• The Darcy Weisbach equation relates the head loss due to frictionin a pipe to the velocity of the fluid ‘U’, friction factor ‘f’ and the diameter ‘d’ of the pipe.

• This is derived by considering a fluid Control Volume (CV) as shown in figure below

Now, Applying Bernoulli’s Principle between the inlet and exit of the control volume,

$\frac{P_1}{ρg}+\frac{v_1^{2}}{2g}+Z_1=\frac{P_2}{ρg}+\frac{v_2^2}{2g+Z_2+h_l$

But, $v_1=v_2=U and Z_1=Z_2$

∴$\frac{P_1}{ρg}+0+0=\frac{P_2}{ρg}+0+0+h_lf$

∴$\frac{P_1}{ρg}-\frac{P_2}{ρg}=h_lf$ (Head loss due to frictional drag)

∴$\frac{P_{1}-P_{2}}{ρg}=h_{lf}$ (Head loss due to frictional drag)

∴$(P_{1}-P_{2}A=ρgh_lf A=\frac{C_{D} ρU^{2} (πdl))}{2}$

∴$ρgh_lf (\frac{π}{4}) d^2 =\frac{C_D ρU^2 (πdl))}{2}$

Rearranging the above equation, we get,

∴$h_lf =\frac{4C_D lU^2}{2gd}$

Substituting $f=4C_D$ (Friction Factor)

∴$h_lf =\frac{flU^2}{2gd}$

…this is the Darcy Weisbach Equation.

Use of the Darcy Weisbach Equation: It is used to obtain the head loss within a pipe due to frictional resistance of a pipe. This equation is applicable to both laminar and turbulent flows.