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Real pipe systems experience head losses from two sources: major losses due to pipe wall friction (Darcy-Weisbach) and minor losses at fittings, valves, bends, and entrances/exits (h_minor = K·V²/2g, where K is a loss coefficient). The extended Bernoulli equation P₁/γ + V₁²/2g + z₁ = P₂/γ + V₂²/2g + z₂ + h_L accounts for all losses. Pipe networks (series, parallel, branching) require simultaneous satisfaction of continuity at junctions and pressure-drop compatibility around loops.
Solve single-pipe problems with both major and minor losses before tackling networks. For parallel pipes, note that pressure drop is equal across parallel paths but flow splits. Use the Hardy-Cross iterative method for complex networks, which systematically corrects flow guesses to satisfy energy compatibility.
Bernoulli's equation, which you know from prerequisites, describes an ideal fluid where no energy is lost: pressure, velocity, and elevation trade off perfectly, and the total head is conserved. Real pipe systems lose energy to friction and local disturbances. Head loss h_L is the quantity that accounts for this: it represents energy per unit weight dissipated by the fluid, and it appears as an additional term on the right side of the extended Bernoulli equation. The total head at the inlet equals the total head at the outlet *plus* all the losses incurred along the way.
Losses come from two sources. Major losses result from friction between the fluid and the pipe wall along the pipe's entire length. The Darcy-Weisbach equation quantifies them: h_f = f(L/D)(V²/2g), where f is the Darcy friction factor (which you get from the Moody chart using the Reynolds number and relative roughness), L is pipe length, D is diameter, and V²/2g is the velocity head. From your turbulent pipe flow work, you know that rougher walls and higher Reynolds numbers increase f, meaning more energy is lost per unit length. A key design insight: halving the diameter quadruples the velocity (from continuity) and increases h_f by a factor of 32 — diameter changes have dramatic effects on losses.
Minor losses arise at valves, bends, tees, contractions, and expansions — anywhere the flow is disturbed from uniform pipe flow. Each fitting is assigned a loss coefficient K, and the loss is h_m = K·V²/2g. Despite the name "minor," these can dominate. A partially closed gate valve can have K > 100, easily exceeding the friction loss in many meters of pipe. The total head loss in a system is the sum of all major and minor contributions, and a designer must account for both.
Pipe networks add another layer of constraint. In a series system, flow rates are equal and head losses add. In a parallel system, head losses across each branch are equal (both paths connect the same two pressure nodes) while flow rates add — the network distributes flow in inverse proportion to resistance. This is the hydraulic analog of electrical resistors in parallel. Real networks with loops and junctions require simultaneous satisfaction of continuity at every node and pressure compatibility around every loop; the Hardy-Cross method iteratively adjusts assumed flows until both are satisfied.
The practical workflow for pipe system design always starts with a sketch: identify source and destination pressures and elevations, enumerate every pipe segment with its L and D, and list every fitting with its K. Then write the extended Bernoulli equation from one end to the other, plug in the head losses, and solve for whatever is unknown — typically the flow rate, required pump head, or pipe diameter. The velocity appears in both major and minor loss terms, so for a known flow rate the solution is straightforward; for an unknown flow rate it requires iteration (since f depends on Re, which depends on V).