In fluid flow, friction loss (or skin friction) is the loss of pressure or “head” that occurs in pipe or duct flow due to the effect of the fluid's viscosity near the surface of the pipe or duct.^[1] In mechanical systems such as internal combustion engines, the term refers to the power lost in overcoming the friction between two moving surfaces, a different phenomenon.

Jean Le Rond d'Alembert, Nouvelles expériences sur la résistance des fluides, 1777

Economics

Friction loss is a significant economic concern wherever fluids are made to flow, whether entirely enclosed in a pipe or duct, or with a surface open to the air.

• Historically, it is a concern in aqueducts of all kinds, throughout human history. It is also relevant to sewer lines. Systematic study traces back to Henry Darcy, an aqueduct engineer.
• Natural flows in river beds are important to human activity; friction loss in a stream bed has an effect on the height of the flow, particularly significant during flooding.
• The economies of pipelines for petrochemical delivery are highly affected by friction loss. The Yamal–Europe pipeline carries methane at a volume flow rate of 32.3 × 10⁹ m³ of gas per year, at Reynolds numbers greater than 50 × 10⁶.^[2]
• In hydropower applications, the energy lost to skin friction in flume and penstock is not available for useful work, say generating electricity.
• Friction loss is a significant economic concern wherever fluids are made to flow, whether entirely enclosed in a pipe or duct, or with a surface open to the air.
  • Historically, it is a concern in aqueducts of all kinds, throughout human history. It is also relevant to sewer lines. Systematic study traces back to Henry Darcy, an aqueduct engineer.
  • Natural flows in river beds are important to human activity; friction loss in a stream bed has an effect on the height of the flow, particularly significant during flooding.
  • The economies of pipelines for petrochemical delivery are highly affected by friction loss. The Yamal–Europe pipeline carries methane at a volume flow rate of 32.3 × 10⁹ m³ of gas per year, at Reynolds numbers greater than 50 × 10⁶.^[2]
  • In hydropower applications, the energy lost to skin friction in flume and penstock is not available for useful work, say generating electricity.
  • Irrigation water is pumped at large yearly volumes of flow, entailing significant expense.
  • HVAC systems pump conditioning air on a widespread basis.
  • In refrigeration applications, energy is expended pumping the coolant fluid through pipes or through the condenser. In split systems, the pipes carrying the coolant take the place of the air ducts in HVAC systems.
  • Wells and domestic water systems must be engineered for effective and economical operation.

  Definition

  In the following discussion, we define volumetric flow rate V̇ (i.e. volume of fluid flowing) V̇ = πr²v

  where

  r = radius of the pipe (for a pipe of circular section, the internal radius of the pipe).

  v = mean velocity of fluid flowing through the pipe.

  A = cross sectional area of the pipe.

  In long pipes, the loss in pressure (assuming the pipe is level) is proportional to the length of pipe involved. Friction loss is then the change in pressure Δp per unit length of pipe L

  ${\displaystyle {\frac {\Delta p}{L}}.}$

  When the pressure is expressed in terms of the equivalent height of a column of that fluid, as is common with water, the friction loss is expressed as S, the "head loss" per length of pipe, a dimensionless quantity also known as the hydraulic slope.

  ${\displaystyle {\frac {\Delta p}{L}}.}$

  When the pressure is expressed in terms of the equivalent height of a column of that fluid, as is common with water, the friction loss is expressed as S, the "head loss" per length of pipe, a dimensionless quantity also known as the hydraulic slope.

  ${\displaystyle S=\frac{}{}}$

  where

  In long pipes, the loss in pressure (assuming the pipe is level) is proportional to the length of pipe involved. Friction loss is then the change in pressure Δp per unit length of pipe L

  ρ = density of the fluid, (SI kg / m³)

  g = the local acceleration due to gravity;

  Characterizing friction loss

  Friction loss, which is due to the shear stress between the pipe surface and the fluid flowing within, depends on the conditions of flow and the physical properties of the system. These conditions can be encapsulated into a dimensionless number Re, known as the Reynolds number

  ${\displaystyle \mathrm {Re} ={\frac {1}{\nu }}VD}Friction loss, which is due to the shear stress between the pipe surface and the fluid flowing within, depends on the conditions of flow and the physical properties of the system. These conditions can be encapsulated into a dimensionless number Re, known as the Reynolds number$

  ${\displaystyle \mathrm{R}\mathrm{e}=\frac{1}{\mathrm{\nu <!--\; \nu \; -->}}}$

  where V is the mean fluid velocity and D the diameter of the (cylindrical) pipe. In this expression, the properties of the fluid itself are reduced to the kinematic viscosity ν

  ${\displaystyle \nu ={\frac {\mu }{\rho }}}$

  where

  μ = viscosity of the fluid (SI kg / m / s)

  where

  μ = viscosity of the fluid (SI kg / m / s)

  Friction loss in straight pipe

  The friction loss in uniform, straight sections of pipe, known as "major loss", is caused by the effects of viscosity, the movement of fluid molecules against each other or against the (possibly rough) wall of the pipe. Here, it is greatly affected by whether the flow is laminar (Re < 2000) or viscosity, the movement of fluid molecules against each other or against the (possibly rough) wall of the pipe. Here, it is greatly affected by whether the flow is laminar (Re < 2000) or turbulent (Re > 4000):^[1]

  • In laminar flow, losses are proportional to fluid velocity, V; that velocity varies smoothly between the bulk of the fluid and the pipe surface, where it is zero. The roughness of the pipe surface influences neither the fluid flow nor the friction loss.
  • In

    Factors other than straight pipe flow induce friction loss; these are known as “minor loss”:

    • Fittings, such as bends, couplings, valves, or transitions in hose or pipe diameter, or
    • Objects intruded into the fluid flow.

    For the purposes of calculating the total friction loss of a system, the sources of form friction are sometimes reduced to an equivalent length of pipe.

    Measurements

    Because of the importance of friction loss in civil engineering and in industry, it has been studied extensively for over a century.

    • Nikuradse, J. (1932). "Gesetzmassigkeiten der Turbulenten Stromung in Glatten Rohren" (PDF). VDI Forschungsheft Arb. Ing.-Wes. 356: 1–36. In translation, NACA TT F-10 359. The data are available in digital form.Measurements

  Hagen–Poiseuille

  Laminar flow is encountered in practice with very viscous fluids, such as motor oil, flowing through small-diameter tubes, at low velocity. Friction loss under conditions of laminar flow follow the Hagen–Poiseuille equation, which is an exact solution to the Navier-Stokes equations. For a circular pipe with a fluid of density ρ and viscosity μ, the hydraulic slope S can be expressed

  ${\displaystyle S={\frac {64}{\mathrm {Re} }}{\frac {V^{2}}{2gD}}={\frac {64\nu }{2g}}{\frac {V}{D^{2}}}}$

  In laminar flow (that is, with Re < ~2000), the hydraulic slope is proportional to the flow velocity.

  Darcy–Weisbach

  In many practical engineering applications, the fluid flow is more rapid, therefore turbulent rather than laminar. Under turbulent flow, the friction loss is found to be roughly proportional to the square of the flow velocity and inversely proportional to the pipe diameter, that is, the friction loss follows the phenomenological Darcy–Weisbach equation in which the hydraulic slope S can be expressed^[10]

  ${\displaystyle S=f_{D}{\frac {1}{2g}}{\frac {V^{2}}{D}}}$

  where we have introduced the Darcy friction factor f_D (but see Confusion with the Fanning friction factor);

  f_D = Darcy friction factor

  Note that the value of this dimensionless factor depends on the pipe diameter D and the roughness of the pipe surface ε. Furthermore, it varies as well with the flow velocity V and on the physical properties of the fluid (usually cast together into the Reynolds number Re). Thus, the friction loss is not precisely proportional to the flow velocity squared, nor to the inverse of the pipe diameter: the friction factor takes account of the remaining dependency on these parameters.

  From experimental measurements, the general features of the variation of f_D are, for fixed relative roughness ε / D and for Reynolds number Re = V D / ν > ~2000,^[a]

  • With relative roughness ε / D < 10⁻⁶, f_D declines in value with increasing Re in an approximate power law, with one order of magnitude change in f_D over four orders of magnitude in Re. This is called the "smooth pipe" regime, where the flow is turbulent but not sensitive to the roughness features of the pipe (because the vortices are much larger than those features).
  • At higher roughness, with increasing Reynolds number Re, f_D climbs from its smooth pipe value, approaching an asymptote that itself varies logarithmically with the relative roughness ε / D; this regime is called "rough pipe" flow.
  • The point of departure from smooth flow occurs at a Reynolds number roughly inversely proportional to the value of the relative roughness: the higher the relative roughness, the lower the Re of departure. The range of Re and ε / D between smooth pipe flow and rough pipe flow is labeled "transitional". In this region, the measurements of Nikuradse show a decline in the value of f_D with Re, before approaching its asymptotic value from below,^[3] although Moody chose not to follow those data in his chart,^[11] which is based on the Colebrook–White equation.
  • At values of 2000 < Re < 4000, there is a critical zone of flow, a transition from laminar to turbulence, where the value of f_D increases from its laminar value of 64 / Re to its smooth pipe value. In this regime, the fluid flow is found to be unstable, with vortices appearing and disappearing within the flow over time.
  • The entire dependence of f_D on the pipe diameter D is subsumed into the Reynolds number Re and the relative roughness ε / D, likewise the entire dependence on fluid properties density ρ and viscosity μ is subsumed into the Reynolds number Re. This is called scaling.^[b]

  The experimentally measured values of f_D are fit to reasonable accuracy by the (recursive) Colebrook–White equation,^[12] depicted graphically in the Moody chart which plots friction factor f_D versus Reynolds number Re for selected values of relative roughness ε / D.

  Calculating friction loss for water in a pipe

  

  Water friction loss (“hydraulic slope”) S versus flow Q for given ANSI Sch. 40 NPT PVC pipe, roughness height ε = 1.5 μm

  In a design problem, one may select pipe for a particular hydraulic slope S based on the candidate pipe's diameter D and its roughness ε. With these quantities as inputs, the friction factor f_D can be expressed in closed form in the Colebrook–White equation or other fitting function, and the flow volume Q and flow velocity V can be calculated therefrom.

  In the case of water (ρ = 1 g/cc, μ = 1 g/m/s^[13]) flowing through a 12-inch (300 mm) Schedule-40 PVC pipe (ε = 0.0015 mm, D = 11.938 in.), a hydraulic slope S = 0.01 (1%) is reached at a flow rate Q = 157 lps (liters per second), or at a velocity V = 2.17 m/s (meters per second). The following table gives Reynolds number Re, Darcy friction factor f_D, flow rate Q, and velocity V such that hydraulic slope S = h_f / L = 0.01, for a variety of nominal pipe (NPS) sizes.

  [9]

  Surface Roughness ε (for air ducts)

  Material	mm	in
  Flexible Duct (wires exposed)	3.00	0.120
  Flexible Duct (wires covered)	0.90	0.036
  Galvanized Steel	0.15	0.006
  PVC, Stainless Steel, Aluminum, Black Iron	0.05	0.0018

  Calculating friction loss

  Hagen–Poiseuille

  Laminar flow is encountered in practice with very viscous fluids, such as motor oil, flowing through small-diameter tubes, at low velocity. Friction loss under conditions of laminar flow follow the Hagen–Poiseuille equation, which is an exact solution to the Navier-Stokes equations. For a circular pipe with a fluid of density ρ and viscosity μ, the hydraulic slope S can be expressed

  ${\displaystyle S=\frac{64}{\mathrm{R}\mathrm{e}}}$

  Laminar flow is encountered in practice with very viscous fluids, such as motor oil, flowing through small-diameter tubes, at low velocity. Friction loss under conditions of laminar flow follow the Hagen–Poiseuille equation, which is an exact solution to the Navier-Stokes equations. For a circular pipe with a fluid of density ρ and viscosity μ, the hydraulic slope S can be expressed

  ${\displaystyle S=\frac{64}{\mathrm{R}\mathrm{e}}\frac{V^2}{2gD}=\frac{64\mathrm{\nu <!--\; \nu \; -->}}{2g}\frac{V}{D^{}}}$

  In laminar flow (that is, with Re < ~2000), the hydraulic slope is proportional to the flow velocity.

  Darcy–Weisbach

  In many practical engineering applications, the fluid flow is more rapid, therefore turbulent rather than laminar. Under turbulent flow, the friction loss is found to be roughly proportional to the square of the flow velocity and inversely proportional to the pipe diameter, that is, the friction loss follows the phenomenological Darcy–Weisbach equation in which the hydraulic slope S can be expressed^[10]

  ${\displaystyle S=f_{D}{\frac {1}{2g}}{\frac {V^{2}}{D}}}$Darcy–Weisbach equation in which the hydraulic slope S can be expressed^[10]

  ${\displaystyle \Delta P_{f}'=CQ'^{2}L'}$