nikuradse weisbach 1.930 colebrook langle prandtl 10.1115 prony rangle laminar δp blasius parametrization poiseuille varepsilon f_ wetted roughness 0.064 afzal kármán dimensionless volumetric δh r_ darcy ρ vortices ν 8538
pressure loss due to friction in a pipeline
Darcy -- Weisbach equation - wikipedia Darcy -- Weisbach equation Jump to : navigation , search In fluid dynamics , the Darcy -- Weisbach equation is a phenomenological equation , which relates the head loss , or pressure loss , due to friction along a given length of pipe to the average velocity of the fluid flow for an incompressible fluid . The equation is named after Henry Darcy and Julius Weisbach . The Darcy -- Weisbach equation contains a dimensionless friction factor , known as the Darcy friction factor . This is also variously called the Darcy -- Weisbach friction factor , friction factor , resistance coefficient , or flow coefficient . Contents ( hide ) 1 Pressure - loss form 2 Head - loss form 2.1 In terms of volumetric flow 3 Shear - stress form 4 Darcy friction factor 4.1 Laminar regime 4.2 Critical regime 4.3 Turbulent regime 4.3. 1 Smooth - pipe regime 4.3. 2 Rough - pipe regime 4.4 Calculating the friction factor from its parametrization 4.4. 1 Direct calculation when friction loss S is known 4.5 Confusion with the Fanning friction factor 5 History 6 Derivation by dimensional analysis 7 Practical application 8 See also 9 Notes 10 References 11 Further reading 12 External links Pressure - loss form ( edit ) 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 : Δ p L = f D ⋅ ρ 2 ⋅ ⟨ v ⟩ 2 D , ( \ displaystyle ( \ frac ( \ Delta p ) ( L ) ) = f_ ( \ mathrm ( D ) ) \ cdot ( \ frac ( \ rho ) ( 2 ) ) \ cdot ( \ frac ( ( \ langle v \ rangle ) ^ ( 2 ) ) ( D ) ) , ) where the pressure loss per unit length Δp / L ( SI units : Pa / m ) is a function of : ρ , the density of the fluid ( kg / m ) ; D , the hydraulic diameter of the pipe ( for a pipe of circular section , this equals the internal diameter of the pipe ) ( m ) ; ⟨ v ⟩ , the mean flow velocity , experimentally measured as the volumetric flow rate Q per unit cross-sectional wetted area ( m / s ) ; f , the Darcy friction factor ( also called flow coefficient λ ) . For laminar flow , this can be rewritten as Δ p L = 128 π ⋅ μ Q D 4 , ( \ displaystyle ( \ frac ( \ Delta p ) ( L ) ) = ( \ frac ( 128 ) ( \ pi ) ) \ cdot ( \ frac ( \ mu Q ) ( D ^ ( 4 ) ) ) , ) where μ is the dynamic viscosity of the fluid ( Pa s = N s / m = kg / ( m s ) ) ; Q is the volumetric flow rate ( m / s ) . Head - loss form ( edit ) The head loss Δh ( or h ) 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 Δ p = ρ ⋅ g ⋅ Δ h , ( \ displaystyle \ Delta p = \ rho \ cdot g \ cdot \ Delta h , ) where Δh is the head loss due to pipe friction over the given length of pipe ( SI units : m ) ; g is the local acceleration due to gravity ( m / s ) . It is useful to present head loss per length of pipe ( dimensionless ) : S = Δ h L = 1 ρ ⋅ g ⋅ Δ p L , ( \ displaystyle S = ( \ frac ( \ Delta h ) ( L ) ) = ( \ frac ( 1 ) ( \ rho \ cdot g ) ) \ cdot ( \ frac ( \ Delta p ) ( L ) ) , ) where L is the pipe length ( m ) . Therefore , the Darcy -- Weisbach equation can also be written in terms of head loss : S = f D ⋅ 1 2 g ⋅ ⟨ v ⟩ 2 D . ( \ displaystyle S = f_ ( \ text ( D ) ) \ cdot ( \ frac ( 1 ) ( 2g ) ) \ cdot ( \ frac ( ( \ langle v \ rangle ) ^ ( 2 ) ) ( D ) ) . ) In terms of volumetric flow ( edit ) The relationship between mean flow velocity ⟨ v ⟩ and volumetric flow rate Q is Q = A ⋅ ⟨ v ⟩ , ( \ displaystyle Q = A \ cdot \ langle v \ rangle , ) where : Q is the volumetric flow ( m / s ) , A is the cross-sectional wetted area ( m ) . In a full - flowing pipe of diameter D , Q = π 4 D 2 ⟨ v ⟩ . ( \ displaystyle Q = ( \ frac ( \ pi ) ( 4 ) ) D ^ ( 2 ) \ langle v \ rangle . ) Then the Darcy -- Weisbach equation in terms of Q is S = f D ⋅ 8 π 2 g ⋅ Q 2 D 5 . ( \ displaystyle S = f_ ( \ text ( D ) ) \ cdot ( \ frac ( 8 ) ( \ pi ^ ( 2 ) g ) ) \ cdot ( \ frac ( Q ^ ( 2 ) ) ( D ^ ( 5 ) ) ) . ) Shear - stress form ( edit ) The mean wall shear stress τ in a pipe or open channel is expressed in terms of the Darcy -- Weisbach friction factor as τ = 1 8 ⋅ f D ⋅ ρ ⋅ ⟨ v ⟩ 2 . ( \ displaystyle \ tau = ( \ tfrac ( 1 ) ( 8 ) ) \ cdot f_ ( \ text ( D ) ) \ cdot \ rho \ cdot ( \ langle v \ rangle ) ^ ( 2 ) . ) The wall shear stress has the SI unit of pascals ( Pa ) . Darcy friction factor ( edit ) Figure 1 . The Darcy friction factor versus Reynolds number for 10 < Re < 10 for smooth pipe and a range of values of relative roughness ε / D. Data are from Nikuradse ( 1932 , 1933 ) , Colebrook ( 1939 ) , and McKeon ( 2004 ) . The friction factor f is not a constant : it depends on such things as the characteristics of the pipe ( diameter D and roughness height ε ) , the characteristics of the fluid ( its kinematic viscosity ν ( nu ) ) , and the velocity of the fluid flow ⟨ v ⟩ . It has been measured to high accuracy within certain flow regimes and may be evaluated by the use of various empirical relations , or it may be read from published charts . These charts are often referred to as Moody diagrams , after L.F. Moody , and hence the factor itself is sometimes erroneously called the Moody friction factor . It is also sometimes called the Blasius friction factor , after the approximate formula he proposed . Figure 1 shows the value of f 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 ( edit ) 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 D = 64 R e , ( \ displaystyle f_ ( \ mathrm ( D ) ) = ( \ frac ( 64 ) ( \ mathrm ( Re ) ) ) , ) where Re is the Reynolds number R e = ρ μ ⟨ v ⟩ D = ⟨ v ⟩ D ν , ( \ displaystyle \ mathrm ( Re ) = ( \ frac ( \ rho ) ( \ mu ) ) \ langle v \ rangle D = ( \ frac ( \ langle v \ rangle D ) ( \ nu ) ) , ) and where μ is the viscosity of the fluid and ν = μ ρ ( \ displaystyle \ nu = ( \ frac ( \ mu ) ( \ rho ) ) ) is known as the kinematic viscosity . In this expression for Reynolds number , the characteristic length D is taken to be the hydraulic diameter of the pipe , which , for a cylindrical pipe flowing full , equals the inside diameter . In Figures 1 and 2 of friction factor versus Reynolds number , the regime Re < 2000 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 ε : the flow velocity in the neighborhood of the pipe wall is zero . Critical regime ( edit ) For Reynolds numbers in the range 2000 < Re < 4000 , 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 ( edit ) Figure 2 . The Darcy friction factor versus Reynolds number for 1000 < Re < 10 for smooth pipe and a range of values of relative roughness ε / D. Data are from Nikuradse ( 1932 , 1933 ) , Colebrook ( 1939 ) , and McKeon ( 2004 ) . For Reynolds number greater than 4000 , the flow is turbulent ; the resistance to flow follows the Darcy -- Weisbach equation : it is proportional to the square of the mean flow velocity . Over a domain of many orders of magnitude of Re ( 4000 < Re < 10 ) , the friction factor varies less than one order of magnitude ( 0.006 < f < 0.06 ) . 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 ( edit ) 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 1 f D = − 1.930 ⋅ log ⁡ ( 1.90 R e f D ) . ( \ displaystyle ( \ frac ( 1 ) ( \ sqrt ( f_ ( \ mathrm ( D ) ) ) ) ) = - 1.930 \ cdot \ log \ left ( ( \ frac ( 1.90 ) ( \ mathrm ( Re ) ( \ sqrt ( f_ ( \ mathrm ( D ) ) ) ) ) ) \ right ) . ) The factors 1.930 and 1.90 are phenomenological ; these specific values provide a fairly good fit to the data . The product Re √ f ( called the `` friction Reynolds number '' ) can be considered , like the Reynolds number , to be a ( dimensionless ) parameter of the flow : at fixed values of Re √ f , the friction factor is also fixed . In the Kármán -- Prandtl resistance equation , f can be expressed in closed form as an analytic function of Re through the use of the Lambert W function : 1 f D = 1.930 ⋅ W ( R e 1.930 ⋅ 1.90 ) ( \ displaystyle ( \ frac ( 1 ) ( \ sqrt ( f_ ( \ mathrm ( D ) ) ) ) ) = 1.930 \ cdot W \ left ( ( \ frac ( \ mathrm ( Re ) ) ( 1.930 \ cdot 1.90 ) ) \ right ) ) 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 Re √ f 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 ( edit ) When 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 ) . 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 R ∗ = 1 8 ⋅ ( R e f D ) ⋅ ε D ( \ displaystyle R_ ( * ) = ( \ frac ( 1 ) ( \ sqrt ( 8 ) ) ) \ cdot ( \ mathrm ( Re ) ( \ sqrt ( f_ ( \ mathrm ( D ) ) ) ) ) \ cdot ( \ frac ( \ varepsilon ) ( D ) ) ) where the roughness height ε is scaled to the pipe diameter D . 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 and Colebrook -- White are shown . It is illustrative to plot the roughness function B : B ( R ∗ ) = 1 1.930 f D + log ⁡ ( 1.90 8 ⋅ ε D ) ( \ displaystyle B ( R_ ( * ) ) = ( \ frac ( 1 ) ( 1.930 ( \ sqrt ( f_ ( \ mathrm ( D ) ) ) ) ) ) + \ log \ left ( ( \ frac ( 1.90 ) ( \ sqrt ( 8 ) ) ) \ cdot ( \ frac ( \ varepsilon ) ( D ) ) \ right ) ) Figure 3 shows B versus R for the rough pipe data of Nikuradse , Shockling , and Langelandsvik . In this view , the data at different roughness ratio ε / D fall together when plotted against R , demonstrating scaling in the variable R . The following features are present : When ε = 0 , then R 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 R < 5 , the data lie on the line B ( R ) = R ; flow is in the smooth pipe regime . When R > 100 , the data asymptotically approach a horizontal line ; they are independent of Re , f , and ε / D . The intermediate range of 5 < R < 100 constitutes a transition from one behavior to the other . The data depart from the line B ( R ) = R very slowly , reach a maximum near R = 10 , then fall to a constant value . A fit to these data in the transition from smooth pipe flow to rough pipe flow employs an exponential expression in R that ensures proper behavior for 1 < R < 50 ( the transition from the smooth pipe regime to the rough pipe regime ) : 1 f D = − 1.930 log ⁡ ( 1.90 R e f D ( 1 + 0.34 R ∗ exp ⁡ − 11 R ∗ ) ) , ( \ displaystyle ( \ frac ( 1 ) ( \ sqrt ( f_ ( \ mathrm ( D ) ) ) ) ) = - 1.930 \ log \ left ( ( \ frac ( 1.90 ) ( \ mathrm ( Re ) ( \ sqrt ( f_ ( \ mathrm ( D ) ) ) ) ) ) \ left ( 1 + 0.34 R_ ( * ) \ exp ( \ frac ( - 11 ) ( R_ ( * ) ) ) \ right ) \ right ) , ) This function shares the same values for its term in common with the Kármán -- Prandtl resistance equation , plus one parameter 0.34 to fit the asymptotic behavior for R → ∞ along with one further parameter , 11 , to govern the transition from smooth to rough flow . It is exhibited in Figure 3 . The Colebrook -- White relation fits the friction factor with a function of the form 1 f D = − 2.00 log ⁡ ( 2.51 R e f D ( 1 + R ∗ 3.3 ) ) . ( \ displaystyle ( \ frac ( 1 ) ( \ sqrt ( f_ ( \ mathrm ( D ) ) ) ) ) = - 2.00 \ log \ left ( ( \ frac ( 2.51 ) ( \ mathrm ( Re ) ( \ sqrt ( f_ ( \ mathrm ( D ) ) ) ) ) ) \ left ( 1 + ( \ frac ( R_ ( * ) ) ( 3.3 ) ) \ right ) \ right ) . ) 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 . 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 ( edit ) See also : Darcy friction factor formulae For turbulent flow , methods for finding the friction factor f include using a diagram , such as the Moody chart , or solving equations such as the Colebrook -- White equation ( upon which the Moody chart is based ) , or the Swamee -- Jain equation . While the Colebrook -- White relation is , in the general case , an iterative method , the Swamee -- Jain equation allows f to be found directly for full flow in a circular pipe . Direct calculation when friction loss S is known ( edit ) In typical engineering applications , there will be a set of given or known quantities . 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 f can be calculated directly from the chosen fitting function . Solving the Darcy -- Weisbach equation for √ f , f D = 2 g S D ⟨ v ⟩ ( \ displaystyle ( \ sqrt ( f_ ( \ mathrm ( D ) ) ) ) = ( \ frac ( \ sqrt ( 2gSD ) ) ( \ langle v \ rangle ) ) ) we can now express Re √ f : R e f D = 1 ν 2 g S D 3 ( \ displaystyle \ mathrm ( Re ) ( \ sqrt ( f_ ( \ mathrm ( D ) ) ) ) = ( \ frac ( 1 ) ( \ nu ) ) ( \ sqrt ( 2g ) ) ( \ sqrt ( S ) ) ( \ sqrt ( D ^ ( 3 ) ) ) ) Expressing the roughness Reynolds number R , R ∗ = ε D ⋅ R e f D ⋅ 1 8 = 1 2 g ν ε S D ( \ displaystyle ( \ begin ( aligned ) R_ ( * ) & = ( \ frac ( \ varepsilon ) ( D ) ) \ cdot \ mathrm ( Re ) ( \ sqrt ( f_ ( \ mathrm ( D ) ) ) ) \ cdot ( \ frac ( 1 ) ( \ sqrt ( 8 ) ) ) \ \ & = ( \ frac ( 1 ) ( 2 ) ) ( \ frac ( \ sqrt ( g ) ) ( \ nu ) ) \ varepsilon ( \ sqrt ( S ) ) ( \ sqrt ( D ) ) \ end ( aligned ) ) ) we have the two parameters needed to substitute into the Colebrook -- White relation , or any other function , for the friction factor f , the flow velocity ⟨ v ⟩ , and the volumetric flow rate Q . Confusion with the Fanning friction factor ( edit ) The Darcy -- Weisbach friction factor f is 4 times larger than the Fanning friction factor f , so attention must be paid to note which one of these is meant in any `` friction factor '' chart or equation being used . Of the two , the Darcy -- Weisbach factor f is more commonly used by civil and mechanical engineers , and the Fanning factor f by chemical engineers , but care should be taken to identify the correct factor regardless of the source of the chart or formula . Note that Δ p = f D ⋅ L D ⋅ ρ ⟨ v ⟩ 2 2 = f ⋅ L D ⋅ 2 ρ ⟨ v ⟩ 2 ( \ displaystyle \ Delta p = f_ ( \ mathrm ( D ) ) \ cdot ( \ frac ( L ) ( D ) ) \ cdot ( \ frac ( \ rho ( \ langle v \ rangle ) ^ ( 2 ) ) ( 2 ) ) = f \ cdot ( \ frac ( L ) ( D ) ) \ cdot ( 2 \ rho ( \ langle v \ rangle ) ^ ( 2 ) ) ) Most charts or tables indicate the type of friction factor , or at least provide the formula for the friction factor with laminar flow . If the formula for laminar flow is f = 16 / Re , it is the Fanning factor f , and if the formula for laminar flow is f = 64 / Re , it is the Darcy -- Weisbach factor f . Which friction factor is plotted in a Moody diagram may be determined by inspection if the publisher did not include the formula described above : Observe the value of the friction factor for laminar flow at a Reynolds number of 1000 . If the value of the friction factor is 0.064 , then the Darcy friction factor is plotted in the Moody diagram . Note that the nonzero digits in 0.064 are the numerator in the formula for the laminar Darcy friction factor : f = 64 / Re . If the value of the friction factor is 0.016 , then the Fanning friction factor is plotted in the Moody diagram . Note that the nonzero digits in 0.016 are the numerator in the formula for the laminar Fanning friction factor : f = 16 / Re . The procedure above is similar for any available Reynolds number that is an 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 ( edit ) Historically this equation arose as a variant on the Prony equation ; this variant was developed by Henry Darcy of France , and further refined into the form used today by Julius Weisbach of Saxony in 1845 . Initially , data on the variation of f with velocity was lacking , so the Darcy -- Weisbach equation was outperformed at first by the empirical Prony equation in many cases . In later years it was eschewed in many special - case situations in favor of a variety of empirical equations valid only for certain flow regimes , notably the Hazen -- Williams equation or the Manning equation , most of which were significantly easier to use in calculations . However , since the advent of the calculator , ease of calculation is no longer a major issue , and so the Darcy -- Weisbach equation 's generality has made it the preferred one . Derivation by dimensional analysis ( edit ) Away from the ends of the pipe , the characteristics of the flow are independent of the position along the pipe . The key quantities are then the pressure drop along the pipe per unit length , Δp / L , and the volumetric flow rate . The flow rate can be converted to a mean flow velocity V by dividing by the wetted area of the flow ( which equals the cross-sectional area of the pipe if the pipe is full of fluid ) . Pressure has dimensions of energy per unit volume , therefore the pressure drop between two points must be proportional to 1 / 2ρ ⟨ v ⟩ , which has the same dimensions as it resembles ( see below ) the expression for the kinetic energy per unit volume . We also know that pressure must be proportional to the length of the pipe between the two points L as the pressure drop per unit length is a constant . To turn the relationship into a proportionality coefficient of dimensionless quantity , we can divide by the hydraulic diameter of the pipe , D , which is also constant along the pipe . Therefore , Δ p ∝ L D ⋅ 1 2 ρ ⟨ v ⟩ 2 . ( \ displaystyle \ Delta p \ propto ( \ frac ( L ) ( D ) ) \ cdot ( \ frac ( 1 ) ( 2 ) ) \ rho ( \ langle v \ rangle ) ^ ( 2 ) . ) The proportionality coefficient is the dimensionless `` Darcy friction factor '' or `` flow coefficient '' . This dimensionless coefficient will be a combination of geometric factors such as π , the Reynolds number and ( outside the laminar regime ) the relative roughness of the pipe ( the ratio of the roughness height to the hydraulic diameter ) . Note that 1 / 2ρ ⟨ v ⟩ is not the kinetic energy of the fluid per unit volume , for the following reasons . Even in the case of laminar flow , where all the flow lines are parallel to the length of the pipe , the velocity of the fluid on the inner surface of the pipe is zero due to viscosity , and the velocity in the center of the pipe must therefore be larger than the average velocity obtained by dividing the volumetric flow rate by the wet area . The average kinetic energy then involves the root mean - square velocity , which always exceeds the mean velocity . In the case of turbulent flow , the fluid acquires random velocity components in all directions , including perpendicular to the length of the pipe , and thus turbulence contributes to the kinetic energy per unit volume but not to the average lengthwise velocity of the fluid . Practical application ( edit ) 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 . 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 . See also ( edit ) Bernoulli 's principle Darcy friction factor formulae Euler number Hagen -- Poiseuille equation Water pipe Notes ( edit ) Jump up ^ The value of the Darcy friction factor is four times that of the Fanning friction factor , with which it should not be confused . Jump up ^ This is related to the piezometric head along the pipe . Jump up ^ The data exhibit , however , a systematic departure of up to 50 % from the theoretical Hagen -- Poiseuille equation in the region of Re > 500 up to the onset of critical flow . Jump up ^ In its originally published form , 1 f D = − 2.00 log ⁡ ( 2.51 1 R e f D + 1 3.7 ε D ) ( \ displaystyle ( \ frac ( 1 ) ( \ sqrt ( f_ ( \ mathrm ( D ) ) ) ) ) = - 2.00 \ log \ left ( 2.51 ( \ frac ( 1 ) ( \ mathrm ( Re ) ( \ sqrt ( f_ ( \ mathrm ( D ) ) ) ) ) ) + ( \ frac ( 1 ) ( 3.7 ) ) ( \ frac ( \ varepsilon ) ( D ) ) \ right ) ) References ( edit ) Jump up ^ Manning , Francis S. ; Thompson , Richard E. ( 1991 ) . Oilfield Processing of Petroleum . Vol. 1 : Natural Gas . PennWell Books . p. 293 . ISBN 0 - 87814 - 343 - 2 . Jump up ^ Brown , Glenn . `` The Darcy -- Weisbach Equation '' . Oklahoma State University -- Stillwater . ^ Jump up to : Rouse , H. ( 1946 ) . Elementary Mechanics of Fluids . John Wiley & Sons . Jump up ^ Incopera , Frank P. ; Dewitt , David P. ( 2002 ) . Fundamentals of Heat and Mass Transfer ( 5th ed . ) . John Wiley & Sons . p. 470 paragraph 3 . ^ Jump up to : Crowe , Clayton T. ; Elger , Donald F. ; Robertson , John A. ( 2005 ) . Engineering Fluid Mechanics ( 8th ed . ) . John Wiley & Sons . p. 379 ; Eq. 10 : 23 , 10 : 24 , paragraph 4 . 1 f D = 2 log ⁡ ( R e f D ) − 0.8 for R e > 3000 . ( \ displaystyle ( \ frac ( 1 ) ( \ sqrt ( f_ ( \ mathrm ( D ) ) ) ) ) = 2 \ log \ left ( \ mathrm ( Re ) ( \ sqrt ( f_ ( \ mathrm ( D ) ) ) ) \ right ) - 0.8 \ quad ( \ mbox ( for ) ) \ mathrm ( Re ) > 3000 . ) Jump up ^ Chaudhry , M.H. ( 2013 ) . Applied Hydraulic Transients ( 3rd ed . ) . Springer . p. 45 . ISBN 978 - 1 - 4614 - 8538 - 4 . Jump up ^ McKeon , B.J. ; Zagarola , M. V ; Smits , A.J. ( 2005 ) . `` A new friction factor relationship for fully developed pipe flow '' ( PDF ) . Journal of Fluid Mechanics . Cambridge University Press . 538 : 429 -- 443 . Bibcode : 2004JFM ... 511 ... 41M . doi : 10.1017 / S0022112005005501 . Retrieved 25 June 2016 . ^ Jump up to : Nikuradse , J. ( 1933 ) . `` Strömungsgesetze in rauen Rohren '' ( PDF ) . V.D.I. Forschungsheft . Berlin . 361 : 1 -- 22 . In translation , NACA TM 1292 . The data are available in digital form . ^ Jump up to : Afzal , Noor ( 2007 ) . `` Friction Factor Directly From Transitional Roughness in a Turbulent Pipe Flow '' . Journal of Fluids Engineering . ASME. 129 ( 10 ) : 1255 -- 1267 . doi : 10.1115 / 1.2776961 . ^ Jump up to : Colebrook , C.F. ( February 1939 ) . `` Turbulent flow in pipes , with particular reference to the transition region between smooth and rough pipe laws '' . Journal of the Institution of Civil Engineers . London . doi : 10.1680 / ijoti. 1939.14509 . Jump up ^ Schlichting , H. ( 1955 ) . Boundary Layer Theory . McGraw - Hill . ^ Jump up to : Shockling , M.A. ; Allen , J.J. ; Smits , A.J. ( 2006 ) . `` Roughness effects in turbulent pipe flow '' . Journal of Fluid Mechanics. 564 : 267 -- 285 . Bibcode : 2006JFM ... 564 ... 267S . doi : 10.1017 / S0022112006001467 . Jump up ^ Langelandsvik , L.I. ; Kunkel , G.J. ; Smits , A.J. ( 2008 ) . `` Flow in a commercial steel pipe '' ( PDF ) . Journal of Fluid Mechanics . Cambridge University Press . 595 : 323 -- 339 . Bibcode : 2008JFM ... 595 ... 323L . doi : 10.1017 / S0022112007009305 . Retrieved 25 June 2016 . Jump up ^ Afzal , Noor ( 2011 ) . `` Erratum : Friction factor directly from transitional roughness in a turbulent pipe flow '' . Journal of Fluids Engineering . ASME. 133 ( 10 ) : 107001 . doi : 10.1115 / 1.4004961 . Jump up ^ Afzal , Noor ; Seena , Abu ; Bushra , A. ( 2013 ) . `` Turbulent flow in a machine honed rough pipe for large Reynolds numbers : General roughness scaling laws '' . Journal of Hydro - environment Research . Elsevier . 7 ( 1 ) : 81 -- 90 . doi : 10.1016 / j. jher. 2011.08. 002 . Jump up ^ Brown , G.O. ( 2003 ) . `` The History of the Darcy - Weisbach Equation for Pipe Flow Resistance '' . In Rogers , J.R. ; Fredrich , A.J. Environmental and Water Resources History . American Society of Civil Engineers . pp. 34 -- 43 . ISBN 978 - 0 - 7844 - 0650 - 2 . Further reading ( edit ) De Nevers ( 1970 ) . Fluid Mechanics . Addison -- Wesley . ISBN 0 - 201 - 01497 - 1 . Shah , R.K. ; London , A.L. ( 1978 ) . `` Laminar Flow Forced Convection in Ducts '' . Supplement 1 to Advances in Heat Transfer . New York : Academic . Rohsenhow , W.M. ; Hartnett , J.P. ; Ganić , E.N. ( 1985 ) . Handbook of Heat Transfer Fundamentals ( 2nd ed . ) . McGraw -- Hill Book Company . ISBN 0 - 07 - 053554 - X . External links ( edit ) 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 ducts Retrieved from `` https://en.wikipedia.org/w/index.php?title=Darcy -- Weisbach_equation&oldid = 826833638 '' Categories : Dimensionless numbers of fluid mechanics Equations of fluid dynamics Piping Hidden categories : All articles with unsourced statements Articles with unsourced statements from July 2016 Talk Contents About Wikipedia Català Deutsch Español فارسی Français 한국어 Bahasa Indonesia Italiano עברית Nederlands 日本 語 Norsk Polski Português Русский Slovenčina Slovenščina Svenska Türkçe Українська 中文 12 more Edit links This page was last edited on 21 February 2018 , at 07 : 09 . 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