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.[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.

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
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
ρ = density of the fluid, (SI kg / m3)
g = the local acceleration due to gravity;

Characterizing friction loss[edit]

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
μ = viscosity of the fluid (SI kg / m / s)
Friction loss in straight pipe[edit]
The friction loss In a 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:
o In the smooth pipe domain, friction loss is relatively insensitive to roughness.
o In the rough pipe domain, friction loss is dominated by the relative roughness and is insensitive
to Reynolds number.
o 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.
Form friction[edit]
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.

Hagen–Poiseuille[edit]
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[edit]
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]
where we have introduced the Darcy friction factor  fD (but see Confusion with the Fanning friction factor);
fD = 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 fD are, for fixed relative
roughness ε / D and for Reynolds number Re = V D / ν > ~2000,[a]

 With relative roughness ε / D < 10−6, fD declines in value with increasing Re in an approximate power

law, with one order of magnitude change in fD 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, fD 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 fD 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 fD 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 fD 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]