Steady incompressible turbulent flow passes through equal-length pipes of diameters $d_1,d_2$ in series. Their friction factor is $f=K\mathrm{Re}^{-n}$ with positive constants K,n. Neglect minor losses. The pressure-drop ratio $\Delta P_1/\Delta P_2$ is
Both pipes carry the same volume flow, so $V\propto d^{-2}$. Reynolds number scales as $Vd\propto d^{-1}$, hence $f\propto d^n$. Darcy pressure drop is proportional to $fLV^2/d\propto d^{n-5}$. Therefore $\Delta P_1/\Delta P_2=(d_1/d_2)^{n-5}=(d_2/d_1)^{5-n}$, A.
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