Computational Model Development for Hybrid Tilting Pad Journal Bearings Lubricated with Supercritical Carbon Dioxide

Featured Application

Power generation systems based on supercritical carbon dioxide.

Abstract

Fluid film bearings lubricated with supercritical carbon dioxide (sCO₂) eliminate the infrastructural requirement for oil lubricant supply and sealing in turbomachinery for sCO₂ power systems. However, sCO₂’s thermohydrodynamic properties, which depend on pressure and temperature, pose a challenge, particularly with computational model development for such bearings. This study develops a computational model for analyzing sCO₂-lubricated tilting pad journal bearings (TPJBs) with external pressurization. Treating sCO₂ as a real gas, the Reynolds equation for compressible turbulent flows solves the pressure distribution using the finite element method, and the Newton−Raphson method determines the static equilibrium position by simultaneously calculating forces, moments, flow rates of externally pressurized sCO₂, and pressure drop due to flow inertia. The finite difference method solves the energy equation for temperature distribution. The density and viscosity of sCO₂ are converged using the successive substitution method. The obtained predictions agree with the previous and authors’ computational fluid dynamics predictions, thus validating the developed model. Hybrid lubrication increases the minimum film thickness and stiffness up to 80% and 65%, respectively, and decreases the eccentricity ratio by up to 65% compared to those of pure hydrodynamic TPJB, indicating significant improvement in the load capacity. The bearing performance is further improved with increasing sCO₂ supply pressure.

1. Introduction

The supercritical carbon dioxide (sCO₂) cycle enables higher thermal efficiency than the conventional steam Rankine cycle, resulting in reduced fuel cost and greenhouse gas production [1]. Thus, the sCO₂ cycle is extensively applied in various power-conversion systems, including concentrated solar thermal and nuclear power [2,3,4,5], waste heat recovery from large engines [6], and geothermal [1].

The high density of sCO₂ at high-pressure can cause compact turbomachinery, critical equipment in circulatory systems for the sCO₂ thermodynamic cycle, at low capital costs [2]. To achieve commercial viability, various types of bearings have been investigated to support the rotors of such high-speed turbomachinery.

Wright et al. [2] replaced angular contact ball bearings initially installed in a turbocompressor for the sCO₂ Brayton cycle with gas-foil bearings (GFBs). The GFBs lubricated with sCO₂, enabled a simple configuration without the sealing mechanism to prevent oil contamination and ensured a long bearing life. During preliminary tests, the authors found that high-pressure sCO₂ in the foil thrust bearing generates significant power loss, which is attributed to the rotational speed. Cho et al. [7] employed GFBs to support the shaft in their 10-kWe-class unrecuperated sCO₂ Brayton cycle [7]. Preliminary operations were successfully conducted at 30 krpm and at a turbine inlet temperature and pressure of 83 °C and 8500 kPa, respectively. Cho et al. [8] also reported a 60-kWe-class recuperated Brayton cycle facility with turbomachinery supported on oil-lubricated tilting-pad bearings. To prevent oil contamination of the process fluid (sCO₂), carbon ring-type mechanical seals are fixed between the turbine and the bearing unit, resulting in a long shaft. The above-cited studies focused on cycle analysis rather than the constituents. Although Ahn et al. [9] and Utamura et al. [10] studied the design of the experimental loop for sCO₂ power cycles, they did not detail the bearings employed for turbomachinery.

To achieve the desired performance of turbomachinery for sCO₂ power cycles, the performance of their bearings is an important design consideration [10]. Thus, researchers have developed computational models to predict the performance of various types of sCO₂-lubricated bearings. Conboy [11] used an isothermal and real gas model to investigate the static performance of a sCO₂-lubricated gas-foil thrust bearing (GFTB). The author used the thermodynamic properties of sCO₂ reported in Ref. [12] and showed that GFTB has a larger load capacity and power loss when lubricated with sCO₂ than with air. Qin et al. [13,14] investigated the static performance of sCO₂-lubricated GFTBs using computational fluid dynamics (CFD). Dousti and Allaire [15] studied a plain sCO₂-lubricated journal bearing using a density model linearized with pressure. The isothermal model considers pressure-dependent density on only the Reynolds equation’s left-hand side (Poiseuille terms). The overly simplified compressible model predicted a greater load capacity of the bearing than did the incompressible model. Heshmat et al. [16] assumed sCO₂ as an isothermal ideal gas and analyzed a gas-foil journal bearing (GFJB) with the same diameter and length as the journal bearing reported in Ref. [15]. The authors predicted a comparable load capacity of GFJB for the plain journal bearing. In addition, they reported that the load capacity obtained using the compressible model is higher than that obtained using the incompressible model, which is inconsistent with the results reported in Ref. [15]. Dimond and Allaire [17] studied sCO₂-lubricated tilting pad journal bearings (TPJBs). The predictive model neglects the most crucial parameters associated with sCO₂, such as the density and viscosity dependency on pressure and temperature. The authors compared oil- and sCO₂-lubricated TPJBs and discovered that using sCO₂ as a TPJB lubricant requires increased bearing size to have dynamic coefficients comparable to those of oil-lubricated TPJBs. Xu and Kim [18] developed a thermoelastohydrodynamic model for CO₂-lubricated externally pressurized GFTB. The lubricant, CO₂, was treated as a real gas. The authors compared laminar and turbulent flow models for CO₂- and R245fa-lubricated bearings. Preuss [19] analyzed hybrid journal and thrust bearings lubricated with sCO₂ and calculated the load capacity using the rule of thumb. To calculate the rotordynamic coefficients, the author used a commercial tool developed by San Andres and Childs [20] for water-lubricated journal bearings and reported that the stiffness coefficient increases with an increase in the bearing diameter for angled injection hybrid journal bearing. Chunxiao et al. [21] reported a detailed dynamic analysis of sCO₂-lubricated plain journal bearings. The turbulent flow model employs the look-up table method to extract the thermodynamic properties of sCO₂ in Ref. [12], similar to that in Ref. [11]. Kim et al. [22] conducted theoretical and experimental studies on the rotor dynamic instability of magnetic journal bearings operating in an sCO₂ ambiance. To understand the unstable shaft levitation at high speeds, sCO₂ pressure forces developed between the shaft and the magnetic bearings were calculated using the one-dimensional Reynolds equation.

Herein, we developed a computational model for hybrid (hydrostatic/hydrodynamic) TPJB lubricated with sCO₂ and predict its static and dynamic performances. In the hybrid TPJB, pressurized sCO₂ is supplied through a hole located at the center of the pad’s recess to enhance the load capacity and dynamic characteristics, as reported for water- and air-lubricated bearings [20,23,24]. The Reynolds equation for turbulent compressible fluid flows solves hydrodynamic pressure generation. The mass flow balance between the orifice flow and the Poiseuille and Couette flows at the pad recess boundaries calculates the recess pressure. The fluid inertia effects at the recess boundaries, which cause a pressure drop, were modeled using the average velocities of film flows. The real gas model is used for sCO₂, whose thermophysical properties are functions of pressure and temperature.

2. Hybrid TPJB Lubricated with sCO₂ and Its Computational Model

Figure 1 shows a schematic of TPJB with four pads and an angular extent of θ_pad in the load-between-pad (LBP) configuration. Each pad has a rectangular recess with a hole at its geometric center to supply pressurized sCO₂ through the pivot support. The supplied sCO₂ flows into the gap between the rotating journal and the pads.

2.1. Thermohydrodynamic Model for Turbulent Compressible Fluid Flows

The Reynolds equation for turbulent and compressible fluid flows governs pressure distribution in the film land area. Note that sCO₂ is a highly compressible fluid, and its large density and low viscosity result in a large Reynolds number:

$$\frac{\partial }{\partial x}\left(\frac{G_x\rho h^3}{\mu }\frac{\partial p}{\partial x}\right)+\frac{\partial }{\partial z}\left(\frac{G_z\rho h^3}{\mu }\frac{\partial p}{\partial z}\right)=\frac{U}{2}\frac{\partial \left(\rho h\right)}{\partial x}+\frac{\partial \left(\rho h\right)}{\partial t}$$

(1)

where ρ, h, µ, p, U, t, x, and z denote the lubricant density, film thickness, viscosity, pressure, journal surface speed, time, circumferential coordinate, and axial coordinate, respectively. G_x and G_z are the turbulent coefficients empirically determined using the Hirs bulk flow model [25], which considers the inertia effects inherent to the flow turbulence:

$$G_x=\frac{1}{2+m_0}{G}_z, G_z=\frac{2^{1+m_0}}{n_0}R{e}^{-(1+m_0)}$$

(2)

where m₀ (=−0.25) and n₀ (=0.066) are empirical constants applicable exclusively to smooth surfaces. Re is the local flow Reynolds number dependent on the local density ρ, viscosity µ, and film thickness h:

$$Re=\frac{\rho Uh}{\mu }$$

(3)

The flow becomes turbulent when Re is greater than or equal to the critical Reynolds number Re_c, which is equal to 1000 [15,26,27]. For Re less than the critical value, G_x and G_z are equal to 1/12 for the laminar flow.

The equation of film thickness is modified from that in Ref. [28] by including the recess depth D_r to account for the depth of the recess:

$$h=C_p+e\cos \left({\theta }_c-\theta \right)-\left(r_p-\zeta \right)\cos \left(\theta -{\theta }_p\right)-\delta \left(R_p+t_p\right)\sin \left(\theta -{\theta }_p\right)+D_r$$

(4)

where C_p (=R_p − R_j), e, θ_c, θ, r_p (=R_p − R_b), ζ, θ_p, δ, t_p, and D_r are the pad radial clearance, journal eccentricity, angle between the line passing through the bearing and journal centers and the horizontal axis X, circumferential coordinate starting from the axis X, preload, pivot deflection, pivot circumferential location, pad tilt angle, pad thickness, and depth of the recess, respectively. R_p, R_b, and R_j are the pad, bearing, and journal radii, respectively. The mathematical model for the pivot deflection ζ is detailed in Refs. [28,29,30]. Note that D_r is null at the film land region.

In the recess region, externally pressurized sCO₂ with pressure p_s is provided through the orifice restrictor, causing a pressure drop from p_s to p_re, which then flows over the land area on each pad’s surface. The gaseous mass flow rate through the orifice restrictor Q_o for the choked and unchoked conditions are derived from the Bernoulli’s equation considering mechanical energy balance between two arbitrary points on a streamline. The relations for both choked and unchoked conditions can be given from [18,31] as follows:

$$\begin{array}{l}\mathrm{Un}-\mathrm{choked}: \frac{p_{re}}{p_s}>{\left(\frac{2}{\kappa +1}\right)}^{\raisebox{1ex}{$\kappa $}\!\left/ \!\raisebox{-1ex}{$\left(\kappa -1\right)$}\right.}\\ Q_o=C_d{A}_o\sqrt{R_g{T}_s}\sqrt{\frac{2\kappa }{\kappa -1}\left[{\left(\frac{p_{re}}{p_s}\right)}^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$\kappa $}\right.}-{\left(\frac{p_{re}}{p_s}\right)}^{\raisebox{1ex}{$\left(\kappa +1\right)$}\!\left/ \!\raisebox{-1ex}{$\kappa $}\right.}\right]}\\ \mathrm{Choked}: \frac{p_{re}}{p_s}\le {\left(\frac{2}{\kappa +1}\right)}^{\raisebox{1ex}{$\kappa $}\!\left/ \!\raisebox{-1ex}{$\left(\kappa -1\right)$}\right.}\\ Q_o=C_d{A}_o\sqrt{R_g{T}_s}\sqrt{\frac{2\kappa }{\kappa +1}}{\left(\frac{2}{\kappa +1}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\left(\kappa -1\right)$}\right.}\end{array}$$

(5)

where C_d, A_o, R_g, and T_s are the discharge coefficient, orifice area, gas constant, and supply temperature, respectively, and κ is the ratio of the specific heat capacity at constant pressure c_p to that at constant volume c_v.

The mass flow balance between the orifice mass flow (M_o = ρQ_o) and Poiseuille and Couette flows at the recess’s circumferential leading (Γ_i) and trailing (${\Gamma }_{i^{\prime }}$) boundaries and axial leading (Γ_j) and trailing (${\Gamma }_{j^{\prime }}$) boundaries gives recess pressure (p_re):

$${\displaystyle \underset{{\Gamma }_i\cup {\Gamma }_{i^{\prime }}}{\overset{}{\int }}\left(\frac{G_x\rho h^3}{\mu }\frac{\partial p}{\partial x}-\frac{\rho Uh}{2}\right)dz}+{\displaystyle \underset{{\Gamma }_j\cup {\Gamma }_{j^{\prime }}}{\overset{}{\int }}\left(\frac{G_z\rho h^3}{\mu }\frac{\partial p}{\partial z}\right)}dx=M_o$$

(6)

The pressure drop due to inertia forces at the boundary edges of the recess is modeled based on Bernoulli type relationships [32,33]. These relationships were derived for step bearings by assuming inertia effect dependency on average velocity instead of local velocity. The equations for turbulent flows can be given as follows:

Circumferential direction:

$$\begin{array}{l}{p}_{re}-p_{dc}=\frac{\rho }{2}{Q}_l{}^2\left(\frac{1}{h_l{}^2}-\frac{1}{h_{re}{}^2}\right)+0.412\rho Q_l{}^2\left(\frac{{\delta }_l}{h_l{}^2}-\frac{{\delta }_{re}}{h_{re}{}^2}\right)-1.725\rho V^2\left({\beta }_l-{\beta }_{re}\right)\\ \mathrm{where} \beta =\frac{0.885}{R{e}^{0.367}}, \delta =\frac{1.95}{R{e}^{0.43}}\hspace{1em}\hspace{1em}l:\mathrm{film} \mathrm{land}, \hspace{1em}re:\mathrm{recess}\end{array}$$

(7)

Axial direction:

$$p_{re}-p_{da}=\frac{\rho }{2}{Q}_l{}^2\left(\frac{1}{h_l{}^2}-\frac{1}{h_{re}{}^2}\right)$$

(8)

where p_dc and p_da are the pressures at the entrances of the film land in the circumferential and axial directions, respectively. Q_l, h_l, and h_re are the flux, film thickness at the land area, and film thickness at the recess area, respectively.

The numerical scheme to find the recess pressure (p_re) and pressure drops (p_re-p_dc, p_re-p_da) at the entrance boundary of the film land for a given supply pressure (p_s) is explained in the next section.

The energy equation calculates the temperature distribution T. In the energy equation, thermal energy transport due to fluid flow advection and heat convection into the bearing and journal surfaces balance the compression work and frictional power dissipation as follows [34]:

$$\begin{array}{l}{c}_p\left[\frac{\partial (\rho hVT)}{\partial x}+\frac{\partial (\rho hWT)}{\partial z}\right]+h_{tb}\left(T-T_b\right)+h_{tj}\left(T-T_j\right)={\beta }_{t}hT\left[V\frac{\partial p}{\partial x}+W\frac{\partial p}{\partial z}\right]+\frac{hU}{2}\frac{\partial p}{\partial x}\\ +\frac{\mu }{h}\left[\frac{1}{G_x}\left(V^2+W^2+\frac{U}{2}V\right)+\frac{U}{G_x}\left(\frac{U}{4}-V\right)\right]\end{array}$$

(9)

where V and W are the bulk flow velocities in the circumferential and axial directions, respectively, and h_tb and h_tj are the convective heat transfer coefficients for the bearing surface at temperature T_b and journal at temperature T_j, respectively. β_t is the volumetric expansion coefficient [34], which can be calculated using the following relation:

$${\beta }_t=-{\left(\frac{1}{\rho }\frac{\partial \rho }{\partial T}\right)}_{\rho }$$

(10)

Notably, β_t is 0 for incompressible liquids and 1 for ideal gases. The thermal energy mixing model for the inlet flow between adjacent pads follows Ref. [28].

2.2. Real Gas Model for sCO₂

CO₂ is a real gas whose density and viscosity vary with temperature and pressure, particularly in the supercritical region. Therefore, the current study takes the density and viscosity models proposed by Wang et al. [35] and Fenghour and Wakeham [36], respectively, to predict the pressure and temperature-dependent density and viscosity of CO₂.

The empirical model for calculating the density of CO₂ is valid for pressure and temperature ranging from 3 to 60 MPa and from 303 to 473 K, respectively. The density can be expressed as follows:

$$\begin{array}{l}\rho =\left(a_1{T}_r{}^3+a_2{T}_r{}^2+a_3{T}_r+a_4\right)p_r{}^6+\left(b_1{T}_r{}^3+b_2{T}_r{}^2+b_3{T}_r+b_4\right)p_r{}^5+\left(c_1{T}_r{}^3+c_2{T}_r{}^2+c_3{T}_r+c_4\right)p_r{}^4\\ +\left(d_1{T}_r{}^3+d_2{T}_r{}^2+d_3{T}_r+d_4\right)p_r{}^3+\left(e_1{T}_r{}^3+e_2{T}_r{}^2+e_3{T}_r+e_4\right)p_r{}^2+\left(f_1{T}_r{}^3+f_2{T}_r{}^2+f_3{T}_r+f_4\right)p_r\\ +\left(g_1{T}_r{}^3+g_2{T}_r{}^2+g_3{T}_r+g_4\right)\end{array}$$

(11)

where T_r (=T/T_c) and p_r (=p/p_c) are the ratios of the local temperature and pressure to the temperature and pressure at the critical point of CO₂, respectively. Note that T_c = 304.1 K, and p_c = 7.37 MPa. See Ref. [35] for the values of the coefficients (a_i, …, g_i, i = 1, …, 4). Note that the value of c4 is mistyped in Ref. [35] and the correct value is −23259.58953 [15].

The viscosity equation is also empirically derived, which models CO₂ as a function of temperature and density:

$$\begin{array}{l}\mu =\frac{1.00697\sqrt{T}}{G^{*}\left(T^{*}\right)}+l_{11}\rho +l_{21}{\rho }^2+\frac{l_{61}{\rho }^6}{T^{*3}}+l_{81}{\rho }^8+\frac{l_{82}{\rho }^8}{T^{*}}\\ \ln G^{*}\left(T^{*}\right)={\displaystyle \sum _{n=0}^4{\lambda }_n{\left(\ln T^{*}\right)}^n},\hspace{1em}{T}^{*}=\frac{\kappa T}{\epsilon },\hspace{1em}\frac{\epsilon }{\kappa }=251.196\end{array}$$

(12)

where T* is the reduced temperature. The coefficients l₁₁, l₂₁, l₈₁, l₈₂, and λ_{n = 0…4} can be found in Ref. [36].

2.3. Boundary Conditions for Pressure and Temperature

The pressure at all the edges of the pad and the pad’s backside is equal to the ambient pressure p_a. The recess pressure p_re is constant throughout the recess region owing to the large recess depth relative to the film thickness in the land area [37]. The bearing and journal surface temperatures are set equal to the supply lubricant temperature [38]. The film flow temperature at the inlet of each pad is calculated using the thermal energy mixing model [28], which considers the mixture of the hot lubricant from the upstream pad with the cold (supplied) lubricant.

3. Numerical Procedure

The current computational model and numerical scheme are based on those for hydrodynamic TPJBs considering pivot stiffness in Ref. [28], which is extended further to consider hydrostatic pressurization at the recess region, flow turbulence, and pressure/temperature-dependent thermochemical properties of the lubricant (sCO₂). The finite element method solves the Reynolds equation for pressure, the finite difference method solves the energy equation for the temperature distribution, and the Newton–Raphson method determines the equilibrium position. Figure 2 shows a flowchart of the numerical procedure.

Each step in Figure 2 is briefly explained as follows:

• The bearing geometry, lubricant properties at the initial condition, the initial guesses for the pad tilting angle δ, pivot deflection ζ, eccentricity ratio ϵ, the angle between the X-axis and the line of the centers θ_c, and the pressure ratios p_ratio (p_re/p_s), p_ratiod (p_dc/p_s), p_ratioa (p_da/p_s) are provided as the input. For the initial guess for δ, Equation (4) assumes zero film thickness at the trailing edge of a pad, which indicates the maximum δ. Half of the maximum δ is then taken as the initial guess for δ of each pad. The initial guess for ζ is 0.1% of the bearing clearance. Similarly, the initial guess for ε, p_ratio, p_ratiod, and p_ratioa must be positive and less than 1, and θ_c is guessed to be 90°, indicating that the rotor is displaced only in the vertical direction.
• The finite element method solves the Reynolds equation to compute the pressure distribution over each pad. The generated hydrodynamic pressure exerts a force F_pad on the surface of the pad, which deflects the supporting pivot and gives rise to a restoring force F_p in it [28]. As shown in Equation (13), TPJB with the number of pads N_pad in the static equilibrium requires that F_p balance F_pad all the moments and bearing horizontal force F_X balance to zero, the bearing vertical force F_Y balance the static load F_p0, and the orifice flow rate Q_o balance the summation of the flow rates at the recess boundaries Q_r. The circumferential and axial pressure drops at the recess edges due to inertia are also calculated iteratively as follows:

  $$\left\{\begin{array}{l}{\left[F_{pad}-F_p\right]}_{n=1\dots N_{pad}}\\ M_{1\dots N_{pad}}\\ {\displaystyle \sum F_X}\\ {\displaystyle \sum F_Y-}{F}_{p0}\\ {\left[Q_r+Q_o\right]}_{n=1\dots N_{pad}}\\ {\left[p_{d{c}_i}-p_{d{c}_{i-1}}\right]}_{n=1\dots N_{pad}}\\ {\left[p_{d{a}_i}-p_{d{a}_{i-1}}\right]}_{n=1\dots N_{pad}}\end{array}\right\}=\left\{\begin{array}{l}0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\end{array}\right\}$$

  (13)

  where i and i − 1 denote the current and previous iterations, respectively. The Newton–Raphson method calculates the equilibrium position.
• Once the static equilibrium position converges, the thermal energy mixing model calculates the fluid’s inlet temperature at the pad’s leading edge. The control-volume finite difference technique with an upwind scheme solves the two-dimensional energy equation (Equation (9)) for the temperature distribution (see Ref. [34] for more details).
• The converged pressure and temperature are then used to calculate the density and viscosity.
• Once the density and viscosity are converged iteratively using the successive substitution method, the journal is perturbed to a new position, and static equilibrium is achieved again.
• The stiffness and damping coefficients are calculated from the ratio of the difference between the forces at the new and old equilibrium positions to the difference in distance and velocity, respectively, at both points. Note that the squeeze velocity term given on the right-hand side of the Reynolds equation accommodates the calculation of stiffness and damping coefficients.

4. Model Validations

The experimental and theoretical performance of hybrid TPJBs lubricated with SCO₂ has not yet been reported. Therefore, we performed comprehensive model validations by comparing the obtained model predictions to the reported predictions or test data available in the literature for density and viscosity models of SCO₂, load-carrying performance of SCO₂-lubricated hydrodynamic journal bearings, and the load-carrying performance of a multi-recess hybrid journal bearing lubricated with low viscosity fluid, including water, operating with significant flow turbulence.

4.1. Density and Viscosity of sCO₂

Figure 3 compares (a) density and (b) viscosity predictions plotted against pressure at a temperature of 37℃ to the predictions from National Institute of Standards and Technology (NIST) [12]. Note that the predicted data from NIST are the most reliable for bearing lubrication models [11,21]. Both density and viscosity increase nonlinearly with an increase in pressure. The density and viscosity calculated using Equations (11) and (12), respectively, agree well with the predictions in Ref. [12].

4.2. Fixed-Pad Hydrodynamic Journal Bearings

Figure 4 compares the predicted static load versus eccentricity ratio for an sCO₂-lubricated plain journal bearing to the predictions in Ref. [16] with incompressible and compressible fluid flows. The analyzed bearing has an axial length and a diameter of 40 mm (L/D = 1) with a radial clearance of 40 µm, and the rotating speed is 60 krpm [16]. CFD model predictions using ANSYS FLUENT software are also compared with both predictions. In the CFD model, the ANSYS design modeler is used to create the fluid film model. The fluid film is meshed with 10 divisions in the radial direction and 900 divisions in the circumferential direction, and it has an element face size of 0.4 mm in the axial direction, resulting in 900,000 hexahedron elements. The interfaces attached to the bearing and journal sides are specified as stationary and rotating no-slip walls, respectively. Three-dimensional (3D) compressible Reynolds-averaged Navier–Stokes (RANS) with an SST k-omega turbulence model is solved using a coupled numerical scheme until the predefined convergence criteria are reached. Note that the CFD tool also uses Equations (11) and (12) to model the density and viscosity of sCO₂, respectively. The static load increases nonlinearly with the eccentricity ratio for all predictions. Compressible fluid flow models show higher load capacity than the incompressible ones, particularly for high eccentricity ratios, because the density and viscosity of compressible fluids (sCO₂) increase with an increase in hydrodynamic pressure under heavy loads (Figure 3). The current bulk flow model predictions agree well with those obtained using CFD. For the compressible fluid flow model, the discrepancies between the current model predictions and those reported in Ref. [16] can be attributed to the difference in the density and viscosity models. Reference [16] treats sCO₂ as an isoviscous ideal gas, whereas the proposed model considers it a real gas with temperature- and density-dependent viscosity. The mean square errors for the current predictions and those in Ref. [16] are 0.13% and 0.35% for incompressible and compressible fluids, respectively.

4.3. Fixed-Pad Hydrostatic and Hybrid Journal Bearings

The test data for the water-lubricated, three-recess hybrid fixed-pad journal bearing reported in Ref. [39] validate the proposed bulk flow model for hydrostatic/hydrodynamic hybrid operations. The analyzed bearing has an axial length and a diameter of 80 mm (L/D = 1) with a radial clearance of 125 µm. The circumferential angle and width of the recess are 90° and 50 mm, respectively, and the recess depth is 10 mm [39]. The low viscosity of water may lead to flow turbulence at high operating speeds and significant fluid inertia at recess edges. Figure 5 compares the predicted static load plotted against the eccentricity ratio with the test data from Ref. [39] at 0 rpm, i.e., pure hydrostatic operation. In general, the predictions agree well with the test data, with a mean square error of 0.3%.

Figure 6 shows the predicted centerline pressure ratio (P/P_s) compared to the test data from Ref. [39] at rotor speeds of 5 and 8 krpm with a static load of 356 N applied at 270°. Note that the figure shows one-third of the bearing geometry and film pressure between two externally pressurized flow supply tubes located circumferentially at 210° and 330°. The predicted pressure ratios are compared to test data measured at 11 locations from 210° to 310° with an increment of 10°. The film pressure ratio is higher at the recess regions. The sudden pressure drop at the entrance of the film land is attributed to the increase in the flow inertia with an increase in rotor speed. The predictions agree well with test data with mean square errors of 0.04% and 0.05% for 5- and 8-krpm rotor speeds, respectively, thus validating the proposed bearing model for hybrid operations.

5. Results

Table 1. Tilting-pad journal bearing (TPJB), pivot, and recess geometries and lubricant properties.

Parameters		Values
	Load configuration	LBP
	Journal diameter, D (mm)	25
	Pad axial length, L (mm)	25
	Number of pads, Npad (-)	4
	Pad arc angle, θpad (deg)	70
TPJB	Pad thickness, tp (mm)	8.0
	Pivot offset (-)	0.5
	Radial pad clearance, Cp (µm)	67
	Preload factor, m (-)	0.5
	Pivot type	Rocker-back
	Pivot’s housing radius, Rh (mm)	21.0
Pivot	Pivot radius, Rp (mm)	18.0
	Rocker-back length, LR (mm)	25.0
	Young’s modulus, E (GPa)	200
	Recess type	Rectangular
	Circumferential length of recess, l (mm)	5.0
Recess	Axial length of recess, b (mm)	5.0
	Depth of recess, Dr (mm)	0.1
	Orifice diameter, do (mm)	0.5
	Orifice discharge coefficient, Cd	0.68
	Lubricant type	sCO2
	Pressure, Pa (MPa)	8.0
Lubricant	Temperature, Ta (°C)	37
	Density, ρ0 (kg/m3)	426.6
	Viscosity, µ0 (µPa·s)	30.4

lists TPJB, pivot, and recess geometries and the properties of sCO₂ at the initial condition. The journal diameter and axial pad length are both equal to 25 mm. The four pads, each with an angular extent of 70°, a pad thickness of 8 mm, a pivot offset of 0.5, and a preload factor of 0.5, constitute the TPJB. The pivot is of the rocker-back shape, a cylindrical type whose radius is 18 mm with a housing radius of 21 mm. The rocker-back pivot length is equal to the bearing length (25 mm). Each pad has a rectangular recess whose geometrical center is concentric with the pad’s pivot. The recess has a circumferential length of 5 mm (22.9°), an axial length of 5 mm, and a depth of 0.1 mm, thus covering about 6.67% of the pad area. The orifice at the center of the recess has a diameter of 0.5 mm, and the orifice discharge coefficient is 0.68 [31]. The bearing is lubricated with sCO₂ with an ambient pressure and temperature of 8.0 MPa and 37 °C, respectively, at which its density and viscosity are 426.6 kg/m³ and 30.4 μPa·s, respectively. The temperature of the high-pressure sCO₂ supplied through the orifice is similar to that of the ambiance. Note that sCO₂ supplied through the orifice must have a higher pressure than the ambient pressure to pressurize the bearing hydrostatically.

Figure 7 shows plots of the 3D distributions of the (a) pressure, (b) temperature, (c) density, and (d) viscosity of the lubricant (sCO₂) over each pad for hybrid TPJB with pressurized recesses. The supply pressure of sCO₂ at the orifice is 10 MPa, and the static load applied at 270° and rotor speed are 100 N and 60 krpm, respectively. Each pad has peak constant pressure at the recess region. The loaded pads (pads 3 and 4) have the highest recess pressure, whereas the unloaded pads (pads 1 and 2) have the lowest values. The temperature is lowest at the pad leading edge and highest at the trailing edge. For each pad, the stepped track from the recess toward the trailing edge is attributed to the cooling effect of the fresh orifice flow. The density and viscosity of sCO₂ are maximum at the recess region. The loaded pads (pads 3 and 4) have the highest values, whereas the unloaded pads (pads 1 and 2) have the lowest values, similar to the case of pressure. Though the density and viscosity of sCO₂ are functions of both pressure and temperature, it mainly follows the trend of the film pressure, as the temperature rise is relatively insignificant owing to the large specific heat of the lubricant.

Figure 8 shows the 3D distributions of (a) pressure, (b) temperature, (c) density, and (d) viscosity of sCO₂ over pads for conventional hydrodynamic TPJB without recesses but with a load and speed similar to those in Figure 7. The loaded pads (pads 3 and 4) have the highest hydrodynamic pressure, whereas the unloaded pads (pads 1 and 2) have the lowest pressure. Note the relatively small film pressure on the unloaded pads for the hydrodynamic TPJB compared to that of hybrid TPJB. The film temperature rises from the lowest value at the leading edge of the unloaded pad (pad 1) to the maximum value at the trailing edge of the loaded pad (pad 4). Note that hydrodynamic TPJB has a higher temperature than hybrid TPJB, particularly at the loaded pads; the hydrostatically pressurized fresh fluid flow from the orifice cools the film for hybrid TPJB. Temperature changes in the axial direction are relatively insignificant, as in hybrid TPJBs.

Figure 9 shows the centerline (a) film thickness, (b) pressure, and (c) temperature plotted against the circumferential coordinate for the supply pressures of 10, 15, and 20 MPa for the hybrid TPJB with a static load of 100 N and rotor speed of 60 krpm. The predictions for the hydrodynamic TPJB without recesses are also presented for comparison purposes. Note that the large film thickness at the recess area for hybrid TPJBs is due to the recess depth of 0.1 mm. The film thickness for hybrid TPJBs is larger than that for the hydrodynamic TPJB at the loaded pads, and it increases with an increase in supply pressure. The centerline pressure for the hybrid TPJB is also greater than that of the hydrodynamic TPJB and increases with an increase in the supply pressure for both loaded and unloaded pads. The pressure drop is attributed to the inertia effects at the boundary edges of the recess, as experimentally obtained in Ref. [39]. The pad tilting angle, pivot radial deflection, and maximum pivot stress for different supply pressures are provided in Appendix A. The pad tilting angle generally decreases, and the pivot deflection and maximum stress increase with an increase in the supply pressure. The predicted values for the pivots are not fatal when considering the pad’s and pivot’s mechanical and material design limits. The predicted centerline temperature for the hybrid TPJB at the unloaded pads is slightly higher than that of the hydrodynamic TPJB. At the loaded pads, hybrid TPJB has a higher temperature than the hydrodynamic TPJB until the trailing edge of the recess of pad 3, after which it becomes lower owing to the influence of externally pressurized sCO₂. The result shows that the freshly supplied sCO₂ cools the hybrid TPJB effectively, particularly in the loaded region.

Figure 10 shows the centerline (a) density and (b) viscosity of an sCO₂ film flow versus the circumferential coordinate for different supply pressures. As expected, the density increases significantly with an increase in supply pressure following the trends of the centerline pressure in Figure 9b. Note that the density at the loaded pads’ trailing edges becomes smaller than the ambient value (426 kg/m³) because of the significant temperature rise in Figure 9c. The viscosity follows the trends of the density with an increase in supply pressure.

Figure 11 shows the journal eccentricity ratio versus rotor speed for different supply pressures at a static load of 100 N. An increase in the supply pressure dramatically reduces the eccentricity ratio. The decrease in journal eccentricity with a supply pressure of 20 MPa is ~65% compared to that for hydrodynamic TPJB at 60 krpm. The journal eccentricity ratio also decreases as the rotor speed increases owing to the increased hydrodynamic effect. However, the effect of the rotor speed on the eccentricity is relatively less significant for hybrid TPJBs, as in Refs. [23,24].

Figure 12 displays the minimum film thickness versus rotor speed for increasing supply pressures. In general, the minimum film thickness is greater for the hybrid TPJB than for hydrodynamic TPJB owing to the decreased eccentricity ratio, and it further increases with an increase in the supply pressure. The minimum film thickness for the hybrid TPJB with a supply pressure of 20 MPa increases by ~80% compared with that for the hydrodynamic case at the lowest rotor speed. The minimum film thickness increases with increasing rotor speed; however, the degree of increment for the hybrid TPJB is lower than that for the hydrodynamic TPJB. The significant increment in the minimum film thickness indicates a considerable improvement in the load capacity due to the external pressurization.

Figure 13 shows the variation of the maximum film pressure with increasing rotor speed for different supply pressures. The maximum pressure increases with an increase in the supply pressure and rotor speed. The hydrodynamic TPJB has the lowest maximum pressure. As shown in Figure 14, the predicted bearing drag power loss increases nonlinearly with the rotor speed. The hybrid TPJB has a higher power loss than the hydrodynamic TPJB because of the increase in the viscosity of sCO₂. The change in power loss as the supply pressure increases from 10 to 20 MPa is small.

Figure 15 shows the direct stiffness coefficient plotted against the rotor speed for the increasing supply pressures. The stiffness coefficient increases with an increase in rotor speed for both hydrodynamic and hybrid lubrication. Most importantly, external pressurization significantly increases the stiffness coefficient. Note that the stiffness increases by ~85% for a supply pressure of 20 MPa compared to that for the hydrodynamic TPJB at the highest rotor speed. As shown in Figure 16, the direct damping coefficient decreases slightly with an increase in rotor speed. The increasing supply pressure increases the damping coefficient, as in Ref. [20]. The hybrid TPJB predicts damping coefficients lower than those of the hydrodynamic TPJB without recesses.

6. Conclusions

sCO₂-lubricated TPJBs enable a simple configuration of turbomachinery for sCO₂ power cycles by eliminating not only the issue of mixing-up of oil lubricant and sCO₂ process fluid but also the need for oil pumps and sealing devices for the lubricant.

The present study develops a computational model for sCO₂-lubricated TPJB with hydrostatic pressurization at the tilting pad recesses to improve its static and dynamic performances. The finite element and finite difference methods solve the Reynolds and energy equations for the pressure and temperature distributions, respectively. The Newton–Raphson method is employed to find the equilibrium position, and the successive substitution method finds the numerical solution for the local density and viscosity of sCO₂, which depend nonlinearly on pressure and temperature.

The models for the density and viscosity of sCO₂, hydrodynamic lubrication with compressible and incompressible fluid flows, and hydrostatic lubrication with a turbulent flow and high Reynolds numbers are validated by comparing their results with predictions and test data in the literature. A parametric study of the hybrid TPJB reveals that an increase in supply pressure significantly increases the pad pressure. At a rotor speed of 60 krpm, with an increase in supply pressure, the journal eccentricity decreases by 65%, and the minimum film thickness increases by 80%, thus enhancing the load capacity compared to that of the hydrodynamic TPJB. Hydrostatic pressurization has the most significant effects on the load capacity at the lowest rotor speed, where hydrodynamic pressure generation is the lowest. The stiffness coefficients increase, but the damping coefficients decrease owing to external pressurization. For the hybrid TPJB, with an increase in the supply pressure, the stiffness significantly increases, but the damping coefficients change slightly.