Load and loss for high-speed lubrication flows of pressurized gases between non-concentric cylinders

Abstract

We examine the high-speed flow of pressurized gases between non-concentric cylinders where the inner cylinder rotates at constant speed while the outer cylinder is stationary. The flow is taken to be steady, two-dimensional, compressible, laminar, single phase and governed by a Reynolds lubrication equation. Approximations for the lubricating force and friction loss are derived using a perturbation expansion for large speed numbers. The present theory is valid for general Navier–Stokes fluids at nearly all states corresponding to ideal, dense and supercritical gases. Results of interest include the observation that pressurization gives rise to large increases in the lubricating force and decreases in the fluid friction. The lubrication force is found to scale with the bulk modulus. Within the context of the Reynolds equation an exact relation between total heat transfer and power loss is developed.

Virial Approximation for Load and Loss in High-Speed Journal Bearings Using Pressurized Gases

“…The Reynolds equation is valid for steady, two-dimensional, compressible, laminar, single-phase flows at thermodynamic states sufficiently far from those of the thermodynamic critical point. Again, we refer the reader to [23,27] for details, derivations, and validation of these results. In the following, we summarize the principal results of [27], which form the basis for the present study.…”

“…Again, we refer the reader to [23,27] for details, derivations, and validation of these results. In the following, we summarize the principal results of [27], which form the basis for the present study. The configuration considered in this study is a two-dimensional, infinitely-long journal bearing as sketched in Figure 2.…”

“…Explicit expressions for the local density, pressure, temperature, and heat flux were given in terms of the bulk modulus, shear viscosity, thermal conductivity, thermal expansivity, Prandtl number, speed number, and the film thickness. In [27], Chien and Cramer used approximations, which are second-order in the speed number, to derive general expressions for the load, loss, and attitude angle for a two-dimensional journal bearing. For convenience, the relevant results of [27] are recorded in the next section.…”

“…In [27], Chien and Cramer used approximations, which are second-order in the speed number, to derive general expressions for the load, loss, and attitude angle for a two-dimensional journal bearing. For convenience, the relevant results of [27] are recorded in the next section. While the results are explicit, the results require the numerical evaluation of a number of integrals if the gas model is more complicated than that of an ideal gas.…”

“…We begin with the general formulas for the load, loss, and attitude angle derived in [27] for large speed number. These formulas were derived by finding the second-order approximations to the Reynolds equation of [23,24] and then integrating the results for pressure over the journal bearing configuration sketched in Figure 2.…”