The steady-flow mechanical energy equation (p₁/ρg + v₁²/2g + z₁ + H_pump = p₂/ρg + v₂²/2g + z₂ + H_turbine + H_loss) extends Bernoulli to include work interactions and irreversibilities. Pump head and turbine head represent useful work transfer; head loss represents energy dissipated as heat by viscous friction. This equation is the foundation for all piping system design.
You already know that Bernoulli's equation is an energy balance along a streamline for an ideal, inviscid fluid: pressure energy, kinetic energy, and potential energy trade off while their sum stays constant. But Bernoulli breaks down when the fluid passes through a machine (pump or turbine) or when friction is significant. The mechanical energy equation is the corrected version: it adds terms for work added by pumps, work extracted by turbines, and energy destroyed by friction — all expressed in the same units of length called head.
Head is the most important concept here. By dividing each energy term by ρg, you convert joules per kilogram into meters — a "height equivalent" of energy. Pressure head (P/ρg) is the height a column of fluid would reach if all pressure energy were converted to elevation. Velocity head (V²/2g) is the equivalent height for kinetic energy. Elevation head z is the actual height. Pump head H_pump is the mechanical energy added to the fluid per unit weight of fluid — it increases the total head at the pump discharge. Turbine head H_turbine is the energy extracted. Head loss H_loss is energy permanently destroyed by viscous friction and converted to heat; it always appears on the right side of the equation because you always lose it, regardless of which way you write the balance.
The equation p₁/ρg + V₁²/2g + z₁ + H_pump = p₂/ρg + V₂²/2g + z₂ + H_turbine + H_loss reads as: total head at inlet, plus any head added by a pump, equals total head at outlet, plus any head extracted by a turbine, plus all head losses in between. This is an accounting statement: every joule of energy that enters a control volume must go somewhere. To use it in a piping system problem, pick two points (usually where conditions are known, like tank surfaces), write the equation, and solve for the unknown — typically pump head, flow rate, or pressure at some point.
The power required by or delivered by a machine follows directly from the head: P = ρgQH, where Q is volumetric flow rate. This connects the hydraulic head concept back to the first-law open-system analysis you learned earlier — power is the rate of energy transfer. Real pumps and turbines have efficiencies less than 1, so the shaft power input to a pump is P_shaft = ρgQH_pump/η_pump, and the shaft power output from a turbine is P_shaft = η_turbine · ρgQH_turbine. Correctly applying this equation is what allows engineers to size pumps for water distribution systems, calculate hydroelectric power output, or determine whether a pipe network can deliver the required flow rate.