Theoretical Model Development and Mixed Lubrication Analyses of Rolling Piston Type Rotary Compressors: A Review

Abstract

Owing to the requirements for high performance and long life, the friction and wear problems of rolling piston type rotary compressors have drawn much attention. Various theoretical models have been developed and improved to reveal the inner state of the compressor and obtain the optimization schemes. However, there remain some disadvantages and research gaps in the corresponding modeling and mixed lubrication analyses, and a comprehensive summary is lacking. To have a better understanding of the research status, this paper reviews the theoretical model development and mixed lubrication analyses of the compressor in the past decades. The determination of compression pressure, the modeling process of moving components, and the key findings are presented in detail. On this basis, some important influencing factors and the problems remaining to be solved are also discussed. This paper provides multifaceted guidance for manufacturers and researchers to conduct further theoretical analysis and optimal design.

1. Introduction

Developed in the 1960s, rolling piston type rotary compressors are the key components in household air conditioners to realize refrigerant pressurization [1,2,3]. They are mounted between the evaporator and the condenser (see Figure 1), and the compression process is achieved with a piston and cylinder arrangement [4,5]. The special working principle of it results in the unique inner structures, which are much different from those of the scroll compressor [6,7], the reciprocating compressor [8,9], and the sliding vane compressor [10,11]. Owing to the advantages in structure, size, and weight, they are taking an increasing share compared to other types of compressors [2,12]. Correspondingly, the volumetric efficiency, consumption power, noise, and vibration of the compressor have been the concern of researchers for a long time [13,14,15,16,17]. In the rolling piston type rotary compressor, the energy losses mainly consist of the friction loss, the motor loss, the compression loss, and the lubricant pumping loss, and the mass flow losses are mainly composed of the leakage loss, the suction gas heating loss, and the clearance volume loss [5]. Among them, the friction loss and the leakage loss play the most significant roles. With the wide applications of household air conditioners, the friction loss within the compressor counts a lot in energy conservation and environmental protection. In the current stage, the annual global production volume of the rotary compressor is around 200 million units, and China is responsible for approximately 90% of that prouction [3]. Meanwhile, higher working frequency and lower viscosity are adopted for high-performance compressors [1]. Severe operation conditions will result in surface wear, thus influencing the reliability and service life of the compressor. Under these circumstances, the friction and wear of rotary compressors have become more and more important, and many tribologists have paid much attention in the latest decades [18,19,20,21,22,23].

On the one hand, the friction and wear of rotary compressors largely result from the complex motions of the components. A rotary compressor contains many specially designed components (see Figure 2), such as a rolling piston, sliding vane, crankshaft, cylinder, main bearing, and sub-bearing [1,24]. Driven by an AC motor, the former three components undergo complex motions during operation, thus leading to the time-varying interacting forces between the components [20,25]. In the working process of the compressor, the complex motions of the moving components will change the lubrication performance of the interacting surfaces, which can enter into a mixed lubrication regime when the load-carrying capacity of the oil film is not enough. As shown in Figure 3, owing to the angular displacement of the vane, the severe wear of the vane slot on the suction side occurs in the region close to the cylinder wall. On the other hand, different types of friction pairs within a compressor with coupling effects increase the complexity of the tribological process. For a typical rotary compressor, there are mainly seven friction pairs (see Figure 4), which can be affected by each other owing to multi-body coupling [2,26,27]. During operation, these different types of friction pairs show different operation performances, and the mutual influence between them is a nonnegligible factor.

In addition, the special working environment of the rotary compressor influences the lubrication characteristics between interacting surfaces. To provide a high-pressure refrigerant, all the friction pairs of the compressor must be in the hermetic space [28,29], which will influence the formation of oil film pressure and cavitation [19]. Exposed to the lubricant oil and the refrigerant vapor [12], the clearance between the mating surfaces may be filled with the mixture instead of the pure oil [30]. The solubility of the refrigerant may greatly influence the tribological performance of the conjunction by changing the rheological properties [31]. However, accurate modeling of the mixture is very difficult, as the solubility of the refrigerant is affected by many factors. For simplification, a constant solubility can be applied, and its influence can be further revealed. In the past, ozone-harming chlorofluorocarbon refrigerants and replacement refrigerants such as R410a and R134a were widely applied [3,32,33]. With the increasing requirements for environmental protection, some new refrigerants such as R32 and R290 are being produced and spread [3,12,34,35].

To understand the working state of the rotary compressor and conduct optimal design, theoretical analysis is often the preference. Though experimental studies have been conducted to some extent, further measurements become extremely difficult considering the high-pressure hermetic space and the compact size [12,36,37,38,39]. By comparison, theoretical analysis is more convenient and can reveal multiphysics characteristics within the compressor. With the development of mixed lubrication theory and tribodynamic modeling, the tribological analysis of the rotary compressor has been conducted for many years and is gradually being improved. Based on simulations, the optimal designs of various structure parameters have been determined and applied [2,18,26]. However, there still remain some disadvantages and research gaps, and a comprehensive summary is lacking. To provide comprehensive guidance for manufacturers and researchers, this paper reviews the theoretical model development and mixed lubrication analysis of rolling piston type rotary compressors. The model development and analysis of the compression pressure, rolling piston, sliding vane, and crankshaft are presented first in detail. Then, the theoretical research status regarding a whole compressor in different stages is further summarized. As the supplement, several influencing factors are discussed, along with the potential problems remaining to be solved.

2. Model Development and Analyses

In engineering applications, the mating surfaces are under a hydrodynamic lubrication regime when the oil film between them is large enough; otherwise, the mixed lubrication regime will occur, resulting in surface wear. In the latest decades, the mixed lubrication theory was developed both in the governing equation [40,41,42] and the asperity contact [43,44,45]. Simplified from the Navier–Stokes equations, the Reynolds equation with the Reynolds cavitation model [46,47] or the Jakobsson–Floberg–Olsson (JFO) cavitation model [48,49] is widely applied to obtain the film characteristics. The mass-conserving JFO cavitation model is more accurate than the Reynolds cavitation model, and its predicted results with are closer to the experimental data [50]. As the JFO cavitation model is more difficult to implement, the Reynolds cavitation model after being verified is often the preference in mixed lubrication analysis [2,19]. In terms of roughness effects, the deterministic method considering roughness distribution in detail is more accurate, especially in scuffing failure analysis [51]. However, considering the time-consuming problem of this method, the average flow model [40,41] and the homogenized model [52,53] were the preferences in mixed lubrication analysis. Correspondingly, the theoretical model and mixed lubrication analysis of the rotary compressor have also been developed and improved. In the early stage, the theoretical analysis of a part of the moving components or a part of the friction pairs was the preference considering the solution efficiency [54,55]. Later on, simulations of the whole compressor considering multibody coupling were presented [20]. Further studies began to consider the effects of elastic deformation [4], thermal performance [56], groove distribution [1,2] and other important influencing factors.

2.1. Determination of Compression Pressure

During operation, the outer surface of the rolling piston is surrounded by the refrigerant, and the refrigerant also applies forces on part of the sliding vane [57]. Therefore, the compression pressure in the cylinder has been studied to obtain the boundary conditions for dynamic analysis and lubrication analysis (see Figure 5).

Owing to the advantage in calculation efficiency, the function expressions of the compression pressure have been proposed and widely applied. Based on the geometrical relationships, Pandeya et al. [5] obtained the volume angle and pressure angle expressions. Considering the limitation of the discharge pressure, Okada et al. [58] presented a piecewise function. In their derivation, the compression pressure remains constant when reaching the discharge pressure. To describe the process after the discharge valve opens, Yanagisawa et al. [59] improved the expression, in which the compression pressure after the discharge angle decreases linearly with the crank angle. To further consider the angle of the discharge valve, the pressure angle relationship was further modified with a three-segment function, which is still widely applied in the current stage [2,54,60]. The three-segment function is expressed as follows [25]:

$$P_c=\left\{\begin{array}{cc}{P}_s{\left(V_s/V_c\right)}^{\kappa }& 0\le {\theta }_1\le {\theta }_c\\ P_d\left[1+c_s\left(2\pi -{\theta }_1\right)/\left(2\pi -{\theta }_c\right)\right]& {\theta }_c<{\theta }_1\le {\theta }_d\\ P_s+\left(P_d-P_s\right)·\left(2\pi -{\theta }_1\right)/\left(2\pi -{\theta }_d\right)& {\theta }_d<{\theta }_1\le 2\pi \end{array}\right.$$

(1)

where $P_c$ and $P_s$ are the compression pressure and the suction pressure, respectively, $P_d$ denotes the discharge pressure, $c_s$ represents the correction coefficient, $V_s$ represents the suction chamber volume, $V_c$ represents the compression chamber volume, ${\theta }_1$ is the crank angle, ${\theta }_c$ is the starting angle of the refrigerant discharge, and ${\theta }_d$ is the angle of the discharge valve.

Apart from the chamber volume, the compression pressure is also influenced by the supercharging effect [61], the thermodynamics [32] and the valve dynamics [62]. Based on a nonlinear gas dynamic model, Liu et al. [63] studied the supercharging phenomenon in a variable-speed compressor and found that it can increase the capacity of the compressor. To consider the relationship between pressure, temperature, and mass, the thermodynamics model was developed, along with the geometry model and the valve flow model [32]. To describe the internal flow more accurately, three-dimensional (3D) computational fluid dynamics (CFD) analysis was also carried out to obtain the pressure–volume diagram [64,65,66,67]. In the period between the opening and closing of the discharge valve, the variation in the compression pressure is dramatically affected by the valve dynamics. For this reason, some researchers have established improved methods to obtain the compression pressure by analyzing the valve motion. By regarding the valve as a beam with varying width, Ooi et al. [68] analyzed the valve dynamics and obtained the chamber pressure considering the valve deflection. To consider the valve motion and the refrigerant flow together, fluid–structure interaction models were further established with commercial software [69,70,71]. However, as these methods and models were much more complex, they were rarely applied in the lubrication analysis and dynamic analysis of moving components.

Figure 5. Determination of compression pressure (Reproduced with permission from [66], Creative Commons CC BY license; reproduced with permission from [71], Copyright 2017, Elsevier).

Figure 5. Determination of compression pressure (Reproduced with permission from [66], Creative Commons CC BY license; reproduced with permission from [71], Copyright 2017, Elsevier).

2.2. Sliding Vane and Rolling Piston

In view of the vane–piston interaction, the sliding vane and the rolling piston were considered together in theoretical modeling and analysis. To achieve decoupling, the translational movements in three directions and the angular displacements in two directions of the crankshaft are ignored, which only rotates along the rotor axis. The vane–slot conjunction and the vane–piston interaction are the main friction pairs for the sliding vane, while those for the rolling piston are the crank–piston conjunction and the vane–piston interaction. For performance improvement, the mixed lubrication performance of these friction pairs has been investigated for a long time.

With the chamber pressure acting as the boundary pressure, kinematic and mixed lubrication analysis of the sliding vane and the rolling piston have been conducted comprehensively (see

Table 1. Theoretical research on rolling piston and vane.

Type	Year	Reference	Research Content and Findings
1	1978	Pandeya et al. [5]	Analyzed the rotation speed of the rolling piston; Proposed formulas to describe piston-related friction;
	1982	Okada et al. [58]	Analyzed the kinematic behaviors of the rolling piston; Rotation directions of piston and crank are the same; Direction of the vane–piston friction is a variable;
	1982	Yanagisawa et al. [59]	Rotation direction of the rolling piston is positive and negative alternately;
	2010	Ito et al. [72]	Piston–vane sliding velocity decreases with decrease in the piston weight;
	2021	Xu et al. [73]	Smaller density of piston material contributes to keeping the rolling state between piston and vane;
2	1999	Yoshimura et al. [27]	Conducted mixed EHL analysis of piston–vane conjunction; Piston–vane friction coefficient changes from 0.04 to 0.08;
	2000	Tanaka et al. [74]	Oil viscosity greatly affects piston–vane friction loss;
	2001	Cho et al. [31]	Solubility of refrigerant dramatically affects piston–vane friction;
	2006	Cho et al. [75]	Vane thickness and position of vane significantly influence the friction force and energy loss between vane and piston;
3	2006	Ito et al. [54]	Conducted mixed lubrication analysis of the vane–slot conjunction; Period of mixed lubrication and friction loss of vane–slot conjunction decrease with increasing vane slot length;
	2009	Ito et al. [77]	Considered the influence of cylinder deformation by using the finite element method; Elastic deformation of the cylinder contributes to the smooth reciprocating motion of the sliding vane;
	2018	Yang et al. [76]	Analyzed the friction power and wear of the vane–slot conjunction;
4	1985	Yanagisawa et al. [78]	Analyzed the friction losses in vane–slot conjunction, piston–vane interaction, and rolling piston bearing; Cylinder with a short length/diameter configuration reduces the total loss of piston and vane;
	2014	Ito et al. [25]	Conducted mixed lubrication analysis of the piston–crank conjunction and the vane–slot conjunction together; Working frequency and chamber pressure differences greatly change piston rotation speed and piston–vane sliding velocity;
5	1993	Padhy [79]	Proposed a mathematical model to analyze sliding vane dynamics and piston dynamics;
	1998	Minami et al. [80]	Built a two-degree-of-freedom tribodynamic model of the rolling piston; Lubrication performance with R410a refrigerant is better than that with R22;
	2019	Geng et al. [81]	Established a two-degree-of-freedom tribodynamic model of the sliding vane; Second-order motion characteristics worsen the lubrication status of the vane–slot conjunction on the suction side;
	2021	Li et al. [18]	Proposed an improved tribodynamic model for vane by considering cylinder deformation and groove distribution; Optimal vertical groove reduces the friction loss and wear between the sliding vane and the vane slot;
	2022	Geng et al. [36]	Analyzed the separation–collision process between the vane tip and the rolling piston;

and Figure 6). In 1978, Pandeya et al. [5] analyzed the rotation speed of the rolling piston and proposed convenient formulas to calculate the vane tip–piston friction force and the piston–cylinder head friction torque. By solving the equation of motion of the rolling piston and the equilibrium equations of the sliding vane together, Okada et al. [58] discussed the kinematic behaviors of the rolling piston. It was found that the rotation direction of the rolling piston is the same as that of the crank, which varies with time periodically. However, the direction of the vane–piston friction was proved to be a variable, thus influencing the rotation speed of the piston. Yanagisawa et al. [59] also analyzed the rotating motion of the rolling piston, and the conclusions verified with the experimental results were drawn. However, the rotation direction of the rolling piston was found to be positive and negative alternately for the cases studied. Further investigation on the rolling piston was conducted by Ito and Xu [72,73] and it was found that the piston–vane sliding velocity can be reduced with a decrease in the piston weight. In Refs. [58,59,72], the piston–crank interaction was determined by solving the traditional Reynolds equation, while the piston–vane conjunction and the vane–slot conjunction were described with the Coulomb friction law. Instead of using the Coulomb friction law, Yoshimura et al. [27] began to apply the mixed elastohydrodynamic lubrication (EHL) theory to analyze piston–vane friction. The one-dimensional average flow model was used to obtain the oil film pressure, and the asperity contact was described with the Greenwood and Tripp contact model. It was found that the mating surfaces are supported by the oil film and the metallic contact, between which the friction coefficient changes from 0.04 to 0.08. Based on the mixed EHL analysis, Tanaka et al. [74] found that the oil viscosity significantly influences the asperity contact friction loss and the viscous friction loss. Later on, Cho et al. [31,75] further revealed the partial EHL characteristics between the sliding vane and the rolling piston with the same theory. After their comparison, the solubility of the refrigerant was proved to dramatically affect the piston–vane friction. A mixed lubrication analysis of the vane–slot conjunction was first implemented by Ito et al. [54], in which the second-order motion of the vane was considered. By coupling the equations of motion of the vane, the average Reynolds equation, and the contact model, a proposed model was applied for optimal design of the vane. It was found that the period of mixed lubrication and the friction loss decreases with the increasing vane slot length. Using the same models, Yang et al. [76] analyzed the friction power and wear of the vane–slot conjunction, which were in good agreement with their durability experiment. In the subsequent work of Ito et al. [77], the influence of the cylinder deformation was considered by using the finite element method. The results showed that the cylinder deformation contributes to the smooth reciprocating motion of the sliding vane. Based on the kinematics of the compressor, Yanagisawa et al. [78] analyzed the friction losses in the vane–slot conjunction, the piston–vane interaction, and the rolling piston bearing. It was found that the cylinder with a short length/diameter configuration is beneficial to reduce the total loss. To improve the model accuracy, Ito et al. [25] further proposed an improved numerical model by conducting mixed lubrication analysis of the piston–crank conjunction and the vane–slot conjunction together. The simulation results indicated that the piston–vane friction force has the most important influence on piston rotation compared to other friction forces. It was also found that the working frequency and the chamber pressure difference greatly change the piston rotation speed and the piston–vane sliding velocity.

Considering the increasing impacts of dynamic processes, tribodynamic models coupling dynamic behaviors and mixed lubrication have become the preferences in the current stage (see Table 1 and Figure 6). Considering the one-dimensional vane dynamics and the one-dimensional piston dynamics together, Padhy [79] proposed a mathematical model to analyze the tribo-dynamic performance of the compressor, which was verified by their experiments [82]. Coupling the momentum equations and the Reynolds equation, a two-degree-of-freedom tribodynamic model of the rolling piston was first built by Minami et al. [80]. By comparison, the results showed that the lubrication performance with R410a refrigerant is better than that with R22. Considering second-order motion, Geng et al. [81] first established a two-degree-of-freedom tribodynamic model of the sliding vane. Based on their dynamic solution, it was found that second-order motion characteristics worsen the lubrication status of the vane–slot conjunction on the suction side. Later on, the mathematical model was modified to analyze the separation–collision process between the vane tip and the rolling piston [36]. The simulation results indicated that the separation–collision behaviors are closely related to the spring stiffness and the rotation velocity of the shaft. In engineering applications, both the vane slot and the sliding vane are processed with the groove structures, such as the spring groove, the vertical groove, and the dovetail groove. To consider groove effects, Li et al. [18] proposed an improved tribodynamic model for the sliding vane, in which the cylinder deformation caused by the chamber pressures is also calculated. In their work, the fluid pressure within the groove was assumed to be equal to the discharge pressure, and the pressure distribution at the boundaries was also defined. As the interacting area between the vane and the slot changed with the crank angle, the pressure boundary conditions were updated before the pressure iteration. It was found that the cylinder deformation has a significant influence on the second-order motion of the vane and the mixed lubrication performance of the vane–slot conjunction (see Figure 7). After their comparison, the optimal vertical groove was proved to reduce the friction loss and wear between the sliding vane and the vane slot. In Geng and Li’s work, it was assumed that the direction of the sliding speed between the vane tip and the rolling piston remained unchanged, and the normal interacting force of the vane–piston conjunction was obtained based on the force equilibrium.

2.3. Crankshaft

During operation, the main bearing, the sub-bearing, the thrust bearing, and the inner surface of the piston interact with the crankshaft [83], and the friction and wear of the corresponding friction pairs have also been studied for a long time. Under the action of an electric current, magnetic forces between the rotor assembly and the stator assembly will occur [84]. To conduct dynamic balance, the upper and lower balancers are mounted on the proper positions of the rotor [19,85]. To make the refrigeration capacity variable, frequency conversion technology has been widely applied [86,87], resulting in the acceleration and deceleration processes of the crankshaft. In addition, with the rotary compressor developing towards high power density, twin-cylinder and three-cylinder schemes began to be applied [88,89,90], which further worsened the operation conditions of the crankshaft.

In the last decades, dynamic analysis of the crankshaft–journal bearing system has been conducted by some researchers, especially at the early stage (see

Table 2. Theoretical research on crankshaft.

Type	Year	Reference	Research Content and Findings
1	1990	Hattori et al. [55]	Established a dynamic model for the rotor in a twin rotary compressor; Whirling locus and elastic deformation of the rotor are smaller than those of the single rotary compressor;
	1998	Dufour et al. [91]	Modification of bearing clearance partially improves the rotor dynamics of the variable-speed compressor;
	2000	Seve et al. [85]	Proposed improved balancing, which satisfies requirements for vibration level and industrial implementation;
	2023	Zhang et al. [84]	Established a two-degree-of-freedom tribodynamic model for vibration analysis; Amplitude of the triplet frequency is higher than that of the double frequency;
2	2016	Fu et al. [92]	Analyzed the lubrication characteristics of the main bearing and the sub-bearing;
	2021	Yu et al. [93]	Conducted numerical analysis and optimization of the main bearing and the sub-bearing considering roughness effects; Optimized width–diameter ratio can increase the load carrying capacity of oil film and reduce the friction power of sub-bearing;
	2006	Hirayama et al. [94]	Optimally smoothed surface roughness improves the ability of the lubrication film formation;
	2006	Xie et al. [95]	Bottom of sub-bearing housing under low speed and top of main bearing housing under high speed were in relatively severe working conditions;
	2010	Kitsunai et al. [96]	An annular depression on the shaft surface reduces the losses and increases the volume efficiency
	2014	Zhang et al. [88]	Analyzed the dynamic behaviors of the crankshaft for both single-cylinder and twin-cylinder compressors; Proper dynamic balance can improve the wear performance of journal bearings;
	2018	Meier et al. [97]	Conducted sensitivity analysis of surface roughness, bearing lengths, and clearances of the friction power losses;
	2020	Wang et al. [4]	Developed a numerical model to evaluate the lubricating condition of the compressor under tested conditions;
	2021	Liu et al. [19]	Constructed a complete coupled journal–thrust bearing model; Considered the bearing deformation and the shaft deformation; High ambient pressure contributes a lot to the wear of the thrust bearing;
	2022	Lyu et al. [26]	Conducted texture optimization for the thrust bearing; Applied particle swarm optimization (PSO), simulated annealing algorithm (SAA) and genetic algorithm (GA);

). To achieve decoupling, the gas forces and unbalance forces caused by the rolling piston need to be determined in advance. To estimate the reliability and conduct the optimal design of the bearing, Hattori et al. [55] established a dynamic model for the rotor in a twin rotary compressor. In their model, the bearing reaction forces were obtained using the short bearing theory, and the unbalanced forces and the gas forces were all considered. It was found that the whirling locus and elastic deformation of the rotor are smaller than those of the single rotary compressor with the same discharging capacity. By developing a finite element model of rotor–crankshaft assembly, Dufour et al. [91] discovered that modification of the bearing clearance partially improves the rotor dynamics of the variable-speed compressor. Based on dynamic simulations, Seve et al. [85] further analyzed the balancing of this type of compressor. By comparing different schemes, they proposed improved balancing, which satisfies the requirements for the vibration level and industrial implementation. Considering the electromagnetic force acting on the motor rotor, Zhang et al. [84] established a two-degree-of-freedom tribodynamic model for vibration analysis of the rotor. It was found that the amplitude of the triplet frequency is higher than that of the double frequency owing to the coupling effects of the bearing force and the electromagnetic force.

With the detailed lubrication simulations, great efforts have been devoted to the tribological performance analysis of the bearings associated with the crankshaft (see Table 2). With the Reynolds equation and the load equation, Fu et al. [92] analyzed the lubrication characteristics of the main bearing and the sub-bearing. It was found that the wear-out may occur at the bottom of the sub-bearing when the oil viscosity is low. Considering the roughness effects, Yu et al. [93] further conducted numerical analysis and optimization of the main bearing and the sub-bearing. After mixed lubrication analysis, the results showed that the optimized width–diameter ratio can increase the load carrying capacity of the oil film and reduce the friction power of the sub-bearing by more than 80%. By coupling the average Reynolds equation, the contact equation, and the elastic deformation, Hirayama et al. [94] proposed an analytical method for the journal bearings. The simulation results indicated that the optimally smoothed surface roughness improves the ability of the lubrication film formation. With the finite element model and the Reynolds equation, Xie et al. [95] also analyzed the lubrication performance of the bearings. It was found that the bottom of the sub-bearing housing under low speed and the top of the main bearing housing under high speed were in relatively severe working conditions. Based on mixed EHL analysis, Kitsunai et al. [96] found that an annular depression on the shaft surface reduces the losses and increases the volume efficiency. Zhang et al. [88] analyzed the dynamic behaviors of the crankshaft for both single-cylinder and twin-cylinder compressors and discussed the configuration of balancers. The simulation results revealed that proper dynamic balance can improve the wear performance of journal bearings. To realize crankshaft optimization, Meier et al. [97] conducted sensitivity analysis of the surface roughness, bearing lengths, and clearances of the friction power losses. By coupling the crankshaft deformation and the mixed lubrication model, Wang et al. [4] developed a numerical model to evaluate the lubricating condition of the compressor under tested conditions. In the tribodynamic models mentioned above, only the journal bearings were considered, and the effects of the thrust bearing were ignored. To further improve the simulation accuracy, Liu et al. [19] constructed a complete coupled journal–thrust bearing model. Their simulation results showed that high ambient pressure contributes a lot to the dynamic performance of the crankshaft and the wear of the thrust bearing (see Figure 8). With three different stochastic optimization algorithms and a similar tribodynamic model, Lyu et al. [26] further conducted texture optimization for the thrust bearing. The simulation results indicated that the optimized textures greatly reduce the contact forces and wear of the thrust bearing under a high working frequency.

2.4. Whole Compressor

In the references mentioned above, only a part of the moving components was considered in theoretical models and analysis. With the rotary compressor developing towards high working frequency, multibody coupling effects become more and more important, and a tribodynamic model for the whole compressor is required. The theoretical research status on the whole compressor is presented in

Table 3. Theoretical research on the whole compressor.

Type	Year	Reference	Research Content and Findings
1	1984	Jorgensen et al. [98]	Developed a mechanical loss model of rotary compressor;
	2002	Jun et al. [99]	Studied the mechanical loss of a two-cylinder type rotary compressor;
2	2004	Han et al. [100]	Applied multibody dynamic analysis method to study the vibration characteristics of the compressor; Mass of the balancer dramatically affects the acceleration
	2014	Venugopal et al. [101]	Investigated vibration performance of rotary compressor by analyzing dynamic behaviors of moving components, constraint forces, and sliding speed; Speed variation of the crankshaft is one of the major factors to cause the compressor vibration;
3	2014	Mi et al. [20]	Established a tribodynamic model, which ignored roughness effects and the vane dynamics;
	2016	Mi et al. [56]	Conducted thermohydrodynamic lubrication analysis of piston–crank conjunction, main bearing, and sub-bearing;
	2023	Wen et al. [2]	Established an eight-degree-of-freedom tribodynamic model for the whole compressor; Revealed the impacts of the oil supply structures processed on the main bearing, the sub-bearing, and the crank;
	2024	Wen et al. [1]	Developed an eleven-degree-of-freedom tribodynamic model for the whole compressor; Analyzed the impacts of suction pressure, conical structure, and vane width;

. In the early stage, the vibration and mechanical loss of the whole compressor were analyzed. In 1984, Jorgensen et al. [98] developed a mechanical loss model of the rotary compressor, which was verified by their experiments. Based on the dynamic model of the moving parts, Jun et al. [99] studied the mechanical loss of the two-cylinder type rotary compressor. Using a multibody dynamic analysis method, Han et al. [100] studied the vibration characteristics of the compressor. It was found that the mass of the balancer dramatically affects the acceleration. Venugopal et al. [101] also investigated the vibration performance of the rotary compressor by analyzing the dynamic behaviors of the moving components, the constraint forces, and the sliding speed. The results showed that the speed variation of the crankshaft is one of the major factors to cause compressor vibration. By analyzing the hydrodynamic lubrication and dynamics of the rolling piston and the crankshaft, as well as the kinematics of the sliding vane, Mi et al. [20] first established a tribodynamic model for the whole compressor. However, the surface roughness effects and the dynamic performance of the vane were ignored, as well as the dynamic of the crankshaft in the axial direction. Later on, with the aid of the dynamic analysis in Ref. [20], they carried out a thermohydrodynamic lubrication analysis of the piston–crank conjunction, the main bearing, and the sub-bearing [56]. By solving the Reynolds equation and the energy equation, the temperature characteristics of the oil film were presented. To improve the model accuracy of the whole compressor, Wen et al. [2] calculated bearing deformation with the compliance matrix method and established an eight-degree-of-freedom tribodynamic model with the average Reynolds equation, the asperity contact equation, and dynamic differential equations. Based on the solution, the impacts of the oil supply structures on the main bearing, the sub-bearing, and the crank were revealed (see Figure 9). To study the dynamic behaviors of the vane, they further developed an eleven-degree-of-freedom tribodynamic model [1]. With this model, the effects of the vane width, the conical structure, and the suction pressure were revealed (see Figure 10). It was found that the proper conical structure on the vane slot of the suction side can greatly reduce the maximum wear load of the vane slot.

3. Influencing Factors and Problems

During the working process to realize refrigerant compression, the tribological performance of the rotary compressor is influenced by many factors. Some influencing factors have been analyzed comprehensively by many researchers, and other influencing factors remain to be discussed.

Though the lubricant oil is provided for the friction pairs with the aid of an oil pick up at the end of the crankshaft [102,103], starved lubrication will also occur, especially for the vane–slot conjunction. Under a starved lubrication regime, the friction and wear between the mating surfaces are much different from that under a fully flooded lubrication regime, thus influencing the service performance of the compressor. However, in the previous studies, it was generally assumed that all the friction pairs were under a fully flooded lubrication regime. To reveal the inner state of the compressor comprehensively, the effects of starved lubrication need to be studied. However, the oil flow and oil distribution within the compressor are extremely complex [102,103,104], and the starved lubrication analysis is full of challenges. In the field of tribology, there are several methods dealing with the starved lubrication problem. By applying the volume of fluid (VOF) model in the CFD method, the distributions of refrigerant vapor and lubricant oil at the inlet can be defined, thus describing the severity of lubricant starvation. In the previous investigations, this model has been frequently applied to obtain the oil distribution and oil flow in the mechanical components [105,106]. The mass-conserving JFO model can also be used to describe the lubricant starvation with the cavity fraction. The JFO model for the journal bearing in the mixed lubrication analysis can be expressed as follows [40,107,108]:

$${\displaystyle \frac{1}{{R_1}^2}}{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }\theta }}\left({\varnothing }_{\theta }{\displaystyle \frac{\rho h^3}{\eta }}{\displaystyle \frac{\mathit{\partial }p}{\mathit{\partial }\theta }}\right)+{\displaystyle \frac{\mathit{\partial }}{\mathit{\partial }x}}\left({\varnothing }_x{\displaystyle \frac{\rho h^3}{\eta }}{\displaystyle \frac{\mathit{\partial }p}{\mathit{\partial }x}}\right)=6{\omega }_s\left[{\varnothing }_c\rho {\displaystyle \frac{\mathit{\partial }\left[\left(1-\gamma \right)h\right]}{\mathit{\partial }\theta }}+\sigma {\displaystyle \frac{\mathit{\partial }\left[\left(1-\gamma \right)\rho {\varnothing }_s\right]}{\mathit{\partial }\theta }}\right]+12{\varnothing }_c{\displaystyle \frac{\mathit{\partial }\left(1-\gamma \right)\rho h}{\mathit{\partial }t}}$$

(2)

where $\gamma$ denotes the cavity fraction; ${\varnothing }_{\theta }$, ${\varnothing }_x$, ${\varnothing }_c$, and ${\varnothing }_s$ are the flow factors [40,41,107]; $\theta$ and $x$ are the position parameters; $p$ is the oil film pressure; $R_1$ is the journal radius; $t$ represents the time; $\rho$ and $\eta$ are the density and viscosity of the lubricant, respectively; $h$ is the oil film thickness, $\sigma$ is the composite roughness, and ${\omega }_s$ represents the rotation speed. In addition, the inlet boundary condition can also be described with the oil supply flow rate, which has been widely applied in the starved lubrication analysis of friction pairs [109,110]. The corresponding expressions in this method are as follows [109]:

$$Q\left(x_{in},t\right)=Q_s\left(x_{in},t\right)$$

(3)

$$Q\left(x_{in},t\right)=-{\varnothing }_x{\displaystyle \frac{h{\left(x_{in},t\right)}^3}{12\eta }}{\displaystyle \frac{\mathit{\partial }p}{\mathit{\partial }x}}+{\displaystyle \frac{U}{2}}\left[h\left(x_{in},t\right)+\sigma {\varnothing }_s\right]$$

(4)

$$Q_s\left(x_{in},t\right)=U{h}_s$$

(5)

where $x_{in}$ represents the inlet position, $Q\left(x_{in},t\right)$ and $Q_s\left(x_{in},t\right)$ denote the lubricant flow rate and the oil supply flow rate, $U$ is the sliding speed, $h_s$ is the supplied lubricant film thickness, and $h\left(x_{in},t\right)$ is the oil film thickness at the inlet.

Owing to the complex interacting forces, the elastic deformation of the compressor components is an important influencing factor. In general, there are two methods to calculate the elastic deformation, namely the finite element method [4] and the compliance matrix method [19]. Compared to the former, the compliance matrix method has advantages in calculation efficiency, in which the finite element analysis only needs to be conducted to obtain the compliance matrix in advance. With the compliance matrix, the elastic deformation is determined as follows [111,112]:

$$\delta \left(i,j\right)={\displaystyle \sum }_{k=1}^M{\displaystyle \sum }_{l=1}^N\left[p\left(k,l\right)+p_c\left(k,l\right)-p_b\right]D_{i,j}^{k,l}∆A$$

(6)

where $D_{i,j}^{k,l}$ denotes the compliance matrix value; $∆A$ denotes the grid area; $\delta \left(i,j\right)$ is the elastic deformation; and $p$, $p_c$, and $p_b$ are the oil film pressure, the contact pressure, and the boundary pressure, respectively. To date, these two methods have been frequently applied to consider the effects of elastic deformation [2,19,88,94]. For stationary components such as the main bearing and the sub-bearing, both methods have enough accuracy [2,94]. However, for a moving component such as the crankshaft, the finite element method is more suitable by considering it as the flexible body [4,19]. In Liu’s work [19], to obtain the elastic deformation of the crankshaft, the compliance matrix method was still applied with the two-dimensional bearing elements of Combi214 providing the motion constraints. However, the constant bearing stiffnesses were used, which in the fact changed obviously with the eccentricity. In the past, the impacts of the elastic deformation on one moving component were generally conducted. As the tribodynamic model of the whole compressor has been established in recent years, coupling analysis of elastic deformation and multibody movements for the whole compressor can be further carried out.

In addition, the thermal characteristics of the rotary compressor have also been given much attention considering their significant effects. The thermal analysis of the rolling piston type rotary compressor can be divided into three aspects: one is for the friction pairs [56,113], another is for the temperature distribution of the components [29], and the third is for the vapor compression [83,114]. In the last decades, thermohydrodynamic lubrication analysis of the journal bearings in the rotary compressor has been presented by Mi et al. [56,113]. However, mixed thermohydrodynamic lubrication analysis of the compressor has rarely been conducted. In addition, temperature distribution simulations of the compressor can be used to provide the boundary conditions for different friction pairs. The coupling analysis of temperature distribution and mixed lubrication remains to be finished in the future.

During variable-speed processes, such as start-ups and stops, the operation performance of the compressor can be greatly affected. For this reason, the starting and stopping characteristics of the compressor have been studied in the last decades. In the early stage, the starting torque of the rotary compressor was given much attention, which was estimated theoretically and experimentally [115,116]. Based on the experiment results, Ishii et al. [117] analyzed the cylinder pressure, the mechanical vibration, and the crankshaft behaviors after the motor was shut down. Later on, some studies have been carried out to reveal the starting performance of the compressor. Lin et al. [118] experimentally investigated the start-up process of the rotary compressor under a low ambient temperature heating condition, while Wu et al. [119] focused on the cold start-up under cooling conditions. To analyze the temperature distribution during start-up, Guo et al. [120] established a transient mathematical model and discussed the heat exchange characteristics between the solid and the fluid. To reveal the start-up performance of the rotary compressor, Zhao et al. [121] proposed a heat and mass transfer model and studied transient characteristics such as the heat exchange rate and the phase change rate. However, in the references mentioned above, the experimental investigations mainly focused on the pressure, the temperature, the heat exchange rate, and the liquid level of the oil sump, and the tribodynamic modeling and the tribological analysis were rarely seen. During the variable-speed process, the operation condition of the mating surface becomes severe, and the surface wear easily occurs. Variation in the tribological performance of the rotary compressor during the start-ups and the stops is an extremely important topic, which needs to be discussed comprehensively in the future.

4. Conclusions

Owing to the requirements for high performance and long life, the tribological problems of rolling piston type rotary compressors have drawn much attention. The stability and service life of the compressor will be affected when the friction pairs suffer a mixed lubrication regime. In the latest decades, various theoretical models have been developed and improved to reveal the inner state of the compressor and obtain the optimization schemes. However, some disadvantages and research gaps still exist, and a comprehensive summary is lacking. To provide the research status and identify future trends, the theoretical model development and mixed lubrication analyses of the rotary compressor are summarized in this paper. The conclusions can be drawn as follows:

(1)

To provide the pressure boundary conditions, the three-segment function of the compression pressure is widely adopted in current models. To consider the effects of supercharging, thermodynamics, and valve dynamics, more comprehensive analyses of the compression process have been conducted. However, the corresponding methods and models are much more complex, limiting their applications in mixed lubrication analysis.

(2)

Considering the disadvantages of the theoretical models of a part of the moving components or a part of the friction pairs, tribodynamic modeling of the whole compressor has been implemented in recent years. To conduct the modeling study of the vane–piston system and the crankshaft separately, decoupling of the moving components needs to be achieved, resulting in the extra assumptions. To improve decoupling model accuracy, second-order motion, elastic deformation, and groove distribution have been considered in the previous studies. With the increasing effects of multibody coupling, the modeling and analysis of the whole compressor is the preference.

(3)

Based on mixed lubrication analyses, various optimal designs of the compressor have been achieved. Optimized textures can greatly reduce the contact forces and wear of the thrust bearing under a high working frequency. An optimal vertical groove can reduce the friction loss and wear of the vane–slot conjunction. Proper conical structure on the vane slot of the suction side can greatly reduce the maximum wear load of the vane slot. In addition, optimally smoothed surface roughness can improve the ability of the lubrication film formation of journal bearings.

(4)

To reveal the service performance of the rotary compressor comprehensively, some important influencing factors need to be further analyzed in the future. Starved lubrication analysis of the compressor is full of challenges and has not been carried out so far. The coupling analysis of elastic deformation and multibody movements for the whole compressor needs to be addressed more. The thermal performance analysis of friction pairs under mixed lubrication regime is also very limited. During variable-speed processes, the lubrication and wear characteristics of the rotary compressor have not been investigated theoretically, which is also an important topic for the future study.