Friction loss

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.
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.
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.
Definition.
In the following discussion, we define volumetric flow rate V̇ (i.e. volume of fluid flowing) 
V̇ = πr2v
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"
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". 
where
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
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
where
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 turbulent (Re > 4000):
For friction.
Factors other than straight pipe flow induce friction loss; these are known as “minor loss”:
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.
Surface roughness.
The roughness of the surface of the pipe or duct affects the fluid flow in the regime of turbulent flow. Usually denoted by ε, values used for calculations of water flow, for some representative materials are:
Values used in calculating friction loss in ducts (for, e.g., air) are:
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
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
where we have introduced the Darcy friction factor "f""D" (but see "Confusion with the Fanning 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,
The experimentally measured values of "f""D" are fit to reasonable accuracy by the (recursive) Colebrook–White equation, 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.
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) 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 gpm"2"ft)" and can be calculated using the following relation:
where formula_8 is the pressure in psi, formula_9 is the flow in "100gpm" and formula_10 is the length of the pipe in "100ft"
Calculating friction loss for air in a duct.
Friction loss takes place as a gas, say air, flows through duct work.
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, Δ"p" / "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. 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 Δ"p" / "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 Δ"p" / "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 Δ"p" / "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 Δ"p" / "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.