Fluid Mechanics and Hydraulics
Core content
Continuity (A1v1 = A2v2 for incompressible flow), the energy equation with losses, and momentum balance are the backbone. Bernoulli: p1/γ + v1²/2g + z1 = p2/γ + v2²/2g + z2 + hL. Head loss from Darcy–Weisbach is hL = f(L/D)(v²/2g), plus minor losses ΣK v²/2g. Reynolds number Re = ρvD/μ: laminar below about 2000, turbulent above 4000; friction factor from the Moody chart or Colebrook, with f = 64/Re in laminar flow. Pumps: the intersection of system curve and pump curve gives the operating point; NPSHa must exceed NPSHr; shaft power = ρgQH/η.
Traps: using radius instead of diameter in Re; comparing pressure when the question asks for head; omitting velocity head when the diameter changes; assuming f is constant across flow regimes; applying Bernoulli across a pump or valve without the added-head or loss term; and confusing volumetric flow (m³/s) with mass flow (kg/s).
Worked example. Water flows at 10 L/s through a 100 mm bore pipe. Area = π(0.1)²/4 = 0.00785 m², so v = 0.010/0.00785 ≈ 1.27 m/s and velocity head v²/2g ≈ 0.083 m. Halve the diameter and the velocity quadruples to about 5.1 m/s, raising velocity head sixteen-fold to roughly 1.32 m - the reason reducers and high velocities are expensive in pumping energy.
Rule of thumb: size typical HVAC water circuits for 1–2 m/s; above 3 m/s expect noise, erosion and excessive pressure drop.