Flow Characteristics of Valves and Restrictors for Thrust Bearings According to a Geometrical Theory Based on Minor Losses

This study presents a unified geometrical-theory framework for predicting the flow-characteristics of nozzle-flapper valves and annular restrictors employed in thrust-bearing applications. Building on the minor-loss formulation introduced in a previous work, the approach separates geometric effects from discharge phenomena, enabling accurate regression of the flow-coefficient K across both incompressible and compressible regimes. The theory incorporates the simulation of choking and a blended discharge-coefficient model that transitions smoothly between exponential and Busemann forms. Parameter identification is performed with a Levenberg–Marquardt algorithm, calibrated against experimental data from the Hayashi (nozzle-flapper) and Belforte (aerostatic restrictor) test cases. Results demonstrate that the unified model reproduces the measured K trends—including the decline of K at high Reynolds numbers and the onset of choking—more faithfully than the conventional equation proposed in literature, while requiring far fewer computational resources than full computational fluid dynamics (CFD) simulations. Sensitivity analyses reveal the dominant influence on both geometric loss coefficients and discharge behavior of the design parameters tuned in the previous work, i.e., flow path parameter and Reynolds number. The validated framework offers a rapid, physics-based tool for designers of precision pneumatic systems and aerostatic bearings, facilitating early-stage sizing, performance optimization, and integration with Reynolds-equation-based load calculations without resorting to intensive numerical modeling.