Performance Analysis of Externally Pressurized Gas Journal Bearing Lubricated with Vapor of R134a in Centrifugal Compressor

Abstract

The performance of an externally pressurized gas journal bearing lubricated with R134a vapor used in an oil-free refrigeration centrifugal compressor is calculated by solving Reynold equations with the finite difference method (CFD). The load capacity, stiffness and mass flow rate of refrigerant vapor under conditions of various operational parameters are obtained and compared with that in air gas bearings. In addition, the influences of the average clearance, eccentricity ratios and supply vapor pressure on the static characteristics of journal bearings are investigated and discussed in detail. It concluded that the optimal operation parameters of gas journal bearings lubricated with R134a vapor are different with air journal bearings. These factors should be considered in the design process of R134a refrigerant vapor gas journal bearings.

1. Introduction

The refrigeration centrifugal compressor is a key component in chiller and heat pump systems, which are widely used in large-scale water chillers. Its operating performance has a large influence on the energy consumption of public buildings [1]. In order to improve the centrifugal compressor efficiency and the performance of the refrigeration system, oil-free variable-speed centrifugal compressors driven by permanent-magnetic synchronous frequency-convertible motors are becoming increasingly popular in refrigeration and heat pump systems. The type of compressor adopts variable frequency drive and oil-less bearings and is more energy efficient for chillers and it is frictionless and noiseless due to the absence of mechanical contact. Further, the whole chiller unit becomes more compact and lightweight because of the elimination of oil and oil management systems [2].

The selection of the appropriate bearing technology is crucial for proper design of an oil-free centrifugal refrigeration compressor. Nowadays, oil lubrication bearings have been widely used in centrifugal compressors and a gearbox and lubrication system is necessary for this type machine. Magnetic bearings have been successfully applied to centrifugal refrigeration compressors in recent years [2]. Although this technology is commercially proven, the control system of magnetic bearings is relatively complicated due to the installation of auxiliary feedback control and touchdown bearings [3].

Gas bearings are another alternative choice due to their advantages of extremely low friction, low power consumption, low noise and high motion accuracy in high-speed working conditions compared with other types of bearings [4]. The gas bearings are widely used spindles [5,6], turbo blowers [7], turbines [8,9] and air cycle machines [10]. Based on the structures and working principles, gas bearings can be divided into aerostatic bearings, aerodynamic bearings and hybrid air bearings. Many theoretical and experimental research studies were conducted to study the performance and application of gas bearings in past decades.

In recent years, there have been some attempts to develop turbogenerators or centrifugal compressors which use gas bearings. Bartosz Kus et al. [3,11] investigated the oil-free CO₂ centrifugal compressor adopting foil bearings for commercial and industrial refrigeration systems. In their analysis, the load capacity of journal bearings and thrust bearings were evaluated through empirical formulas according to Dykas [12] and Wright et al. [13]. A high-speed centrifugal compressor supported by herringbone-grooved journal bearings has been tested through experiments by Schiffmann and Favrat [14,15]. The bearings are lubricated with the refrigerant R134a, and the compressor rotational speeds and pressure ratio can reach 210 krpm and 3.4, respectively. The internal isentropic compressor efficiency is about 79%. Schiffmann [16] pointed out that the load capacity of gas bearings and the power consumption of compressors are heavily dependent on the selection of working fluids. The herringbone-grooved journal bearing and gas foil bearing were investigated in their research. Demierre et al. [17] carried out experimental research on a prototype of an Organic Rankine cycle (ORC) with an oil-free compressor turbine unit, in which a single-stage radial inflow turbine and a single-stage centrifugal compressor were directly coupled through a shaft supported by gas bearings. Wu et al. [18] conducted experimental research on a radial inflow turbogenerator with aerostatic bearings for an ORC system and carbon dioxide (CO₂) was used as the working fluid. The Gauge pressure ratio approach was employed to design the aerostatic bearings [8]. They proved that optimizing turbine design and enhancing bearings’ reliability are of great significance from the viewpoint of machine reliability.

A lot of literature on gas bearings has been published in the last few decades. Theoretically and experimentally, investigations have been conducted to investigate and enhance the performance of gas bearings. Generally, the Reynolds equation describes the flow of a thin lubricant film between two surfaces. Finite element methods (FEM) [19], finite difference methods(FDM) [20] and Computational Fluid Dynamics (CFD) [5] methods are commonly employed for solving the nonlinear Reynolds equation to obtain the bearing performance.

However, most of literature on gas bearings is taking air as the lubricant. Little work focused on gas bearings lubricated with refrigerant vapor. Garcia, M. et al. [21] investigated refrigerant-lubricated foil bearings’ behavior including the likelihood of two-phase flow in the lubricant. Bouchehit et al. [22] theoretically studied the performance of gas foil bearings in a refrigerant vapor environment, which takes into consideration the vapor/liquid lubricant transition. Schiffmann, J. et al. [23] investigated the real gas effect on herringbone-grooved bearings and rotors. Their research showed that the rotors supported by bearings would become unstable when they operate close to the saturation line or real gas.

For the studied oil-less refrigeration centrifugal compressor in this work, the working fluid R134a vapor in the refrigeration system can be used as a lubricant for the gas bearing. The use of gas bearings in a centrifugal compressor provides an oil-free solution. Developing the oil-free refrigeration centrifugal compressor supported with a gas bearing is of great significance. However, the traditional empirical design method for air bearings cannot completely satisfy the requirements of the working conditions when a refrigerant vapor is used as working fluid. The design and selection of gas bearings is important to guarantee the high speed and stability of the rotor system in a centrifugal compressor. Therefore, it is quite necessary to have a scientific understanding and accurate analysis on gas journal bearings lubricated with R134a vapor applied to a centrifugal refrigeration compressor.

The aim of this work is to evaluate the performance of externally pressurized gas journal bearings lubricated with vapor of R134a used in an oil-free refrigeration centrifugal compressor. The load capacity, stiffness and mass flow rate of the gas bearing is calculated by solving the Reynolds equation with the finite difference method (FDM). Comparison analysis is carried out to find out the difference between the gas bearing lubricated with air and R134a. In addition, the effect of average clearance, eccentricity ratio and supply vapor pressure on the static characteristics of gas journal bearings are also investigated and discussed.

2. Mathematical Model

2.1. Governing Equation

To calculate the performance of a gas bearing, the pressure distribution of gas film in bearing must be obtained first. The following assumptions are made to simply the calculation:

(a)

The fluids are assumed to be compressible and isothermal [4].

(b)

The mass flow in and out of the bearing equals the mass flow into the orifice [20].

(c)

The air flow status through the clearance is steady and the flow in the z direction is ignored [24,25].

For different supply pressures, the viscosity is different according to the gas equation of state. The value of viscosity of R134a vapor is obtained based on the Refprop working fluid database. The Reynolds equation is utilized to describe the pressure distribution in the gas film between the shaft and the bearing.

$$\frac{\partial }{\partial x}\left(\frac{\rho h^2}{\mu }\frac{\partial p}{\partial x}\right)+{\left(\frac{2R}{L}\right)}^2\frac{\partial }{\partial z}\left(\frac{\rho h^2}{\mu }\frac{\partial p}{\partial z}\right)=6u\frac{\partial }{\partial x}\left(\rho h\right)+6w\frac{\partial }{\partial z}\left(\rho h\right)+12\frac{\partial }{\partial t}\left(\rho h\right)$$

(1)

Its dimensionless version can be written as [26].

$$\frac{\partial }{\partial \theta }\left(\overline{p}{\overline{h}}^3\frac{\partial \overline{p}}{\partial \theta }\right)+\frac{\partial }{\partial \overline{z}}\left(\overline{p}{\overline{h}}^3\frac{\partial \overline{p}}{\partial \overline{z}}\right)+\overline{Q}{\delta }_i=\mathrm{\Lambda }\frac{\partial }{\partial \theta }\left(\overline{p}\overline{h}\right)$$

(2)

where μ is the dynamic viscosity of working fluid, u is the circumferential velocity of the journal and w is the axial velocity of the journal. In general, the axial velocity w is neglected in the calculation process. $\overline{Q}$ is the dimensionless mass flow factor of the orifice. $\mathrm{\Lambda }$ is the bearing number. It is noted that $\overline{Q}=1$ at the orifice entrance and $\overline{Q}=0$ at the orifice exit. h is the clearance or gas film thickness and it can be obtained by:

$$h=h_0+e\cos \theta$$

(3)

where h₀ is the average clearance and e is the eccentricity ratio.

The dimensionless parameters are defined as follows:

$$\overline{z}=\frac{z}{0.5L}; x=R\theta ; \overline{h}=\frac{h}{h_0} \overline{p}=\frac{p}{p_a} \mathrm{\Lambda }=\frac{6\mu \omega R^2}{p_a{h}_0^2} \overline{Q}=\frac{12\mu L^2{p}_a}{h_0^3{p}_s^2{\rho }_a}\rho \tilde{v}$$

where p_a is the ambient pressure.

2.2. Numerical Analysis

The finite difference method is utilized to discretize the Reynolds equation and the iterative process is employed in this analysis. The structure parameters of the journal are shown in Figure 1. Both the length and the diameter of the journal bearing are 50 mm. Two inlet columns are located at L/4 and 3L/4 of the bearing, respectively. Every column contains 12 orifices arranged symmetrically around the circumference of the bearing. The gas journal bearings operate with externally pressurized gas. When a journal gas bearing is in operation and the bearing has an eccentricity, the upper and lower position of the clearance are different. The pressure at a small clearance is larger than that at a large clearance. So, the load capacity can be formed due to the pressure difference in the gas film.

For the finite difference mesh as shown in Figure 2, the orifices are centered at the grid nodes. Using the second-order centered finite difference formula [26], the discretization form of Equation (2) can be expressed as follows:

$$\begin{array}{l}{A}_{i,j}{\overline{p}}_{i+1,j}^2+B_{i,j}{\overline{p}}_{i-1,j}^2+C_{i,j}{\overline{p}}_{i,j+1}^2+D_{i,j}{\overline{p}}_{i,j-1}^2-E_{i,j}{\overline{p}}_{i,j}^2\\ =\mathrm{\Lambda }\Delta \theta \left({\overline{p}}_{i+1,j}{h}_{i+1,j}-{\overline{p}}_{i+1,j}{h}_{i-1,j}\right)\end{array}$$

(4)

where

$$A_{i,j}=h_{i+1/2,j}^3$$

$$B_{i,j}=h_{i-1/2,j}^3$$

$$C_{i,j}={\left(\frac{2R}{L}\frac{\Delta \theta }{\Delta z}\right)}^2{h}_{i,j+1/2}^3$$

$$D_{i,j}={\left(\frac{2R}{L}\frac{\Delta \theta }{\Delta z}\right)}^2{h}_{i,j-1/2}^3$$

$$E_{i,j}=A_{i,j}+B_{i,j}+C_{i,j}+D_{i,j}$$

According to conservation of mass, the mass inflow rate through an orifice must reach equilibrium within the neighborhood around this orifice as shown in Figure 2. The mass outflow rate can be calculated by [26].

$${\dot{m}}_{out}={\dot{m}}_{\theta 1}+{\dot{m}}_{\theta 2}+{\dot{m}}_{z1}+{\dot{m}}_{z2}$$

(5)

$$\{\begin{array}{l}{\dot{m}}_{\theta 1}+{\dot{m}}_{\theta 2}={\displaystyle {\int }_{B1}^{}\left(-\frac{uh}{2}+\frac{h^3}{12\mu }\frac{\partial p}{\partial x}\right)\rho dz+{\displaystyle {\int }_{B2}^{}\left(-\frac{uh}{2}+\frac{h^3}{12\mu }\frac{\partial p}{\partial x}\right)\rho dz}}\hfill \\ {\dot{m}}_{z1}+{\dot{m}}_{z2}={\displaystyle {\int }_{B3}^{}\left(\frac{h^3}{12\mu }\frac{\partial p}{\partial z}\right)\rho dx+{\displaystyle {\int }_{B4}^{}\left(\frac{h^3}{12\mu }\frac{\partial p}{\partial z}\right)\rho dx}}\hfill \end{array}$$

(6)

The gas flow through the orifice is assumed to be an adiabatic process and non-viscous, and the mass flow rate into the orifice is [21]:

$$Q_m=A{p}_s\varphi \sqrt{\frac{2{\rho }_s}{p_s}}\psi$$

(7)

$$\psi =\{\begin{array}{ll}{\left[\frac{k}{2}{\left(\frac{2}{k+1}\right)}^{k+1/k-1}\right]}^{1/2}\hfill & ,\frac{p_d}{p_s}\le {\beta }_k\hfill \\ {\left\{\frac{k}{k-1}\left[{\left(\frac{p_d}{p_s}\right)}^{2/k}-{\left(\frac{p_d}{p_s}\right)}^{k+1/k}\right]\right\}}^{1/2}\hfill & ,\frac{p_d}{p_s}>{\beta }_k\hfill \end{array}$$

(8)

where A represents the cross-section area of the orifice, k is the ratio of the specific heat of working fluid and ρ_s is the density of gas supply pressure. p_s is the supply pressure and ϕ is the coefficient of the mass flow rate through the orifice and in general equals 0.8.

$${\beta }_k=\frac{p_c}{p_s}={\left(\frac{2}{k+1}\right)}^{k/k-1}$$

(9)

2.3. Boundary Conditions

The atmosphere boundary condition and periodic boundary conditions are defined as follows, respectively.

$$\overline{p}\left(1:m+1,1\right)=\overline{p}\left(1:m+1,n+1\right)=1$$

(10)

$$\overline{p}\left(1,1:n+1\right)=\overline{p}\left(m+1,1:n+1\right)$$

(11)

The horizontal and vertical load capacity of the bearing can be obtained by the integration of film pressure.

$$\{\begin{array}{c}{\overline{W}}_n=2{\displaystyle {\int }_0^{2\pi }{\displaystyle {\int }_0^1\left(\overline{p}-1\right)sin\theta d\theta d\overline{y}}}\\ {\overline{W}}_t=2{\displaystyle {\int }_0^{2\pi }{\displaystyle {\int }_0^1\left(\overline{p}-1\right)cos\theta d\theta d\overline{y}}}\end{array}$$

(12)

So, the load capacity can be obtained as follows:

$$\overline{W}=\sqrt{{\left({\overline{W}}_n\right)}^2+{\left({\overline{W}}_t\right)}^2}$$

(13)

$$W=0.5LR{p}_a\overline{W}$$

(14)

The stiffness of the bearing is represented as the ratio of load and displacement and is given by:

$$K=\Delta W/\Delta e$$

(15)

The calculation flow is presented in Figure 3. The iterative procedure is used to check the convergence of gas film pressure and mass flow rate of every node. Mesh grid size in the calculation is taken to be 100 × 320 according to the structural parameters. The convergence criteria of pressure are $\sqrt{{\displaystyle {\sum }_i^m{\displaystyle {\sum }_j^n{\left(\frac{\Delta {\overline{p}}_{i,j}^n}{{\overline{p}}_{i,j}^n}\right)}^2}}} < {10}^{-6}$, and the mass flow rate convergence criteria are $\frac{\left|{\dot{m}}_{in}-{\dot{m}}_{out}\right|}{{\dot{m}}_{in}}\ll {10}^{-5}$. During the calculation, when node pressure and mass flow rate converge to the above criteria, the calculation terminates.

3. Results and Discussion

This work researches the impacts of average clearance (gas film thickness), orifice diameter, gas supply pressure and eccentricity ratio on the load capacity, stiffness and mass flow rate of a journal bearing operating at zero rotational speed. For centrifugal compressors used in chillers, the supply pressure is mainly related to the pressure of the available gas source in the centrifugal chiller application and generally varied from 0.55 to 1.2 MPa determined by the condensing temperature and pressure. The average clearance varied from 10 to 50 μm and the ambient temperature was set at 278 K in the analysis, which is the evaporation temperature of the refrigeration system. The corresponding pressure of refrigerant vapor at this temperature is about 0.35 MPa, which is the working ambient pressure for gas bearings in this analysis.

Figure 4 illustrated the pressure distributions of gas journal bearings for ε = 0, with supply pressure p_s = 0.95 MPa, average clearance h₀ = 20 μm and diameter of orifice d = 0.3 mm. The pressure at each of the 12 orifices is equal for air and R134a bearings when there is no eccentricity. The pressure distribution of gas journal bearings when ε = 0.5 is shown in Figure 5. The pressure difference exists at a different location of the journal bearing with different clearance. The simulation results in this work are consistent with the results in reference [20].

3.1. Influence of Average Clearance

Figure 6 displays the impact of average clearance on load capacity, stiffness and mass flow rate under different orifice diameters for the fixed supply pressure of 0.95 MPa and an eccentricity ratio of ε = 0.5. There exists an optimal clearance which enables the load capacity and stiffness maximum under given operational conditions. This is similar with the results in reference [27]. It can be further observed that the obtained maximum values of load capacity are almost the same under different orifice diameters. The optimum average clearance which enables the maximum value of capacity increases as the diameter of the orifice increases. Specially, the maximum load capacity of the R134a bearing can be obtained at a smaller average clearance compared with air gas bearings. When d = 0.3 mm, the optimal average clearance range, which makes the maximum load capacity, is about 25–29 μm for air bearings. For the 134a bearing, the value of the optimal average clearance decreases and is about 16–20 μm. This difference should be considered when designing an R134a refrigerant gas bearing.

It is observed from Figure 6b that the maximum value of stiffness increases when the diameter of orifice decreases. Under the same working conditions, the clearance for the R134a bearing that could obtain the maximum stiffness is smaller than that in air bearings. At the same time, the maximum value of stiffness of the R134a bearing is higher than that of air bearings. For example, in the case of d = 0.3 mm, the maximum value of stiffness of air bearings is about 37.28 N/μm and the corresponding clearance is about 20 μm. For the R134a bearing, the maximum value of stiffness is about 53.03 N/μm and the corresponding clearance is about 14 μm. For R134a journal bearings, the average clearance and diameter of orifice should be designed smaller than that of the air bearing to obtain larger load capacity and stiffness.

For the centrifugal refrigeration compressor supported by gas bearings lubricated with R134a vapor, the supplied refrigerant vapor can be taken from the high-pressure side of the system or the discharge refrigerant vapor. The mass flow rate of refrigerant vapor supplied for bearings has an adverse effect on the performance of the refrigeration system since it has no cooling effect. Therefore, it is important to evaluate the value of the mass flow rate consumed by the gas bearing and provide a base to evaluate the refrigeration performance. From Figure 6c, it follows that the mass flow rate increases rapidly at first and then rises gradually for both air and R134a bearings as clearance increases. This is due to the choke flow that occurs in the orifice when clearance is increasing. The mass flow rate of R134a is larger than that of air bearings due to the large density of R134a compared to air. The diameter of orifice has a large influence on the mass flow rate. The mass flow rate increases as the diameter of orifice increases.

Figure 7 shows the influence of supply pressure on load capacity, stiffness and mass flow rate for fixed diameter d = 0.3 and eccentricity ratio ε = 0.5, respectively. It is observed that the load capacity increases with the increase of supply pressure. The optimal clearance decreases slightly as supply pressure increases. It is noted that the optimal average clearance is different between air and R134a gas bearings as discussed in Figure 6a. Load capacity with higher supply pressure may be lower than that with lower supply pressure when average clearance is larger than 30 μm. This phenomenon will be explained in the following figure, which displays the effect of supply pressure on load capacity. The load capacity is a combined effect of orifice diameter, average clearance and supply pressure. As the supply pressure increases, the bearing stiffness increases as well. It is seen from Figure 7c that the mass flow rate of air and R134a increases rapidly at first and then reaches a constant value. It indicates that choke flow occurs both for air and R134a gas bearings when the average clearance is large.

Figure 8 shows the effect of the eccentricity ratio on journal bearing characteristics. As the eccentricity ratios increases, the maximum value of load capacity and stiffness increases as well. The optimal average clearance is unchanged when the eccentricity ratio increases. Moreover, the eccentricity ratio has little impact on mass flow rate when other operational parameters are kept constant as shown in Figure 8c.

3.2. Influence of Supply Pressure

For a centrifugal refrigeration compressor supported with gas bearings, which are lubricated with the externally pressurized refrigerant vapor, the supply vapor pressure is a crucial factor in the operation of gas bearings. It is possible to use the working fluid of the refrigeration system to lubricate and cool the bearing system. Then, no additional vapor supply system is required. Therefore, it is important to investigate the effect of supplied vapor pressure on the performance of the bearings. Figure 9 illustrates the impact of supply pressure on the load capacity and mass flow rate with orifice diameters d = 0.3 and 0.5 mm, for the fixed average clearance h₀ = 25 μm and eccentricity ratio ε = 0.5. From Figure 9a, it follows that the load capacity of air bearings increases linearly as the air supply pressure increases when d = 0.5 mm. When orifice diameter is 0.3 mm, the load capacity is increasing linearly at first and then decreasing with the increase of supply pressure. In addition, the load capacity of d = 0.3 mm is larger than that of d = 0.5 mm when the supply pressure is lower than 1.1 MPa. This can be observed from Figure 6a when the average clearance is 25 μm.

For R134a bearings, there exists an optimal supplied pressure. It means that the supplied pressure of R134a vapor has suitable value and it has the relationship with the orifice diameter and average clearance. As the diameter of orifice decreases, the value of optimal supplied pressure which enables the maximum load capacity decreases. For example, when d = 0.5 mm, the optimal supplied pressure is about 1.05 MPa. The value of optimal supplied pressure is decreased to 0.9 MPa as the orifice diameter is reduced to 0.3 mm. The value of load capacity is determined by the combined effect of orifice diameter and average clearance. Figure 9b shows the mass flow rate variation with supplied pressure when orifice diameter is 0.3 and 0.5 mm. The mass flow rate almost increases linearly with the increase of supply vapor pressure. Mass flow rate in R134a bearing is larger than that in air bearing.

3.3. Influence of Eccentricity Ratio

Figure 10 presents the effect of eccentricity ratios on journal bearing characteristics with different supplied pressures for the fixed orifice diameter d = 0.3 mm and the average clearance of 25μm. It is seen that the load capacity increases rapidly with the increase of the eccentricity ratio at first and then has a decreasing trend when the eccentricity ratio and supplied pressure are large. These calculated results are compared with the results in reference [24], and they have the same variation trend. It should be noted that, for R134a bearings, the load capacity is decreased when the vapor supply pressure is larger than 0.9 MPa. This can be explained from the previous Figure 9a. This phenomenon is obtained when the diameter of orifice is d = 0.3 mm and average clearance is h₀ = 25 μm. For other operation and structural parameters, there exists a similar change trend as discussed above.

Figure 11 shows the mass flow rate variation with the eccentricity ratio under different supply pressures. It can be observed that the mass flow rate decreases slightly as the eccentricity ratio increases. The supply pressure has a larger impact on mass flow rate than the eccentricity ratio. The higher the supply pressure, the larger the mass flow rate. Due to the thermophysical properties of air and R134a being different, the pressure distributions and velocity fields of journal bearings between air and R134a are also different. The mass flow rate is mainly influenced by the density of working fluids.

4. Conclusions

The performance of externally pressurized gas journal bearings lubricated with R134a vapor used in oil-free refrigeration centrifugal compressors is calculated by solving Reynold equations with the finite difference method. The journal bearing performance including load capacity, stiffness and mass flow rate of refrigerant vapor is compared with that in air gas bearings. The main conclusions can be drawn as follows:

(1)

There is an ideal average clearance with the highest load capacity and gas bearing stiffness. The corresponding value of average clearance for R134a gas journal bearings is smaller than that in air gas bearings. For R134a journal bearings, the average clearance and diameter of orifice should be designed smaller than air bearings to obtain a larger load capacity and stiffness.

(2)

For R134a gas journal bearings, there is an ideal supplied pressure that allows for the maximum load capacity. The optimal value of the supply pressure is determined by the diameter of orifice and average clearance under the fixed eccentricity ratio. Compared with air gas bearings, the optimal supply pressure is lower than that of air bearings.

(3)

The mass flow rate of R134a gas journal bearings is larger than that of air bearings. The diameter of orifice has a larger impact on the mass flow rate consumed by gas bearings than supply pressure and eccentricity ratio.

In summary, the performance of gas bearings, including load capacity, stiffness and mass flow rate, is determined by a combined effect of bearing parameters such as the orifice diameter, average clearance or gas film thickness, supply pressure and eccentricity ratio. Journal bearing performance can be further improved by optimizing the operational parameters. It also can be concluded that the optimal operation parameters of gas journal bearings lubricated with R134a vapor are different compared to air journal bearings. This should be taken into consideration in the design of R134a refrigerant vapor gas journal bearings.