Derivation of a reinterpreted and generalized Bernoulli’s equation using the total chemical potential

The work presents a reinterpreted and generalized version of Bernoulli’s equation (BE), which gives consistent results for entire pipe flow system varying geometry and design, in line with empirical data and fundamental principles and laws of physics. Reinterpreted since the role played by the internal water pressure is crucially changed compared to BE and generalized since viscous pressure is included. The framework developed uses thermodynamics, i.e. chemical potential, which consequently implies that BE should be formulated as a pressure balance between external supply and internal demands. External supply is delivered e.g. by a pump whereas internal demands are represented by changes in dynamic-, hydrostatic-, and viscous pressure. The external-internal distinction together with the requirement that entire systems must be considered, changes the way BE should be interpreted and applied since it conventionally only considers local subsections of the system. Entire systems must be considered since external pressure supplied depends on internal flow resistance terms, which further depends on the entire system. The pressure terms should additionally be applied according to respective physical content. High dynamic pressure value (high speed) is e.g. not the cause of decrease in internal water pressure as conventionally assumed. The generalized BE will be especially useful for design and engineering purposes but could also impact on how the Euler- and Navier–Stokes equations should be interpreted and applied. A condition for the cavitation phenomenon is additionally discussed.