Effects of leakage and friction on the miniaturization of a Wankel compressor

Abstract

This paper presents a numerical simulation of the performance of a meso-scale Wankel compressor and discusses the factors affecting its miniaturization. The discussion is related to the effect of leakage and friction on the design limit (cooling capacity and dimension) of the meso Wankel compressor. In the simulation, the main leakage comes from the gaps between the rotor and the endplates as well as between the seal apex and the cylinder. The largest friction originates from the clearance among the end face of the eccentric shaft, the end faces of the rotor, and the endplates. The decreasing cooling capacity of the meso Wankel compressor increases the proportion of leakage to displacement and causes the coefficient of performance COP and the mechanical efficiency to decrease. The rational design cooling capacity limit for the meso-scale Wankel compressor is approximately 4 W.

Keywords

meso-scale / Wankel compressor / leakage / friction

Introduction

Highly efficient mini and meso refrigeration systems are widely needed in the field of electronic cooling. The Wankel compressor has good potential for the meso vapor compression refrigeration system because of its advantages, such as simple structure, high efficiency, low vibration, and low noise.

The Wankel compressor is similar to the Wankel engine and rotary compressor in terms of structure and operation; it consists of a rotor, a cylinder, a shaft, a pair of gears and apex seals, and three enveloped chambers (V₁, V₂, V₃) (Fig. 1). The driving structure of a Wankel compressor is made up of a gear pair and an eccentric shaft, which drives the rotor to make planetary motion in the cylinder of the compressor. The angular speed of the eccentric shaft is three times that of the rotor; thus, two refrigeration cycles are accomplished when the eccentric shaft completes a revolution.

The Wankel machine is a kind of rotary machine. Great progress has been made in terms of research about Wankel and rotary machines in the last century. Numerical simulations have contributed greatly in the research and development of Wankel and rotary machines. Pennock and Beard [1] derived equations for the radial and transverse components of the acceleration of an apex seal in the rotor of a Wankel rotary compressor and made a dynamic force analysis of the seal, including the friction forces between the tips of the seal and the chamber as well as between the side of the seal and the rotor. Heppner et al. [2-4] analyzed the leakage flow and the friction loss of rotary engine and compressor and established design parameters for micro engine sealing systems. Prater and William [5, 6] described the methodology and results of an experiment to measure the fundamental undamped natural frequency and damping ratio for the discharge reed valve in a rolling piston rotary compressor. Hsiao et al. [7-10] presented the simulation of a rotary compressor and its performance comparison with measured results.

Several factors, such as friction loss, leakage, the rotor and cylinder profiles, and the positions of the inlet and exhaust ports, should be considered in designing a meso Wankel compressor. The introduction of computer simulation can greatly shorten the design period and reduce design cost. Therefore, numerical simulation is necessary to obtain the rational design dimensions. The influences of some factors, such as leakage and the friction loss, on the performance of a meso Wankel compressor differ from the effects on a conventional Wankel compressor. Based on current manufacturing and lubrication technologies, the influences of leakage and friction loss increase continuously with the gradual decrease of the design dimensions and the cooling capacity. The displacement and the mechanical efficiency drop greatly when the design dimensions and the cooling capacity reach certain values. Thus, the limit of rational design dimension should be analyzed in designing a meso Wankel compressor.

This paper presents a numerical simulation of the performance of a meso-scale Wankel compressor and discusses the design limit of the cooling capacity and dimensions for a meso Wankel compressor. The simulation model is focused on the cycle performance. The mechanical optimization simulation is performed to search for the optimum dimensions of a meso Wankel compressor. The design limit has been obtained based on the analysis of leakage and friction loss of the meso Wankel compressor.

Mechanical optimization

Friction in a Wankel compressor

Friction loss usually plays an important role in compressor performance. In meso and mini machines, the ratio of surface by volume to the contact surfaces with friction are large and cannot be lubricated adequately as normal scale machines. In this simulation model, eight friction pairs were taken into account and listed as follows:

1) The friction loss between the main shaft and the main bearing, L₁,

(1)L_{\mathrm{1}}={\displaystyle \frac{\mathrm{2}\mathrm{\pi }\mu {\omega }^{\mathrm{2}}{R}_{\mathrm{s}}^{\mathrm{3}}{l}_{\mathrm{m}}}{c_{\mathrm{m}}}}\mathrm{.}

2) The friction loss between the main shaft and the accessory bearing, L₂,

(2)L_{\mathrm{2}}={\displaystyle \frac{\mathrm{2}\mathrm{\pi }\mu {\omega }^{\mathrm{2}}{R}_{\mathrm{s}}^{\mathrm{3}}{l}_{\mathrm{m}}^{\prime }}{c_{\mathrm{m}}^{\prime }}}.

3) The friction loss between the eccentric bearing and the eccentric shaft, L₃,

(3)L_{\mathrm{3}}={\displaystyle \frac{\mathrm{2}\mathrm{\pi }\mu {(\omega -{\omega }_{\mathrm{r}})}^{\mathrm{2}}{R}_{\mathrm{e}}^{\mathrm{3}}{l}_{\mathrm{e}}}{{\delta }_{\mathrm{3}}}}.

4) The friction loss between the seal apex and the internal surface of the cylinder, L₄ [A.10],

(4)L_{\mathrm{4}}={\displaystyle \frac{z}{\mathrm{540}}}·\left[{\displaystyle \sum _{\partial =\mathrm{1}°}^{\mathrm{270}°}{F}_{t\mathrm{2}}{\displaystyle \frac{v_{\mathrm{T}}\sin \psi }{\sin (\theta +\psi )}}+{\displaystyle \sum _{\partial =\mathrm{271}}^{\mathrm{540}}{F}_{t\mathrm{1}}{\displaystyle \frac{v_{\mathrm{T}}\sin \psi }{\sin (\theta -\psi )}}}}\right].

5) The friction loss between the apex seal and the sealing groove, L₅ [A.13],

(5)L_{\mathrm{5}}={\displaystyle \frac{z}{\mathrm{540}}}{\displaystyle \sum _{\partial =\mathrm{1}°}^{\mathrm{540}°}{f}_{\mathrm{2}}(p_{\mathrm{g}\mathrm{s}}-F_{\mathrm{T}})v_{\mathrm{R}}}.

6) The friction loss among the end face of the eccentric shaft, the end faces of the rotor, and the endplates, L₆,

(6)L_{\mathrm{6}}=\mathrm{\pi }\mu \left\{{\displaystyle \frac{{\omega }_{\mathrm{r}}^{\mathrm{2}}[(R_{\mathrm{r}}^{\mathrm{4}}-R_{\mathrm{g}\mathrm{e}}^{\mathrm{4}})+(R_{\mathrm{r}}^{\mathrm{4}}-R_{\mathrm{e}}^{\mathrm{4}})]}{{\delta }_{\mathrm{6}\mathrm{a}}}}+{\displaystyle \frac{{\omega }^{\mathrm{2}}(\mathrm{2}{R}_{\mathrm{e}}^{\mathrm{4}}-R_{\mathrm{s}}^{\mathrm{4}})}{{\delta }_{\mathrm{6}\mathrm{b}}}}\right\}.

7) The friction loss between the gears, L₇ [A.17].

(7)L_{\mathrm{7}}={\displaystyle \frac{f_{\mathrm{7}}{l}_z{z}_{\mathrm{1}}Me{\omega }^{\mathrm{3}}}{\mathrm{2}\mathrm{\pi }\sin \beta }}.

In this work, the friction losses of L₄, L₅, and L₇ were derived on the basis of friction principle. Detailed derivation is depicted in the Appendix, and the friction loss of L₁, L₂, L₃, and L₆ are cited from [3, 4, 9-11].

Optimization to reduce friction in a Wankel compressor

Given that friction is unavoidable in a Wankel compressor, it is necessary to search for the optimum parameters based on the minimum friction loss at given operational conditions (Table 1). The complex optimization method was used in this study. The objective function is to search for the minimum total friction loss of the Wankel compressor.

Here, nine design variables were chosen as optimal parameters based on the analysis of friction losses (L₁-L₇) in Eqs. (1)-(7). The design variables included the thickness and height of apex seal, the shape factor, the offset, the radius of the main shaft, the radius and height of the eccentric shaft, the number of tooth of the external gear, and the modulus of the gear.

The optimization can be described as

The objective function is

(8)F\to F_{\min }=L_1+L_2+L_3+L_4+L_5+L_6+L_7=f\left(b_s,h_s,K,a,R_r,R_e,h_e,z_1,m_g\right)\mathrm{a}\mathrm{n}\mathrm{d}

is subjected to the following constraints:

Explicit constraints:

(9)a_j\le x_j\le b_j(j=\mathrm{1},\mathrm{2},\mathrm{...},\mathrm{9}).

Implicit constraints:

(10)g_i(X)\ge \mathrm{0}(i=\mathrm{1},\mathrm{2},\mathrm{...},\mathrm{5}).

In Eq. (9), the x_j (j=1,2,…,9) are the explicit constrains represented by the parameters b_{\mathrm{s}},h_{\mathrm{s}},K,a,R_{\mathrm{s}},R_{\mathrm{e}},h_{\mathrm{e}},z_{\mathrm{1}}, and m_{\mathrm{g}}, respectively. The values of a_j and b_j (Table 2) are the range limits. In Eq. (10), g_i (i=1,2,…,5) is the implicit constraint, and i is the number of implicit constraints. Table 3 lists the limit values of the implicit constraints.

The optimum results in terms of dimensions and friction losses obtained from the optimization are shown in Tables 4 and 5, respectively. Results show a total friction loss of 11.03 W; the largest friction loss comes from L₆, which is equivalent to approximately 63.29%.

The design dimensions (Table 4) shows the creation of a Wankel compressor prototype and the experiment conducted on friction loss under normal atmosphere. Friction losses were obtained by detaching the friction parts of the Wankel compressor prototype through a step-by-step process, after which the corresponding output power of the motor was measured. The predicted and the experimental results are compared in Table 6. The relative error of total friction loss is about -5.53%, indicating that the rationality of the friction loss equations and the optimization model of Wankel compressor are feasible.

Thermodynamic analysis of a meso Wankel compressor

Leakage of a Wankel compressor

Gas leakage occurs in all the gaps connected to the chamber (Fig. 2). The leakage in a Wankel compressor can be divided into internal and external leakages. The internal leakage in a Wankel compressor may occur in the four gaps between the seal apex and the cylinder (m_scy), the seal sides and the endplates (m_sc), the seal and the sealing groove (m_sg), and the rotor and the endplates (m_rc). The external leakage mainly occurs in the gap between the main bearing and the main shaft (m_bs) and can be neglected by the rational shaft seal design.

The gaps of leakage m_scy are very short; therefore, the leakage process is simplified as a gas flow through the convergent nozzle. The leakage can be calculated by

(11)m=\varphi Ag(p_{\mathrm{i}},{\rho }_{\mathrm{i}},p_{\mathrm{o}}),

where

(12)

The other three leakage gaps employ long narrow passages compared with their height and are simplified as a convergent nozzle with equal section straight pipe. The mass flow can be expressed as [11]:

(13)m=\delta H{p}_{\mathrm{e}}{v}_{\mathrm{e}}/R_{\mathrm{g}}{T}_{\mathrm{e}}.

Analysis of compression process

The optimum dimensions and the operation condition are presented in Tables 4 and 1, respectively. Predicted results, such as the variation of volume, pressure, temperature, and mass in three chambers in a Wankel compressor with different eccentric angle, were obtained by numerical simulation. Other performance parameters, such as cooling capacity, leakage, intake mass, exhaust mass, and so on, were obtained as well. The typical simulation results are demonstrated in Fig. 3. The variations of parameters in the three chambers (V₁, V₂, and V₃) are the same. The V₃ chamber was taken in this study as a sample to be analyzed in detail.

Figure 3(a) shows the variation of pressure in three chambers of the Wankel compressor. The pressure (p_V₃) remained almost unchanged during the intake process. The compression process commenced when the volume started to increase until the volume reached the maximum. The pressure increased continually with the volume decrease until the pressure reached the back pressure (the eccentric angle is approximately 360° in this paper). The pressure was almost stable at a certain level in the exhaust process until the exhaust valve shut off (at this point, the eccentric angle was approximately 450°). The operation process transitioned to the expansion process at the closing of the exhaust valve and ended at the opening of the intake valve, thereby completing the whole cycle.

The variation of gas temperature shown in Fig. 3(b) is proportional to the gas pressure in the chamber. In the intake process, the intake gas was heated by the high-temperature gas leaked from the other two chambers and the high-temperature wall of the rotor, cylinder, and endplates, resulting in a slight increase of gas temperature in the intake process. In the compression process, the temperature increased rapidly with the increase of pressure, and the gas temperature decreased continually in the exhaust and the expansion process.

Figure 3(c) illustrates the variation of the gas mass in three chambers of the Wankel compressor. The gas mass maintained constant increase in the intake process and decreased continually in the exhaust process. The gas mass had a slight increase in the early stage of the compression process and a slight decrease in the later stage of the compression process. This condition was conversed in the expansion process. This mechanism is attributed to the gas pressure in chamber V₃ being the lowest of the three chambers at the beginning of compression, causing the increased mass to come out from the other two chambers through the leakage gaps. With the continued compression process, the gas pressure in chamber V₃ increased continually, and the gas mass decreased when the mass exchange with the other two chambers became negative. The mass variation in the expansion process is similar to the compression process.

The variation of the intake mass and the exhaust mass are shown in Fig. 3(d). The intake mass flow increased at first and then decreased in the intake process. This is a phenomenon, which is related with the volume variation of chambers. The exhaust mass flow decreased continuously in the exhaust process. The exhaust rate was higher than the intake rate because the gas density in the exhaust chamber was higher than that in the intake chamber.

Figures 3(e) and 3(f) show the variation of leakage. The main leakage ways are shown by the gap between the rotor and endplates and that between the seal apex and cylinder, accounting for 42% and 28% of the total leakage, respectively.

Design limit of a meso Wankel compressor

The impact of leakage and friction loss on compressor performance gradually increases with the decrease of the meso Wankel compressor dimension. Accordingly, there are two factors determining the design limit of the meso Wankel compressor. First, with an increasingly serious leakage situation, lesser compressed gas is pushed out of the exhaust valve, making it difficult for the gas pressure in the compression chamber to reach the back pressure. Therefore, one design limit for the meso Wankel compressor was that the exhaust valve cannot be opened during the entire operation. Friction loss served as another determinant of the design limit. The proportion of friction loss to shaft power continuously increased, and the mechanical efficiency gradually decreased with the decrease of compressor dimension. Another design limit was set when the mechanical efficiency reached below a certain value (50% in this paper).

In the calculation, the meso Wankel compressor has the same operation conditions except the cooling capacity (as presented in Table 1) and machining tolerance (5 μm). In this work, the initial cooling capacity of the Wankel compressor was estimated on the basis of mathematical model without considering leakage and friction loss; in addition, the volumetric efficiency of the Wankel compressor was assumed as 80%. Table 7 lists a set of optimization dimensions obtained by the optimization model of the meso Wankel compressor. The corresponding performance parameters obtained by the simulation model are respectively presented in Tables 8 and 9.

Table 8 shows that the proportion of the leakage to the gas displacement increases with the decrease of the cooling capacity of the meso Wankel compressor. The deviation between the actual cooling capacity and the initial design value is magnified gradually. Under 5 μm machining tolerance, the leakage between the three chambers is higher than the displacement when the initial cooling capacity reaches 10 W, and the actual cooling capacity is only approximately 4 W. The compressed gas cannot reach the back pressure, and the exhaust valve cannot be opened when the initial cooling capacity is approximately 5 W; otherwise, the meso Wankel compressor cannot work normally.

Table 9 shows the variation of friction loss, shaft power, coefficient of performance (COP), and mechanical efficiency with different cooling capacities of the meso Wankel compressor. The COP and the mechanical efficiency gradually decrease with the decrease of the cooling capacity; both parameters drop to 2.1 and 44.6%, respectively, when the initial cooling capacity of the meso Wankel compressor reaches approximately 10 W.

From the above analysis, the rational cooling capacity limitation obtained for the meso Wankel compressor is approximately 4 W.

Conclusion

Based on the predicted results of the model for the meso Wankel compressor, the variation of pressure, temperature, mass, and leakage in chambers are analyzed in detail in this paper. The analysis has shown that the main leakage stems from the gap between the rotor and the endplates and that between the seal apex and the cylinder.

The optimization for the meso Wankel compressor is discussed based on the friction loss analysis. There are seven kinds of friction loss (L₁-L₇) discussed in this paper. The greatest friction loss comes from L₆, which contributes to approximately 63.29% of the total friction loss.

The feasibility of the system model and the optimization model of the meso Wankel compressor has been proven by the friction loss experiment at normal pressure on a meso Wankel compressor prototype. The relative error of the total friction loss between the predicted and the experimental results is approximately -5.53%.

The impacts of leakage and friction loss are mainly considered in analyzing the design limit of the meso Wankel compressor. With the decrease of the cooling capacity, the proportion of leakage to displacement has been found to gradually increase, and the COP and the mechanical efficiency gradually decreased. The rational cooling capacity limit for the meso Wankel compressor has been found to be approximately 4 W, while those for COP and mechanical efficiency are 2.1% and 44.6%, respectively.

The simulation model has been used in assisting the design of the meso Wankel compressor for microsystems. It also provides a way for more comprehensive simulation studies and for possible overall computer optimization design study.

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