\nwhere\nRe\nis the [Reynolds number](https://bliptext.com/articles/reynolds-number)\nand where μ is the viscosity of the fluid and\nis known as the [kinematic viscosity](https://bliptext.com/articles/kinematic-viscosity).\nIn this expression for Reynolds number, the characteristic length D is taken to be the [hydraulic diameter](https://bliptext.com/articles/hydraulic-diameter) of the pipe, which, for a cylindrical pipe flowing full, equals the inside diameter.\nIn Figures 1 and 2 of friction factor versus Reynolds number, the regime\nRe < 2000\ndemonstrates laminar flow; the friction factor is well represented by the above equation.\nIn 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.\nIn 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.\nNote that the friction loss is insensitive to the pipe roughness height ε: the flow velocity in the neighborhood of the pipe wall is zero.\n\n## Critical regime\nFor Reynolds numbers in the range\n2000 < Re < 4000\n, the flow is unsteady (varies grossly with time) and varies from one section of the pipe to another (is not \"fully developed\").\nThe flow involves the incipient formation of vortices; it is not well understood.\n\n## Turbulent regime\n[[File:Darcy Friction factor for Re between 10E3 and 10E8 for values of relative roughness.svg|thumb|**Figure 2.** The [[Darcy friction factor]] versus Reynolds number for\n1000 < Re < 108\nfor smooth pipe and a range of values of relative roughness ε⁄D.\nData are from Nikuradse (1932, 1933), Colebrook (1939), and McKeon (2004).]]\nFor 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.\nOver a domain of many orders of magnitude of\nRe\n(\n4000 < Re < 108\n), the friction factor varies less than one order of magnitude (\n0.006 < fD < 0.06\n). 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.\n\n### Smooth-pipe regime\nWhen 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\nThe numbers 1.930 and 0.537 are phenomenological; these specific values provide a fairly good fit to the data.\nThe product\nRe√fD\n(called the \"friction Reynolds number\") can be considered, like the Reynolds number, to be a (dimensionless) parameter of the flow: at fixed values of\nRe√fD\n, the friction factor is also fixed.\nIn the Kármán–Prandtl resistance equation,\nfD\ncan be expressed in closed form as an analytic function of\nRe\nthrough the use of the [Lambert W function](https://bliptext.com/articles/lambert-w-function):\nIn this flow regime, many small vortices are responsible for the transfer of momentum between the bulk of the fluid to the pipe wall.\nAs the friction Reynolds number\nRe√fD\nincreases, the profile of the fluid velocity approaches the wall asymptotically, thereby transferring more momentum to the pipe wall, as modeled in [Blasius boundary layer](https://bliptext.com/articles/blasius-boundary-layer) theory.\n\n### Rough-pipe regime\nWhen the pipe surface's roughness height 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).\nIn this regime, the resistance to flow varies according to the square of the mean flow velocity and is insensitive to Reynolds number.\nHere, it is useful to employ yet another dimensionless parameter of the flow, the roughness Reynolds number\nwhere the roughness height ε is scaled to the pipe diameter D.\n[[File:Figure 3 (B vs R*).svg|thumb|alt=Roughness function B vs. friction Reynolds number\nR∗\nR∗\n. The data fall on a single trajectory when plotted in this way.\nThe regime\nR∗ < 1\nis effectively that of smooth pipe flow.\nFor large\nR∗\n, the roughness function B approaches a constant value.\nPhenomenological functions attempting to fit these data, including the Afzal and Colebrook–White are shown.]]\nIt is illustrative to plot the roughness function B:\nFigure 3 shows B versus\nR∗\nfor the rough pipe data of Nikuradse, Shockling, and Langelandsvik.\nIn this view, the data at different roughness ratio ε⁄D fall together when plotted against\nR∗\n, demonstrating scaling in the variable\nR∗\n. The following features are present:\n, then\nR∗\nis identically zero: flow is always in the smooth pipe regime.\nThe data for these points lie to the left extreme of the abscissa and are not within the frame of the graph.\nR∗ < 5\n, the data lie on the line\nB(R∗) = R∗\nR∗ > 100\n, the data asymptotically approach a horizontal line; they are independent of\nRe\n,\nfD\n, and ε⁄D.\n5 < R∗ < 100\nconstitutes a transition from one behavior to the other.\nThe data depart from the line\nB(R∗) = R∗\nvery slowly, reach a maximum near\nR∗ = 10\n, then fall to a constant value.\nAfzal's fit to these data in the transition from smooth pipe flow to rough pipe flow employs an exponential expression in\nR∗\nthat ensures proper behavior for\n1 < R∗ < 50\n(the transition from the smooth pipe regime to the rough pipe regime):\nand\nThis 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\nR∗ → ∞\nalong with one further parameter, 11, to govern the transition from smooth to rough flow.\nIt is exhibited in Figure 3.\nThe friction factor for another analogous roughness becomes\nand\nThis 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\nR∗ → ∞\nalong with one further parameter, 26, to govern the transition from smooth to rough flow.\nThe Colebrook–White relation fits the friction factor with a function of the form\nThis relation has the correct behavior at extreme values of\nR∗\n, as shown by the labeled curve in Figure 3: when\nR∗\nis small, it is consistent with smooth pipe flow, when large, it is consistent with rough pipe flow.\nHowever its performance in the transitional domain overestimates the friction factor by a substantial margin.\nColebrook acknowledges the discrepancy with Nikuradze's data but argues that his relation is consistent with the measurements on commercial pipes.\nIndeed, 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.\n\n## Calculating the friction factor from its parametrization\nFor [turbulent flow](https://bliptext.com/articles/turbulent-flow), methods for finding the friction factor\nfD\ninclude using a diagram, such as the [Moody chart](https://bliptext.com/articles/moody-chart), or solving equations such as the [Colebrook–White equation](https://bliptext.com/articles/colebrook-white-equation) (upon which the [Moody chart](https://bliptext.com/articles/moody-chart) is based), or the [Swamee–Jain equation](https://bliptext.com/articles/swamee-jain-equation).\nWhile the Colebrook–White relation is, in the general case, an iterative method, the Swamee–Jain equation allows\nfD\nto be found directly for full flow in a circular pipe.\n\n### Direct calculation when friction loss S is known\nIn typical engineering applications, there will be a set of given or known quantities.\nThe 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.\nIf as well the head loss per unit length S is a known quantity, then the friction factor\nfD\ncan be calculated directly from the chosen fitting function.\nSolving the Darcy–Weisbach equation for\n√fD\n,\nwe can now express\nRe√fD\nExpressing the roughness Reynolds number\nR∗\n,\nwe have the two parameters needed to substitute into the Colebrook–White relation, or any other function, for the friction factor\nfD\n, the flow velocity\n⟨v⟩\n, and the volumetric flow rate Q.\n\n## Confusion with the Fanning friction factor\nThe Darcy–Weisbach friction factor\nfD\nis 4 times larger than the [Fanning friction factor](https://bliptext.com/articles/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.\nOf the two, the Darcy–Weisbach factor\nfD\nis 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.\nNote that\nMost charts or tables indicate the type of friction factor, or at least provide the formula for the friction factor with laminar flow.\nIf the formula for laminar flow is\n, it is the Fanning factor f, and if the formula for laminar flow is\n, it is the Darcy–Weisbach factor\nfD\n.\nWhich friction factor is plotted in a Moody diagram may be determined by inspection if the publisher did not include the formula described above:\n.\n.\nThe procedure above is similar for any available Reynolds number that is an integer power of ten.\nIt is not necessary to remember the value 1000 for this procedure—only that an integer power of ten is of interest for this purpose.\n\n# History\nHistorically this equation arose as a variant on the [Prony equation](https://bliptext.com/articles/prony-equation); this variant was developed by [Henry Darcy](https://bliptext.com/articles/henry-darcy) of France, and further refined into the form used today by [Julius Weisbach](https://bliptext.com/articles/julius-weisbach) of [Saxony](https://bliptext.com/articles/saxony) in 1845.\nInitially, data on the variation of\nfD\nwith velocity was lacking, so the Darcy–Weisbach equation was outperformed at first by the empirical Prony equation in many cases.\nIn later years it was eschewed in many special-case situations in favor of a variety of [empirical equations](https://bliptext.com/articles/empirical-equation) valid only for certain flow regimes, notably the [Hazen–Williams equation](https://bliptext.com/articles/hazen-williams-equation) or the [Manning equation](https://bliptext.com/articles/manning-equation), most of which were significantly easier to use in calculations.\nHowever, since the advent of the [calculator](https://bliptext.com/articles/calculator), ease of calculation is no longer a major issue, and so the Darcy–Weisbach equation's generality has made it the preferred one.\n\n# Derivation by dimensional analysis\nAway from the ends of the pipe, the characteristics of the flow are independent of the position along the pipe.\nThe key quantities are then the pressure drop along the pipe per unit length,\nΔp⁄L\n, and the volumetric flow rate.\nThe flow rate can be converted to a mean flow velocity V by dividing by the [wetted area](https://bliptext.com/articles/hydraulic-diameter) of the flow (which equals the [cross-sectional](https://bliptext.com/articles/cross-section-geometry) [area](https://bliptext.com/articles/area) of the pipe if the pipe is full of fluid).\nPressure has dimensions of energy per unit volume, therefore the pressure drop between two points must be proportional to the [dynamic pressure](https://bliptext.com/articles/dynamic-pressure) q. 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.\nTo 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.\nTherefore,\nThe proportionality coefficient is the dimensionless \"[Darcy friction factor](https://bliptext.com/articles/darcy-friction-factor-formulae)\" or \"flow coefficient\".\nThis 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](https://bliptext.com/articles/roughness-height) to the [hydraulic diameter](https://bliptext.com/articles/hydraulic-diameter)).\nNote that the dynamic pressure is not the kinetic energy of the fluid per unit volume, for the following reasons.\nEven in the case of [laminar flow](https://bliptext.com/articles/laminar-flow), where all the [flow lines](https://bliptext.com/articles/streamlines-streaklines-and-pathlines) 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.\nThe average kinetic energy then involves the [root mean-square velocity](https://bliptext.com/articles/root-mean-square-speed), which always exceeds the mean velocity.\nIn the case of [turbulent flow](https://bliptext.com/articles/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.\n\n# Practical application\nIn a [hydraulic engineering](https://bliptext.com/articles/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.\nThe 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.\nMeanwhile, the head loss is decreased by a factor of 32 (about a 97% reduction).\nThus the energy consumed in moving a given volumetric flow of the fluid is cut down dramatically for a modest increase in capital cost.\n\n# Advantages\nThe Darcy-Weisbach's accuracy and universal applicability makes it the ideal formula for flow in pipes.\nThe advantages of the equation are as follows:","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-25T18:45:19.686Z","updated_at":"2025-01-25T18:45:19.686Z","search_vector":"'darci':1 'equat':3 'weisbach':2","search_content":" \n \n\n# \nIn [fluid dynamics](https://bliptext.com/articles/fluid-dynamics), the **Darcy–Weisbach equation** is an [empirical](https://bliptext.com/articles/empirical-research) equation that relates the [head loss](https://bliptext.com/articles/head-loss), or [pressure](https://bliptext.com/articles/pressure) loss, due to [friction](https://bliptext.com/articles/friction) along a given length of pipe to the average velocity of the fluid flow for an incompressible fluid.\nThe equation is named after [Henry Darcy](https://bliptext.com/articles/henry-darcy) and [Julius Weisbach](https://bliptext.com/articles/julius-weisbach).\nCurrently, there is no formula more accurate or universally applicable than the Darcy-Weisbach supplemented by the [Moody diagram](https://bliptext.com/articles/moody-diagram) or [Colebrook equation](https://bliptext.com/articles/colebrook-equation).\nThe Darcy–Weisbach equation contains a [dimensionless](https://bliptext.com/articles/dimension-analysis) friction factor, known as the **Darcy friction factor**.\nThis is also variously called the Darcy–Weisbach friction factor, friction factor, resistance coefficient, or flow coefficient.\n\n# Historical background\nThe Darcy-Weisbach equation, combined with the [Moody chart](https://bliptext.com/articles/moody-chart) for calculating head losses in pipes, is traditionally attributed to [Henry Darcy](https://bliptext.com/articles/henry-darcy), [Julius Weisbach](https://bliptext.com/articles/julius-weisbach), and [Lewis Ferry Moody](https://bliptext.com/articles/lewis-ferry-moody).\nHowever, the development of these formulas and charts also involved other scientists and engineers over its historical development.\nGenerally, the [Bernoulli's equation](https://bliptext.com/articles/bernoulli-s-equation) would provide the head losses but in terms of quantities not known a priori, such as pressure.\nTherefore, empirical relationships were sought to correlate the head loss with quantities like pipe diameter and fluid velocity.\nJulius 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.\n[Antoine Chézy](https://bliptext.com/articles/antoine-ch-zy) (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](https://bliptext.com/articles/hydraulic-radius).\nHowever, Chézy's formula was lost until 1800, when [Gaspard de Prony](https://bliptext.com/articles/gaspard-de-prony) (a former student of his) published an account describing his results.\nIt is likely that Weisbach was aware of Chézy's formula through Prony's publications.\nWeisbach's formula was proposed in 1845 in the form we still use today:\nwhere:\nHowever, the friction factor f was expressed by Weisbach through the following empirical formula:\nwith \\alpha and \\beta depending on the diameter and the type of pipe wall.\nWeisbach's work was published in the [United States of America](https://bliptext.com/articles/united-states-of-america) in 1848 and soon became well known there.\nIn contrast, it did not initially gain much traction in [France](https://bliptext.com/articles/france), where [Prony equation](https://bliptext.com/articles/prony-equation), which had a polynomial form in terms of velocity (often approximated by the square of the velocity), continued to be used.\nBeyond the historical developments, Weisbach's formula had the objective merit of adhering to [dimensional analysis](https://bliptext.com/articles/dimensional-analysis), resulting in a dimensionless friction factor f. The complexity of f, dependent on the mechanics of the [boundary layer](https://bliptext.com/articles/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.\nIt 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](https://bliptext.com/articles/reynolds-number) (and thus the velocity) only in the case of rough pipes in a turbulent flow regime (Prandtl-von Kármán equation).\n\n# Pressure-loss equation\nIn a cylindrical pipe of uniform diameter D, flowing full, the pressure loss due to viscous effects\nΔp\nis proportional to length L and can be characterized by the Darcy–Weisbach equation:\nwhere the pressure loss per unit length\nΔp⁄L\n(SI units: [Pa](https://bliptext.co[m](https://bliptext.com/articles/metre)/articles/pascal-unit)/[m](https://bliptext.com/articles/metre)) is a function of:\nD\nDH = 4A/P\nfor a pipe of cross-sectional area A and perimeter P) (m); For [laminar flow](https://bliptext.com/articles/laminar-flow) in a circular pipe of diameter\nD_c\n, the friction factor is inversely proportional to the [Reynolds number](https://bliptext.com/articles/reynolds-number) alone (\n) 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\nμ\nis the [dynamic viscosity](https://bliptext.com/articles/dynamic-viscosity) of the [fluid](https://bliptext.com/articles/fluid) (Pa·s = N·s/m2 = kg/(m·s));\nQ\nis the [volumetric flow rate](https://bliptext.com/articles/volumetric-flow-rate), used here to measure flow instead of mean velocity according to\n(m3/s).\nNote that this laminar form of Darcy–Weisbach is equivalent to the [Hagen–Poiseuille equation](https://bliptext.com/articles/hagen-poiseuille-equation), which is analytically derived from the [Navier–Stokes equations](https://bliptext.com/articles/navier-stokes-equations).\n\n# Head-loss formula\nThe [head loss](https://bliptext.com/articles/hydraulic-head)\nΔh\n(or\nhf\n) expresses the [pressure](https://bliptext.com/articles/pressure) loss due to friction in terms of the equivalent height of a column of the working fluid, so the [pressure](https://bliptext.com/articles/pressure) drop is\n** where: **\nΔh\n''' = The head loss due to pipe friction over the given length of pipe (SI units: m);\nIt is useful to present head loss per length of pipe (dimensionless):\nwhere L is the pipe length (m).\nTherefore, the Darcy–Weisbach equation can also be written in terms of head loss:\n\n## In terms of volumetric flow\nThe relationship between mean flow velocity\n \nand volumetric flow rate Q is\n**where:**\nIn a full-flowing, circular pipe of diameter D_c,\nThen the Darcy–Weisbach equation in terms of Q is\n\n# Shear-stress form\nThe mean [wall shear stress](https://bliptext.com/articles/wall-shear-stress) τ in a pipe or [open channel](https://bliptext.com/articles/open-channel) is expressed in terms of the Darcy–Weisbach friction factor as\nThe wall [shear stress](https://bliptext.com/articles/shear-stress) has the [SI unit](https://bliptext.com/articles/si-unit) of [pascals](https://bliptext.com/articles/pascal-unit) (Pa).\n\n# Darcy friction factor\n[[File:Darcy Friction factor for Re between 10 and 10E8 for values of relative roughness.svg|thumb|**Figure 1.** The [[Darcy friction factor]] versus [Reynolds number](https://bliptext.com/articles/reynolds-number) for\n10 100\n, the data asymptotically approach a horizontal line; they are independent of\nRe\n,\nfD\n, and ε⁄D.\n5 < R∗ < 100\nconstitutes a transition from one behavior to the other.\nThe data depart from the line\nB(R∗) = R∗\nvery slowly, reach a maximum near\nR∗ = 10\n, then fall to a constant value.\nAfzal's fit to these data in the transition from smooth pipe flow to rough pipe flow employs an exponential expression in\nR∗\nthat ensures proper behavior for\n1 < R∗ < 50\n(the transition from the smooth pipe regime to the rough pipe regime):\nand\nThis 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\nR∗ → ∞\nalong with one further parameter, 11, to govern the transition from smooth to rough flow.\nIt is exhibited in Figure 3.\nThe friction factor for another analogous roughness becomes\nand\nThis 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\nR∗ → ∞\nalong with one further parameter, 26, to govern the transition from smooth to rough flow.\nThe Colebrook–White relation fits the friction factor with a function of the form\nThis relation has the correct behavior at extreme values of\nR∗\n, as shown by the labeled curve in Figure 3: when\nR∗\nis small, it is consistent with smooth pipe flow, when large, it is consistent with rough pipe flow.\nHowever its performance in the transitional domain overestimates the friction factor by a substantial margin.\nColebrook acknowledges the discrepancy with Nikuradze's data but argues that his relation is consistent with the measurements on commercial pipes.\nIndeed, 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.\n\n## Calculating the friction factor from its parametrization\nFor [turbulent flow](https://bliptext.com/articles/turbulent-flow), methods for finding the friction factor\nfD\ninclude using a diagram, such as the [Moody chart](https://bliptext.com/articles/moody-chart), or solving equations such as the [Colebrook–White equation](https://bliptext.com/articles/colebrook-white-equation) (upon which the [Moody chart](https://bliptext.com/articles/moody-chart) is based), or the [Swamee–Jain equation](https://bliptext.com/articles/swamee-jain-equation).\nWhile the Colebrook–White relation is, in the general case, an iterative method, the Swamee–Jain equation allows\nfD\nto be found directly for full flow in a circular pipe.\n\n### Direct calculation when friction loss S is known\nIn typical engineering applications, there will be a set of given or known quantities.\nThe 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.\nIf as well the head loss per unit length S is a known quantity, then the friction factor\nfD\ncan be calculated directly from the chosen fitting function.\nSolving the Darcy–Weisbach equation for\n√fD\n,\nwe can now express\nRe√fD\nExpressing the roughness Reynolds number\nR∗\n,\nwe have the two parameters needed to substitute into the Colebrook–White relation, or any other function, for the friction factor\nfD\n, the flow velocity\n⟨v⟩\n, and the volumetric flow rate Q.\n\n## Confusion with the Fanning friction factor\nThe Darcy–Weisbach friction factor\nfD\nis 4 times larger than the [Fanning friction factor](https://bliptext.com/articles/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.\nOf the two, the Darcy–Weisbach factor\nfD\nis 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.\nNote that\nMost charts or tables indicate the type of friction factor, or at least provide the formula for the friction factor with laminar flow.\nIf the formula for laminar flow is\n, it is the Fanning factor f, and if the formula for laminar flow is\n, it is the Darcy–Weisbach factor\nfD\n.\nWhich friction factor is plotted in a Moody diagram may be determined by inspection if the publisher did not include the formula described above:\n.\n.\nThe procedure above is similar for any available Reynolds number that is an integer power of ten.\nIt is not necessary to remember the value 1000 for this procedure—only that an integer power of ten is of interest for this purpose.\n\n# History\nHistorically this equation arose as a variant on the [Prony equation](https://bliptext.com/articles/prony-equation); this variant was developed by [Henry Darcy](https://bliptext.com/articles/henry-darcy) of France, and further refined into the form used today by [Julius Weisbach](https://bliptext.com/articles/julius-weisbach) of [Saxony](https://bliptext.com/articles/saxony) in 1845.\nInitially, data on the variation of\nfD\nwith velocity was lacking, so the Darcy–Weisbach equation was outperformed at first by the empirical Prony equation in many cases.\nIn later years it was eschewed in many special-case situations in favor of a variety of [empirical equations](https://bliptext.com/articles/empirical-equation) valid only for certain flow regimes, notably the [Hazen–Williams equation](https://bliptext.com/articles/hazen-williams-equation) or the [Manning equation](https://bliptext.com/articles/manning-equation), most of which were significantly easier to use in calculations.\nHowever, since the advent of the [calculator](https://bliptext.com/articles/calculator), ease of calculation is no longer a major issue, and so the Darcy–Weisbach equation's generality has made it the preferred one.\n\n# Derivation by dimensional analysis\nAway from the ends of the pipe, the characteristics of the flow are independent of the position along the pipe.\nThe key quantities are then the pressure drop along the pipe per unit length,\nΔp⁄L\n, and the volumetric flow rate.\nThe flow rate can be converted to a mean flow velocity V by dividing by the [wetted [area](https://bliptext.com/articles/area)](https://bliptext.com/articles/hydraulic-diameter) of the flow (which equals the [cross-sectional](https://bliptext.com/articles/cross-section-geometry) [area](https://bliptext.com/articles/area) of the pipe if the pipe is full of fluid).\nPressure has dimensions of energy per unit volume, therefore the pressure drop between two points must be proportional to the [dynamic pressure](https://bliptext.com/articles/dynamic-pressure) q. 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.\nTo 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.\nTherefore,\nThe proportionality coefficient is the dimensionless \"[Darcy friction factor](https://bliptext.com/articles/darcy-friction-factor-formulae)\" or \"flow coefficient\".\nThis 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](https://bliptext.com/articles/roughness-height) to the [hydraulic diameter](https://bliptext.com/articles/hydraulic-diameter)).\nNote that the dynamic pressure is not the kinetic energy of the fluid per unit volume, for the following reasons.\nEven in the case of [laminar flow](https://bliptext.com/articles/laminar-flow), where all the [flow lines](https://bliptext.com/articles/streamlines-streaklines-and-pathlines) 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.\nThe average kinetic energy then involves the [root mean-square velocity](https://bliptext.com/articles/root-mean-square-speed), which always exceeds the mean velocity.\nIn the case of [turbulent flow](https://bliptext.com/articles/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.\n\n# Practical application\nIn a [hydraulic engineering](https://bliptext.com/articles/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.\nThe 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.\nMeanwhile, the head loss is decreased by a factor of 32 (about a 97% reduction).\nThus the energy consumed in moving a given volumetric flow of the fluid is cut down dramatically for a modest increase in capital cost.\n\n# Advantages\nThe Darcy-Weisbach's accuracy and universal applicability makes it the ideal formula for flow in pipes.\nThe advantages of the equation are as follows:","revisionCount":0},"hasMore":false,"nextCursor":null}