In a closed-loop water distribution network, what fundamental law states that the algebraic sum of head losses around any closed loop must equal zero?

The fundamental law stating that the algebraic sum of head losses around any closed loop in a closed-loop water distribution network must equal zero is the principle of conservation of energy. This principle, when applied to fluid flow, dictates that energy cannot be created or destroyed, only transformed or transferred. In a fluid system, this means that the total energy of the water at any given point, represented by its total head, must be consistent throughout the system.

A closed loop refers to any continuous path of pipes within the network that begins and ends at the same physical point. Imagine tracing a route through the pipe system where you eventually return to your original starting location without repeating any pipe segment in the same direction.

Head loss is the reduction in the total energy of the water as it flows due to friction with the pipe walls, changes in direction, and resistance from valves and fittings. This lost energy is primarily converted into heat. Head is a measure of energy per unit weight of fluid and is typically expressed as an equivalent height of a column of water.

The algebraic sum means that all head changes—both losses due to friction and any gains from energy-imparting devices like pumps—are added together, taking into account their respective signs. When traversing a closed loop, if you start at a specific point with a certain total head (which includes pressure head, velocity head, and elevation head) and follow the loop back to that identical starting point, the total head must be the same. Consequently, any energy added to the water (a head gain, such as from a pump) must be exactly balanced by the energy dissipated (head losses due to friction) along that loop. Therefore, the net change in total head around any closed loop must be zero, ensuring that the total energy is conserved. For example, if water flows through a loop containing a pump and several pipe segments, the head gained by the pump must be precisely equal to the sum of all friction losses experienced by the water as it travels through the pipes and fittings in that loop to return to the initial energy state at the starting point.