Deriving the governing equation for a shock damper to model the unsteady flow caused by sudden valve closure and sudden demand change

The focus of this work is on deriving the governing equations for the 'shock damper', an instrument used in water flow systems. The shock damper is a mechanical tool with a single degree of freedom system that includes a tank, connecting pipe, mass, spring and damper. This tool serves as a boundary condition for characteristic line equations. Additional equations are included by considering the conservation of mass, momentum and energy. This system of equations is then explicitly solved at each time step. In order to illustrate how the shock damper performs, two gravity feed systems with and without a damper are considered. In the first system, the control valve suddenly shuts, and in the second system, rapid demand change occurs in one of the nodes to impose a water hammer condition. Subsequently, unsteady flow parameters such as minimum/maximum flow velocity and pressure are evaluated and a sensitivity analysis is carried out. The results demonstrate that despite being simple and economically efficient, the shock damper is highly capable of moderating unsteady flow characteristics.