Numerical Modeling and Experimental Validation of a Supercritical CO₂-Lubricated Hydrodynamic Journal Bearing

Supercritical carbon dioxide (sCO₂) turbines are increasingly being investigated for use in a variety of compact power generation applications for their high power density. Typical designs, such as the 10 MW sCO₂ Brayton cycle investigated by Sandia National Laboratory (SNL) [1], call for rotors operating near 1250 rev/s (75,000 rpm) and shaft diameters of the order of centimeters, necessitating the use of gas bearings due to CO₂'s much lower viscosity compared to liquid lubricants like oil. At these relatively large power levels and shaft diameters, gas foil bearings are preferable to tilt-pad bearings due to their greater ease of manufacturing and because machines of this size can tolerate the greater bypass leakage inherent to foil bearings compared to tilt-pad bearings, the SNL loop uses gas foil bearings with labyrinth seals and bottoming pumps to maintain the lubricant pressure in these bearings at around 1.379 MPa, well below the operating pressure of the system as their system encountered high parasitic bearing drag at high pressures. Conboy [2] reports that turbulence in the lubricant flow within foil thrust bearings causes increased drag at the operating conditions of the Sandia loop. Multiple studies have attempted to model the thermohydrodynamic properties of CO₂-lubricated bearings while considering fluid compressibility and real gas effects within the bearing resulting from a nonuniform pressure distribution. Xu and Kim [3] present a thermo-elastohydrodynamic analysis of a foil thrust bearing operating in CO₂ at 1.5 MPa and 300 °C using real gas properties and finds that, far from the critical point, the real gas effects are small compared to the effects of turbulence. Qin et al. [4] presents a computational model of a foil thrust bearing lubricated by CO₂ at the bearing operating conditions of the Sandia test loop and of air at atmospheric pressure. The three-dimensional computational fluid dynamics (CFD) model, based on the compressible Navier–Stokes equations and using a lookup table to determine fluid properties, indicates that the bearing behavior and fluid properties in high-pressure CO₂ differ from the ideal gas assumption by less than 10% and that the predicted bearing behavior in atmospheric pressure air closely matches experimental results.

Lubricating the bearings with sCO₂ at the system operating pressure would significantly reduce the overall turbomachinery complexity by eliminating the need for labyrinth seals and bottoming pumps between the bearing housing and the sCO₂ loop. For operation in this regime to be stable, bearings operating near the critical point must not encounter either hydrodynamic instabilities or significant degradation of the system performance. sCO₂ lubrication is particularly attractive for small- to midsize turbomachines (10's to 100's of kW) that might be used in space applications where reliability and simplicity are a paramount consideration and the reduction in efficiency due to windage that will be incurred by operating the bearings with the high-density working fluid may be acceptable. Because the carbon dioxide in the bearing cavity will potentially be in the pseudo-critical region, it is likely that the fluid properties will exhibit large changes with both pressure and temperature. Thatte et al. [5] examined the performance of a hybrid (hydrostatic/hydrodynamic) gas bearing lubricated with sCO₂ at 8 MPa and 500 °C using a coupled fluid–structure interaction model and validated their model with experimental results. However, this experiment was carried out using a bearing operating at 700 kPa in air, which is in the ideal gas region. The modeling indicated that, compared to a bearing lubricated by atmospheric air, the sCO₂ bearing had an enhanced hydrodynamic effect and a larger minimum film thickness, and stiffness followed the same scaling relationship with respect to rotor speed.

For a midsize space application, hydrodynamic tilt-pad journal bearings may perform better than foil bearings as the narrow clearance and more highly controlled surfaces prevent flow turbulence within the bearing and enable operation with less runout, while reducing bypass leakage, which increasingly degrades bearing performance as shaft diameter decreases. In laminar flow, the bearing drag is expected to follow Petroff's law [6] for a lightly loaded bearing and thus scale linearly with lubricant viscosity, which does not increase much with pressure in the subcritical region. Few studies have been conducted on hydrodynamic bearing operation in the near-critical or supercritical regime. Dousti and Allaire [7] use the compressible Reynold's equation to model isothermal sCO₂-lubricated bearings and find large deviations from the incompressible solution for a bearing operating in 320 K, 8 MPa CO₂ at 1000 rev/s and low parasitic loss compared to oil bearings. Chien et al. [8] also employed the Reynolds equation with an analytical equation of state to develop a thermohydrodynamic model of a hydrodynamic journal bearing lubricated with sCO₂. The results of the model were compared to a full Navier–Stokes solution and found to be reasonably accurate. The authors conclude that the incompressible Reynolds equation is adequate for modeling the pressure within the bearing. However, they also indicated that considering the highly nonlinear properties of the lubricant is essential near the critical point. No studies found in the literature have modeled sCO₂-lubricated hydrodynamic bearing stiffness or experimentally demonstrated hydrodynamic bearing operation in the near-critical or supercritical regime. The purpose of this work is to address this gap in the literature related to the theoretical and experimental performance of hydrodynamic bearings when the working fluid is near the critical point.

To investigate the scaling behavior of hydrodynamic bearing performance near the critical point, a numerical model is developed based on the compressible Reynolds equations. The goal of the model is to examine the bearing stiffness when the lubricant is near its critical point. In this regime, fluid properties like density and viscosity cannot be assumed constant, and sections of the bearing may experience local pressures that rise above the pseudo-critical point resulting in sudden changes in the fluid properties.

The geometry of the bearing that is considered in the analysis below is a two-dimensional converging wedge, shown in Fig. 1. The gap has an initial (inlet) clearance, H₀, of 10 μm. The clearance linearly decreases to 5 μm along the length of the bearing, L, which is 1 mm. The bearing clearance is small compared to a typical shaft radius for an sCO₂ turbine and therefore both sides of the bearing clearance are modeled as flat plates. One of the plates is moving with a velocity u_p of 6.28 m/s, which is equivalent to a 1 mm radius shaft rotating at 1000 rev/s. These parameters are approximately consistent with the experimental facility discussed in the experimental validation section and all results presented here are obtained using these parameters. x denotes the direction parallel to the fluid velocity and y denotes the direction perpendicular to fluid velocity.

The following assumptions are used to simplify the Navier–Stokes equations:

• body forces are negligible,

• the flow is laminar and steady,

• the no-slip condition exists at each bearing surface,

• the bearing gap is much smaller than the length of the bearing, so the geometry of the round bearing is approximated using Cartesian coordinates,

• the scale of the y-momentum equation is much smaller than that of the x-momentum equation so that the pressure does not vary across the thickness of the bearing (in the y-direction),

• the bearing is infinite in the transverse direction so there is no flow or flow gradient perpendicular to the direction of relative motion of the bearing surfaces, and

• the temperature is assumed to be constant throughout the domain as the heat generated from viscous dissipation is small compared to the heat capacity of the flow and so can be ignored. This assumption is supported by a calculation of the viscous dissipation below.

However, as shown by Chien et al. [9], which examines the compressible Reynolds equation in high-pressure gases near the critical point by comparing a numerical model to the full Navier–Stokes equations, the Reynolds equation is not valid at critical inlet conditions due singularities in the Prandtl number, specific heat, and thermal expansivity, which causes convection to be non-negligible and therefore the problem cannot be simply reduced to one-dimensional, i.e., $μ=μ(x,y)$ in Eqs. (1)–(4). As the temperature gradient across the channel is expected to be very small as shown in the calculations of viscous dissipation below, this effect is likely minimal for the geometry and bearing speed under consideration.

In order to estimate the pressure distribution over the length of the bearing for a given inlet temperature $T1$ and pressure $P1$⁠, the bearing is discretized into N nodes as shown in Fig. 1. The carbon dioxide viscosity and density are provided by the fluid property interpolation tables (FIT) software library developed by Northland Numerics [10]. FIT uses a piecewise biquintic interpolation of Helmholtz free energy on a temperature-density domain and calculates other thermodynamic properties using partial derivatives.

The pressure gradient at node 2 can be found using Eq. (6) and the process is repeated across the length of the bearing. A nonlinear root-finding algorithm is employed to find the inlet pressure gradient that causes the inlet and outlet pressures to be equal.

This study is primarily interested in the effect that the high fluid property gradients near the critical point may have on bearing performance. For instance, if the bearing inlet pressure is just below the critical pressure, the pressure within the bearing may locally rise above the critical pressure, causing a large increase in viscosity and density, as shown in Fig. 2, that would not be captured by a simpler, constant-property model (CPM).

Figure 3(a) shows the peak bearing Reynolds number for a range of inlet conditions near the critical point and Fig. 3(b) shows the peak bearing Reynolds number as a function of pressure at the critical temperature over a wider pressure range. The highest Reynolds number is 335 so the flow is expected to be laminar throughout the range of interest.

where $W$ is an external radial load on the bearing and $Z$ is the axial length of the bearing. Figure 4 illustrates the geometry discussed in Eqs. (12) and (13).

The fluid film is very thin and the bearing surfaces are not actually adiabatic, and so conduction is likely to dominate the temperature distribution in a real bearing. However, the above calculation is useful for supporting the isothermal assumption.

Figure 5(a) shows a map of the bearing frictional power loss for a range of inlet conditions near the critical point and Fig. 5(b) shows the bearing drag as a function of inlet pressure at the critical temperature for a wider range of pressures, for a bearing with an axial length of $L$ = 2.6 mm, the length of the bearings studied in the experimental section below. The power loss increases sharply by a factor of 1.5 near the critical pressure but only increases by approximately 50% from atmospheric pressure to 7 MPa, indicating that, for laminar operation, power loss should not be a strong function of inlet pressure for subcritical CO₂ except for very near the critical pressure. However, the magnitude of the calculated power loss is very low compared to the power rating for a turbomachine, which uses a bearing of this size, typically of the order of 1 kW, so journal bearing frictional power loss is not expected to be the significant source of losses in an operational sCO₂ turbomachine of this size.

Figure 6(a) shows a contour map of the temperature rise across the bearing associated with viscous dissipation across a range of inlet conditions near the critical point and Fig. 6(b) shows the temperature rise as a function of inlet pressure at the critical temperature for a wider range of pressures. The increased viscous dissipation shown in Fig. 5 is outweighed by the increased mass flow through the bearing because the fluid density scales more quickly with pressure than the fluid viscosity, so the temperature rise decreases with increasing inlet pressure for this geometry. The temperature rise from viscous dissipation is approximately 0.0001 K near the critical point and increases with decreasing inlet pressure below the critical point up to 0.4 K at atmospheric pressure due to the lower mass flow rate at those conditions.

Figure 7 shows that the numerical model produces the same pressure distribution as the analytical solution when implemented with constant properties for an inlet pressure of 7381.5 kPa and a temperature of 304.16 K, thus validating the modeling approach.

where $Pmax,cp$ is the peak pressure within the bearing assuming constant properties and $Pmax,vp$ is the peak pressure assuming variable properties. A contour map of the difference in maximum predicted pressure is shown in Fig. 8 for a range of conditions near the pseudo-critical line where the discrepancy is expected to be the largest.

This maximum deviation occurs at an inlet pressure of 7.382 MPa and a temperature of 304.16 K; the velocity, density, viscosity, and pressure along the length of the bearing are predicted by the VPM and CPM are shown in Fig. 9 at these conditions. In this worst-case scenario, the pressure rise predicted by the VPM exceeds the CPM prediction by approximately 12%. Therefore the macroscopic behavior of the bearing is not expected to vary dramatically from what would be expected using a constant-property analysis.

where $F2$ and $F1$ represent the total integral of the pressure distribution within the bearing in the displaced and initial geometry, respectively, and $ΔH0$ is the clearance decrease between the displaced and initial geometry. Figure 10 shows a plot of the stiffness and Fig. 11 shows a plot of the nondimensional stiffness predicted using the numerical model as a function of inlet temperature and pressure near the critical point. For inlet pressures more than approximately 2 kPa below the critical pressure, stiffness is fairly constant but decreases by approximately 20% between 1 and 2 kPa below the critical pressure. At critical inlet conditions, the stiffness sharply increases with inlet pressure until reaching a plateau that is approximately 50% higher than the subcritical stiffness. The nondimensional bearing stiffness decreases by up to 20% for inlet pressures approximately 1 to 3 kPa below the critical pressure, while for inlet pressures from 0 to 1 kPa below the critical pressure, the nondimensional bearing stiffness rises by 10%. At all conditions more than 3 kPa away from critical pressure, the constant-property assumption approximates the bearing behavior within 1%, as shown in Fig. 12, indicating that the CPM adequately predicts the bearing stiffness for those conditions.

These sharp variations are mostly the result of the large density gradients near the critical pressure. Figure 13 shows the pressure, viscosity, density, and mean velocity profiles along the bearing compared to the constant-property case for an inlet pressure of 7.3812 MPa and inlet temperature of 304.16 K, which is approximately 1.6 kPa below critical pressure. As seen in Fig. 13(a), the pressure rise within the bearing at these conditions is approximately 1.8 kPa so parts of the bearing will reach critical pressure. The region of the bearing in which the lubricant exceeds the critical pressure experiences a sharp increase in the viscosity and density as seen in Figs. 13(b) and 13(c). Equation (7) indicates that the pressure gradient increases directly with fluid viscosity and fluid velocity and inversely with the square of the bearing clearance, while fluid velocity is inversely proportional to fluid density and bearing clearance due to mass continuity, as stated in Eq. (9).

Therefore, when the bearing is compressed as described above to calculate stiffness, the decreased clearance relative to the uncompressed bearing causes an increased pressure gradient across the entire bearing due to the inverse dependence on clearance in Eq. (7); therefore, a greater portion of the bearing experiences supercritical pressure, as shown by in Fig. 13(a). The fluid velocity will be lower over a greater portion of the bearing in the compressed case due to the increased density in the supercritical regime, as shown in Fig. 13(c), thus decreasing the pressure gradient in the supercritical region of the bearing due to the pressure gradient's dependence on fluid velocity. In the near-critical inlet case shown in Fig. 13, the negative effect of the density increase outweighs the positive effect of the viscosity increase on the pressure gradient, as evidenced by the VPM predicting a lower maximum pressure than the CPM in both the compressed and uncompressed cases, shown in Fig. 13(a). As more of the bearing experiences supercritical pressure in the compressed case, the negative impact of the lubricant being supercritical on the pressure gradient counteracts the positive impact of the decreased clearance. Therefore, the difference in the bearing load between the compressed and uncompressed geometries, i.e., $F2−F1$ in Eq. (20), and hence the stiffness, is less than the CPM predicts.

Conversely, when the inlet pressure is just below the critical pressure, the nondimensional stiffness is greater than one, indicating that the CPM overpredicts the stiffness. In this case, for both the compressed and uncompressed geometries, the majority of the bearing experiences supercritical conditions and the portion of the bearing in the supercritical regime does not increase much when the bearing is compressed, as shown in Fig. 14. Because most of the bearing is in the supercritical regime while the inlet conditions are subcritical, the VPM predicts higher pressure gradients, and thus a higher bearing load, due to the higher viscosity throughout the bearing. In contrast, the CPM assumes that the inlet conditions persist throughout. Therefore, the decrease of the clearance between the uncompressed and compressed cases results in a larger change in the bearing load than is predicted by the CPM, as shown in Fig. 14(a).

It is possible that the abrupt change in the stiffness that occurs as the bearing transitions through the critical point may significantly affect bearing performance or in some way negatively affect the turbomachine taken as a whole. Additionally, the peak pressure occurs farther from the bearing inlet for near-critical inlet conditions, which might cause other rotordynamic issues. Rapid changes in bearing pressure are unlikely during normal operation but may result from an accidental depressurization of the sCO₂ loop. However, during normal operation, it is not expected that these small changes in stiffness would cause noticeable adverse behavior or instability.

Experiments were carried out to verify that hydrodynamic bearings operating in CO₂ in the region near the critical point continue to function without issue. These experiments also indirectly measured the drag coefficients of these bearings through coast-down tests at high subcritical pressures to determine if trends observed at lower pressures and bearing Reynolds numbers reported in the literature continue in this regime as the modeling predicts. These tests were performed at the University of Wisconsin-Madison Thermal Hydraulics Lab in an autoclave shown in Fig. 15.

The test section is constructed from a 2.5 in. schedule 80 316SS pipe with a 2.5 in. Grayloc flange welded to one end that has been hydrostatically pressure-tested to 24.13 MPa. A 316 stainless steel endcap allows for data and power transfer using high-pressure feedthroughs. A turbomachine assembly was provided by Creare LLC of Hanover, NH and is described in detail by Zaragola et al. [12]. The assembly consists of a shaft supported by two flexure-pivot self-acting tilt-pad hydrodynamic gas journal bearings. The bearing geometry and clearance are proprietary but nominally consistent with the geometry used in the model. The rotor and bearings both have a radius of 1 mm but the rotor lacks the aerodynamic features of a functional turbomachine. The test assembly includes an alternator that is used to spin the rotor via interaction with a permanent magnet mounted within the shaft. The magnetic interaction between the permanent magnet and the electrical stator also provides a weak axial centering force that is also used to maintain the axial location of the shaft. Capacitance-based shaft displacement probes are used to measure the instantaneous separation between the bearings and the shaft during operation to monitor speed and stability. The signals from these probes are passed to a signal conditioner and then to an oscilloscope and spectrum analyzer in order to allow the displacement to be examined in both the time and frequency domain in real-time. Two type-K thermocouples are installed in the facility. One is affixed to the turbomachine housing and the other is located far from the turbomachine to measure the ambient chamber temperature. A detailed rendering of the turbomachine assembly is shown in Fig. 16. Figure 17 shows a rendering of the turbomachine housed inside the test section. Wires running from the turbomachine through the feedthroughs carry probe and thermocouple data out to the signal conditioner and transmit electric power for the motor/alternator.

The ambient temperature sensor is used to provide feedback to a PID control system that energizes a tape heater wound around the test section in order to maintain a temperature of 308 K in the chamber during testing. Industrial grade (99%) CO₂ was pulled from a bottle by a high-pressure pump and fed into the test section. A pressure gauge provided feedback to a second PID controller which controlled the pump, allowing precise pressure control in the chamber. A vacuum pump was used to evacuate air from the test section prior to testing to ensure a nearly pure CO₂ atmosphere once refilled.

The test section was filled with CO₂ to atmospheric pressure. The turbomachine was activated at rotational speeds of 500, 1000, 1500, 2000, 2500, and 3000 rev/s. The temperature data were recorded at each rotational speed and the capacitance sensor trace was monitored for indications of instability. The pressure was then slowly (at a rate of less than 69 kPa/min) raised in 689 kPa (100 psi) intervals and the test procedure was repeated for pressures up to 5.62 MPa (800 psi).

When attempting to restart the turbomachine at 5.62 MPa, the sensor traces behaved erratically. Therefore, the pressure was reduced to 4.24 MPa (600 psi) at which point the turbomachine was able to start properly and the traces appeared normal. Pressure was then slowly increased while the rotational speed was held at a constant 1000 rev/s and the traces and spectrum analyzer were monitored for signs of instability. Every 345 kPa (50 psi), recordings of the inverter power and temperature were taken up to 7.343 MPa (1050 psi), where the pressure rise was stopped again because erratic capacitance probe data was observed. Again, when the pressure was reduced to 5.62 MPa the turbomachine could be restarted without issue and operated smoothly. The smooth restart of the turbomachine was evidence that no bearing contact had occurred as this likely would have damaged the bearing to the point of inoperability. Further investigation showed that erratic capacitance probe signals occurred while the turbomachine was not powered as well, providing further evidence that the issue did not arise from bearing instability but likely was related to an issue with the sensor itself. The test conditions are shown on a T–s diagram in Fig. 18 (the circled points indicate the test conditions at which erratic signals were observed).

After restarting the rotor at 5.62 MPa, coast down tests were performed to estimate the drag on the rotor. These tests were carried out by shutting off the drive signal at an initial rotational speed of 2000 rev/s. To measure the coast down time, the oscilloscope trace showing the capacitance probe data was recorded using high-speed video during the coast down. Once the frequency could no longer be determined from the oscilloscope, i.e., when the period became greater than the window width, the last obtainable frequency and the time between motor shut off and the time of the last obtainable frequency was recorded. Coast down test data are shown in Table 1 in the Results section.

The bearings were operated in a pressure and density regime beyond what has previously been reported in the literature for contactless hydrodynamic gas bearings and above that expected in a typical sCO₂ turbine [1]. Throughout the bearing tests, the turbomachine ran smoothly and showed no signs of instability such as half-frequency whirl from atmospheric pressure to 7.343 MPa, nearly reaching the critical pressure of CO₂ of 7.377 MPa at 308 K. This result supports the modeling work which indicates that operation in the near-critical regime should not introduce adverse bearing behavior. Due to the very localized nature of the effect of fluid property gradients on bearing stiffness, the maximum expected deviation from the stiffness predicted using constant properties is 0.1% for the test conditions.

Coast down test data are shown in Table 1 below along with the calculated coefficient of friction from the coast down data and the predicted coefficient of friction based on applying Petroff's law to the bearing geometry.

Figure 19 shows the coefficient of friction as calculated from the coast down times as a function of the Bearing Reynolds number and the system pressure. The coefficient of friction increases linearly with bearing Reynolds number throughout the test regime, although the prediction from Petroff's law predicts only a small increase proportional to the increased fluid viscosity. It is unclear what this discrepancy stems from, and may be caused by the turbulent drag on the turbomachine shaft or residual torque from the electric motor after power is cut off as our analysis assumes instantaneous loss of torque from the motor. As our data collection apparatus did not enable a time series of shaft angular speed to be recorded during the coastdown tests, we are unable to isolate transient effects adequately. Overall, these experiments demonstrate that gas hydrodynamic bearings operate up to bearing Reynolds numbers of 400 and near the critical point of CO₂. Also, bearing drag increases linearly with bearing Reynolds number and system pressure, which is consistent with existing data at lower pressures. However, we were unable to determine if stable operation was possible above the critical point due to testing limitations.

A simplified numerical model of a hydrodynamic journal bearing that accounts for variations in fluid properties caused by operating near the critical point was developed to assess the scaling relationships between inlet conditions and bearing performance in this regime. The model predicts that the sharp fluid property gradients that occur when transitioning from subcritical to supercritical pressure do not significantly affect bearing performance and that, for inlet conditions more than 3 kPa from critical, bearing behavior closely matches the behavior predicted by simpler models that assume constant fluid properties. At inlet pressures 0–1 kPa below the critical pressure, the predicted bearing stiffness increases by up to 20% compared to the CPM while for inlet conditions from 1 to 3 kPa below critical pressure, predicted bearing stiffness decreases by up to 15% compared to the CPM. Bearing drag is predicted to rise roughly linearly with inlet pressure in the subcritical regime resulting from increased fluid viscosity consistent with Petroff's law and increase by 50% at the critical pressure.

A pair of hydrodynamic journal bearings were mounted in a test chamber and operated in a high-pressure CO₂ atmosphere. No bearing instability was detected up to 7.34 MPa and bearing drag measurements from 0.1 to 5.62 MPa indicate that the drag increases linearly with pressure in this regime, but with a stronger dependence than predicted by Petroff's law. A more sophisticated data collection method that would enable a time series of shaft angular speed to be recorded during the coastdown tests could provide additional insight and eliminate sources of error in these measurements.

The modeling results indicate that fluid property gradients within the bearing arising from operation in near-critical inlet conditions should have only minor effects on bearing stability and are only non-negligible within 3 kPa of the critical pressure. The experimental data show that stable operation in CO₂ up to within 50 kPa of the critical pressure is possible and that the bearing drag scales with pressure in the same manner at these high-subcritical conditions as at low pressure, thus supporting the findings of the model. This work indicates that sCO₂ may function well as a lubricant for hydrodynamic gas journal bearings in the laminar regime. The use of sCO₂ bearings would greatly reduce the complexity of current turbomachinery assemblies used in sCO₂ loops, which require complex seals and bottoming pumps to maintain a low bearing pressure.

• U.S. Department of Energy (DOE) Nuclear Energy University Programs Graduate Fellowship Award No. DE-NE0008679 (Funder ID: 10.13039/100000015).

$C$ =

constant

$cp$ =

specific heat capacity (J/kg K)

$f$ =

coefficient of friction

F =

load supported by bearing (N)

$H$ =

bearing clearance (m)

$I$ =

moment of inertia (kg m²)

$k$ =

bearing stiffness (N/m)

$K$ =

constant

$k¯$ =

nondimensional bearing stiffness

$L$ =

bearing length (m)

$m˙$ =

mass flow rate (kg/s)

$P$ =

pressure (Pa)

$Q$ =

total frictional power loss per meter of axial bearing length (W/m)

$q′$ =

differential frictional power loss per meter of axial bearing length (W/m²)

$R$ =

bearing radius (m)

S =

specific entropy (kJ/kg K)

T =

temperature (K)

$u$ =

velocity (m/s)

$W$ =

bearing radial load (N)

x =

direction parallel to fluid flow

y =

direction perpendicular to fluid flow

$Z$ =

axial bearing length (m)

Greek Symbols

$μ$ =

dynamic viscosity (Pa·s)

$ρ$ =

density (kg/m³)

$T′$ =

torque per meter of bearing length (N/m)

$ω$ =

angular velocity (rad/s)

Subscripts

$B$ =

bearing

$cp$ =

constant properties

$f$ =

final

$m$ =

mean

max =

maximum

$p$ =

plane

$rad$ =

radial

$vp$ =

variable properties

$x$ =

x-direction

$y$ =

y-direction

$z$ =

z-direction

0 =

initial

1 =

value at node 1

100 =

time at which bearing speed reaches 100 rev/s

2 =

value at node 2

Nondimensional Numbers

Re =

Reynolds number (ρuH/μ)

Acronyms and Abbreviations Widely Used in Text

CPM =

constant-property model

FIT =

fluid property interpolation tables

sCO₂ =

supercritical carbon dioxide

VPM =

variable-property model