Hydraulic effort and the efficiencies of pump and motors with compressible fluid

When hydraulic fluid is compressible, the usual formula of hydraulic power being the product of pressure (p) and volumetric flow Q, which accounts only for the flow work, is not sufficient. By deriving an explicit formula for the stored compressible energy in accordance with the possibly pressure-dependent bulk modulus, a hydraulic effort (with symbol Φ) is defined to be the sum of the fluid pressure and the compressible energy density at that pressure. It is shown that Φ is the conjugate variable to volumetric flow (Q) such that that the compressible hydraulic power flow is the product Φ(p)Q. With proper understanding of compressible hydraulic power flow, the ideal relationships of pumps and motors with compressible fluid are then derived. These differ from incompressible formulae by replacing p by Φ and identifying an appropriate displacement. From these, definitions for mechanical efficiency, volumetric efficiency, and power efficiency are obtained. Unlike the previous attempts, these definitions are mutually consistent in that the product of mechanical and volumetric efficiencies is indeed the power efficiency. Furthermore, they are exact and are suitable even if the bulk modulus is pressure dependent. To gain insights into these new efficiency definitions, various modes of non-ideal effects in a piston pump/motor with variable valve timing are modeled and their effects on the mechanical, volumetric and power efficiencies are obtained. These non-ideal effects include piston-bore friction, leakage, valve throttling, and non-ideal valve timing.