Darcy weisbach equation derivation || fluid mechanics || 

DARCY WEISBACH EQUACTION DERIVATION || fluid mechanics ||
In fluid dynamics, the Darcy-Weisbach equation is an empirical 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.
Pressure-loss form
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:[2]
{\displaystyle {\frac {\Delta p}{L}}=f_{\mathrm {D} }\cdot {\frac {
ho }{2}}\cdot {\frac {{\langle v
angle }^{2}}{D}},} {\displaystyle {\frac {\Delta p}{L}}=f_{\mathrm {D} }\cdot {\frac {
ho }{2}}\cdot {\frac {{\langle v
angle }^{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/m3);
D, the hydraulic diameter of the pipe (for a pipe of circular section, this equals the internal diameter of the pipe; otherwise D ≈ 2√A/π for a pipe of cross-sectional area A) (m);
⟨v⟩, the mean flow velocity, experimentally measured as the volumetric flow rate Q per unit cross-sectional wetted area (m/s);
fD, the Darcy friction factor (also called flow coefficient λ[3][4]).
For laminar flow in a circular pipe of diameter {\displaystyle D_{c}} D_{c}, the friction factor is inversely proportional to the Reynolds number alone (fD =
64
/
Re
) which itself can be expressed in terms of easily measured or published physical quantities (see section below). Making this substitution the Darcy-Weisbach equation is rewritten as
{\displaystyle {\frac {\Delta p}{L}}={\frac {128}{\pi }}\cdot {\frac {\mu Q}{D_{c}^{4}}},} {\displaystyle {\frac {\Delta p}{L}}={\frac {128}{\pi }}\cdot {\frac {\mu Q}{D_{c}^{4}}},}
where
μ is the dynamic viscosity of the fluid (Pa·s = N·s/m2 = kg/(m·s));
Q is the volumetric flow rate, used here to measure flow instead of mean velocity according to Q =
π
/
4
Dc2⟨v⟩ (m3/s).
Note that this laminar form of Darcy-Weisbach is equivalent to the Hagen-Poiseuille equation, which is analytically derived from the Navier-Stokes equations
Head-loss form
The head loss Δh (or hf) 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
{\displaystyle \Delta p=
ho g\,\Delta h,} {\displaystyle \Delta p=
ho g\,\Delta h,}
where
Δh is the head loss due to pipe friction over the given length of pipe (SI units: m);[b]
g is the local acceleration due to gravity (m/s2).
It is useful to present head loss per length of pipe (dimensionless):
{\displaystyle S={\frac {\Delta h}{L}}={\frac {1}{
ho g}}\cdot {\frac {\Delta p}{L}},} {\displaystyle S={\frac {\Delta h}{L}}={\frac {1}{
ho 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:[5]
{\displaystyle S=f_{\text{D}}\cdot {\frac {1}{2g}}\cdot {\frac {{\langle v
angle }^{2}}{D}}.} {\displaystyle S=f_{\text{D}}\cdot {\frac {1}{2g}}\cdot {\frac {{\langle v
angle }^{2}}{D}}.}
In terms of volumetric flow
The relationship between mean flow velocity ⟨v⟩ and volumetric flow rate Q is
{\displaystyle Q=A\cdot \langle v
angle ,} {\displaystyle Q=A\cdot \langle v
angle ,}
where:
Q is the volumetric flow (m3/s),
A is the cross-sectional wetted area (m2).
In a full-flowing, circular pipe of diameter {\displaystyle D_{c}} D_{c},
{\displaystyle Q={\frac {\pi }{4}}D_{c}^{2}\langle v
angle .} {\displaystyle Q={\frac {\pi }{4}}D_{c}^{2}\langle v
angle .}
Then the Darcy-Weisbach equation in terms of Q is
{\displaystyle S=f_{\text{D}}\cdot {\frac {8}{\pi ^{2}g}}\cdot {\frac {Q^{2}}{D_{c}^{5}}}.} {\displaystyle S=f_{\text{D}}\cdot {\frac {8}{\pi ^{2}g}}\cdot {\frac {Q^{2}}{D_{c}^{5}}}.}
Shear-stress form
The mean wall shear stress τ in a pipe or open channel is expressed in terms of the Darcy-Weisbach friction factor as[6]
{\displaystyle \tau ={\frac {1}{8}}f_{\text{D}}
ho {\langle v
angle }^{2}.} {\displaystyle \tau ={\frac {1}{8}}f_{\text{D}}
ho {\langle v
angle }^{2}.}
The wall shear stress has the SI unit of pascals (Pa)
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