Friction loss

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 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 turbulent flow, losses are proportional to the square of the fluid velocity, V²; here, a layer of chaotic eddies and vortices near the pipe surface, called the viscous sub-layer, forms the transition to the bulk flow. In this domain, the effects of the roughness of the pipe surface must be considered. It is useful to characterize that roughness as the ratio of the roughness height ε to the pipe diameter D, the "relative roughness". Three sub-domains pertain to turbulent flow:
  • In the smooth pipe domain, friction loss is relatively insensitive to roughness.
  • In the rough pipe domain, friction loss is dominated by the relative roughness and is insensitive to Reynolds number.
  • In the transition domain, friction loss is sensitive to both.
• For Reynolds numbers 2000 < Re < 4000, the flow is unstable, varying with time as vortices within the flow form and vanish randomly. This domain of flow is not well modeled, nor are the details well understood.

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.

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.

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,[lower-alpha 1]

• 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.[lower-alpha 2]

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.

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.

Note that the cited sources recommend that flow velocity be kept below 5 feet / second (~1.5 m/s).

Also note that the given f_D in this table is actually a quantity adopted by the NFPA and the industry, known as C, which has the imperial units psi/(100 gpm²ft) and can be calculated using the following relation:

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

where ${\displaystyle \Delta P_{f}'}$ is the pressure in psi, ${\displaystyle Q'}$ is the flow in 100gpm and ${\displaystyle L'}$ is the length of the pipe in 100ft

A graphical depiction of the relationship between Δp / L, the pressure loss per unit length of pipe, versus flow volume Q, for a range of choices for pipe diameter D, for air at standard temperature and pressure. Units are SI. Lines of constant Re√f_D are also shown.[17]

Friction loss takes place as a gas, say air, flows through duct work.[17] The difference in the character of the flow from the case of water in a pipe stems from the differing Reynolds number Re and the roughness of the duct.

The friction loss is customarily given as pressure loss for a given duct length, Δp / L, in units of (US) inches of water for 100 feet or (SI) kg / m² / s².

For specific choices of duct material, and assuming air at standard temperature and pressure (STP), standard charts can be used to calculate the expected friction loss.[9][18] The chart exhibited in this section can be used to graphically determine the required diameter of duct to be installed in an application where the volume of flow is determined and where the goal is to keep the pressure loss per unit length of duct S below some target value in all portions of the system under study. First, select the desired pressure loss Δp / L, say 1 kg / m² / s² (0.12 in H₂O per 100 ft) on the vertical axis (ordinate). Next scan horizontally to the needed flow volume Q, say 1 m³ / s (2000 cfm): the choice of duct with diameter D = 0.5 m (20 in.) will result in a pressure loss rate Δp / L less than the target value. Note in passing that selecting a duct with diameter D = 0.6 m (24 in.) will result in a loss Δp / L of 0.02 kg / m² / s² (0.02 in H₂O per 100 ft), illustrating the great gains in blower efficiency to be achieved by using modestly larger ducts.

The following table gives flow rate Q such that friction loss per unit length Δp / L (SI kg / m² / s²) is 0.082, 0.245, and 0.816, respectively, for a variety of nominal duct sizes. The three values chosen for friction loss correspond to, in US units inch water column per 100 feet, 0.01, .03, and 0.1. Note that, in approximation, for a given value of flow volume, a step up in duct size (say from 100mm to 120mm) will reduce the friction loss by a factor of 3.

Note that, for the chart and table presented here, flow is in the turbulent, smooth pipe domain, with R* < 5 in all cases.