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I've come across two forms of the Darcy-Weisbach Equation, and each one seems to yield a different result when I am solving for $Q$ (flow).

$$h_{f}=\frac{\lambda Lv^{2}}{2gD}$$

$$h_{f}=\frac{f LQ^{2}}{3D^{5}}$$

How do these equations differ, and what (if any) is the relationship between $f$ and $\lambda$ ?

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I think Wikipedia explains it.

Lambda is also called the Darcy friction factor, and equals 4 times the Fanning friction factor, f. The two equations are equivalent, with the first written in the "pressure loss" form, and the second in the "head loss" form.

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The first equation uses the Darcy friction factor $\lambda$. The second one I don't know. You could rewrite the first equation. $$ v=\frac{Q}{A}=\frac{4Q}{\pi D^2}$$ $$ \to h_f=\lambda\frac{8Q^2L}{\pi^2gD^5}$$ If you equate the two $h_f$ and solve for $f$: $$ f=\frac{8\lambda}{3\pi^2g} $$ $f$ might be a different friction coefficient, that holds true in certain hydraulic conditions, as $\pi$, $g$ and a constant factor $\frac{8}{3}$ are included in $f$. But I have no clue, where exactly $\frac{1}{3}$ comes from.

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