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Pressure drop follows Darcy-Weisbach: deltaP = f L rho v^2 / (2 D g). The friction factor comes from the Reynolds number — laminar below 2300 uses 64/Re, and turbulent flow uses a Blasius-type correlation. Because pressure drop scales with velocity squared and inversely with diameter to the fifth power, diameter selection dominates pumping cost.
Darcy-Weisbach
deltaP = f x L x rho x v^2 / (2 x D x g)
Friction factor
Laminar: f = 64/Re; turbulent: f = 0.316 / Re^0.25
For fixed flow, velocity falls with the square of diameter and pressure drop with roughly the fifth power. Upsizing one pipe size can halve pumping power.
Whenever pipe roughness matters — in commercial steel pipe at high Reynolds number the friction factor becomes roughness-dominated, and the Blasius approximation under-predicts.