Real-gas effects on aerodynamic bearings

Motivated by the use of aerodynamic bearings lubricated with high-pressure gases in energy conversion cycles, the Reynolds equation is adapted in order to include effects of real-gas and turbulence. Three geometries (Rayleighstep slider, plain and herringbone-grooved journal bearing) serve to investigate real-gas effects on the static and dynamic properties with a wide variety of lubricants and nondimensional operating conditions. Computational results show a depreciation of the load capacity of journal bearings, with cases reaching a reduction of 50% with unequally affected force components. Stability can be affected both positively and negatively. Some stability losses reach nearly 100%, while improvements of several orders of magnitude with the grooved bearing are reported. Results are fluid-independent for similar reduced pressure and temperature.


Introduction
that the real-gas consideration reduces the peak-pressure in the fluid film, 27 however in moderated proportions due to the high temperature of the fluid 28 film in the investigated cases. Fairuz and Jahn [9] investigated the real-gas 29 effects on dry seals using a bulk-flow model and found that the real-gas ef-30 fects were beneficial for the sealing properties thank to the increased density. 31 In general, prior work lacks of generalization regarding the role of both the 32 lubricant and the operating conditions. In addition, an in-depth description 33 of the effects of real gas on the bearing performance, and their consequences 34 on bearing design are missing.  terms is given as follows: The first two terms correspond to the Poiseuille flow contribution, whereas the third one includes the driving Couette flow. The last term describes the squeeze film effect. G ⊥ and G are the turbulence correction terms.
Correlations suggested by Constantinescu [10] are used in this work since they are valid for both smooth and discontinuous surfaces: where Re = U ρh/µ. This correlation is valid for 1000 < Re < 30000. For where ρ a corresponds to the density at ambient conditions, U the surface 69 velocity and h 0 the nominal clearance.

70
In the historical approach of solving this non-linear differential equation, 71 the ideal gas law P = ρrT is usually applied under the assumption of isother-72 mal compression [5], thus simplifying the approximation of a solution. In 73 order to account for real gas effects, ρ can be computed from the value of 74 P and T using a fluid database, which will indeed integrate the desired ef-75 fects, however, without highlighting the dominating parameters involved in 76 the real-gas behavior. Hence, the following substitution is proposed for the 77 pressure derivative terms in the Poiseuille flow terms of Reynolds equation: the parameter (∂ ρ P ) T is linked to the bulk modulus: The Reynolds equation is nondimensionalized using the following substitu-80 tions: leading to the nondimensionalized Reynolds equation that includes both 82 turbulence and real-gas effects (Figure A.1): Λ and σ are the compressibility and squeeze numbers respectively, defined as follows for journal bearings: Note that equation 8 is free from pressure terms and contains only the 84 density and the derivative term (∂ρP ) T incorporated in the bulk modulusβ.

85
This last term accounts for the real gas effects and equals 1 for an ideal gas: Furthermore, this parameter equals 0 for a real gas at its critical point. Under is solved iteratively starting from a guessed nondimensional density field, 98 usually assumed to be 1 for each node. At each iteration, the system of 99 equation is solved to find the density as follows: where 0 < κ ≤ 1 is a numerical relaxation coefficient easing conver-101 gence. The terms in matrix M ( ρ) are updated based on the last density 102 field. The terms (∂ρP ) T are computed for each node of the domain using a fluid database [13]. The iterative process is stopped when a convergence 104 threshold is reached: where N is the number of nodes in the domain. In order to compute the 106 bearing reaction forces, the resulting density field is converted to pressure as 107 follows: where the compressibility factor Z evaluated at each node by using the fluid 109 database.

110
In the case of journal bearings, stability is a major design objective that 111 has to be satisfied, which often dominates load capacity as design constraint. 112 A metric to characterize the bearing stability is the critical mass. Its com- coordinates is recalled: The clearance is perturbed by an infinitesimal harmonic motion 1x and 118 1y in the x and y directions respectively [14]: 119h =h 0 + 0x cos θ + 0y sin θ + 1x cos θe it + 1y sin θe it (17) where 0x and 0y are the static equilibrium eccentricity ratio. The harmonic 120 clearance perturbation results in a perturbation of the densityρ, the bulk 121 modulusβ and the turbulence correction terms: All terms of order higher than 1 are discarded and only terms of order 123 0 and 1 are retained. Equations 22 and 23 group zeroth-order terms and 124 first-order terms with respect to 1x . The first-order perturbed equation in 125 the y direction is obtained following the same method.
The zeroth-order equation is solved using the numerical procedure described The load capacity W is computed from the static pressure field from the reaction force acting on the bearing in both directions: The equations regrouping terms of first order are linear with respect to 134 the perturbed density. After discretization following a central finite difference 135 scheme, the systems of linear equations in the x and y directions can be 136 written as follows and solved straightforwardly: The perturbed pressure field is obtained from the density following as follows: The bearing impedances Z a,b = K a,b + iωC a,b are obtained as follows: The critical mass is computed by searching for the excitation frequency 140 ω canceling the imaginary part of the equivalent impedance Z of the system:

141
Im(Z(ω)) = 0 where Z is given as follows: At the particular excitation frequency ω c , the following is obtained: where M c is the critical mass of the system.

144
The same method for obtaining the static and dynamic properties can be 152 where the geometry is presented in Figure The boundary conditions areρ = 1 atx = 0 andx = 1, wherex = X/L x .

166
The lubricant is R134a (a typical heat pump working fluid) at reduced ambi-167 ent conditions corresponding to T r = T a /T c = 1 and P r = P a /P c = 0.5. The 168 pressure distributions for ideal gas and real gas are compared in Figure A.4. 169 The real gas consideration leads to a redistribution of the pressure, with a 170 lowered peak pressure compared to the pressure when the ideal-gas law is 171 assumed. For Λ = 3, the pressure of the real gas reaches a slightly higher 199 solution in real-gas lubrication based on the ideal-gas case.

201
This bearing configuration is simulated for a wide range of ambient con-202 ditions with the following parameter serving as a metric to compare the real-203 and ideal-gas load capacity:   conditions where real-gas effects yield to an increased stability compared to 294 the ideal-gas lubrication case.

295
The same behavior is observed for the 10 working fluids mentioned previ-296 ously. Figure