·New head equation for centrifugal pumps (additional version) \The Century Equation of Centrifugal Pumps: Euler’s Model Has Reached Its Limit!

· This paper proposes a major reformulation of Euler’s classical head equation for centrifugal pumps by integrating a synthetic head formula that reflects both the hydraulic limits and design insights. The proposed model reconstructs the H–Q curve as a superposition of two quadratic states, corresponding to Q = 0 and Q = Qₐₓ, with the final expression: H = Hₐₓ⋅(1 − Q/Qₐₓ)². This unifies empirical data and mechanistic understanding into a comprehensive equation that addresses legacy constraints and modern performance needs.