# Factors affecting the cylinder displacement of a Wankel compressor in a micro-cooling system.

The factors that influence the performance of a Wankel compressor mainly include the compression ratio and dimensions (shape factor,
eccentric distance, and cylinder height). In this study, theoretical
equations are derived to analyze the effect of these factors and carry
out a transient simulation of the non-steady-state behavior of a
meso-Wankel compressor using a dynamic mesh to support the theoretical
equations. The simulation and theoretical results agree well with each
other. Cylinder displacement gradually increases with an increasing
shape factor, whereas cylinder displacement per unit cross-sectional
area gradually decreases. Both the cylinder displacement and cylinder
displacement per unit cross-sectional area increase with different
eccentric distances, and this growth follows a converse trend. The
optimum shape factor is approximately 6 for the Wankel compressor
design. On the basis of the design parameters in this article, optimum
cooling capacities of 418 and 352 W are obtained from the theoretical
and simulation results, respectively. The relative deviation is about
16%.

Introduction

With the increasing requirement for temperature control of meso- and micro-facilities, the demand for micro-cooling systems has become increasingly urgent in recent years. Meso- and micro-compressors act as the core parts of a cooling system, and Wankel-type units have become prime choices for micro-cooling systems because of their unique advantages, including simplicity in structure, high efficiency, and low noise. Some achievements in research on the micro-Wankel engine and micro-Wankel compressor include prototype design and manufacturing, as well as corresponding experimental research (Heppner et al. 2007; Lee et al. 2004; Fu et al. 2001). Martinez et al. (2003) and Swanger et al. (2004) improved the sealing mode of the Wankel engine, thereby increasing the sealing efficiency by 10% to 50%. A numerical simulation of the performance of a meso-scale Wankel compressor was conducted, and the factors affecting the miniaturization of the compressor (leakage, friction loss, and design limit) were discussed (Zhang and Wang 2011).

In decreasing the dimensions of a Wankel compressor to satisfy the miniaturization requirements of a micro-cooling system, the surface-to-volume ratio becomes larger, and the surfaces in relative motion cannot be adequately lubricated as in the normal scale. Because of these effects, the performance of meso- and micro-Wankel compressors tremendously decreases, and high efficiency becomes difficult to achieve. Therefore, strategies for improving the performance of a Wankel compressor have become an important issue, especially for micro--and meso-Wankel compressors.

Many factors affect the performance of a Wankel compressor. Such factors mainly include the compression ratio and the dimensions affecting the profile and volume of the working chamber. The dimensions are the shape factor and eccentric distance, which affect the profile of the chamber, as well as the height of the cylinder.

In the current work, the effects of these factors on the performance of a Wankel compressor are analyzed in detail. Related equations are also obtained. The theoretical analysis results are supported by the simulation. The variations in flow field, cylinder displacement, exhaust time, cooling capacity, and cooling capacity per unit cross-sectional area of the cylinder are analyzed under the corresponding operational conditions. Finally, a Wankel compressor prototype is designed based on the optimum dimensions obtained from the theoretical and simulation analyses.

Performance of a Wankel compresssor

The P-V diagram of the thermal process of a Wankel compressor is shown in Figure 1. The sequence 1-2-3-4-1 represents the ideal thermal process of a Wankel compressor. Sequence 1-2 illustrates the compression process, in which the gas pressure in the compression chamber gradually increases with decreasing volume. The gas pressure continues to increase until it reaches the point of back pressure, after which the operation turns into an exhaust process, as shown in sequence 2-3. The gas pressure is kept stable during the exhaust process until the minimum volume of the exhaust chamber is achieved. At this time, the exhaust valve is shut, and the expansion process begins. The gas pressure declines with increasing clearance volume until the pressure reaches the intake process, as illustrated in sequence 3-4. The operation process transforms into the intake process (sequence 4-1), and the pressure is kept unchanged until the volume of the intake chamber reaches its maximum.

[FIGURE 1 OMITTED]

Cylinder displacement of a Wankel compressor

The cylinder displacement of a Wankel compressor is calculated by the difference between the compression chamber volume at the end of the compression process ([V.sub.2]) and the exhaust chamber volume at the end of the exhaust process ([V.sub.3]) (Figure 1):

[V.sub.d] = [V.sub.2]- [V.sub.3]. (1)

The compression process in the compression chamber is assumed to be a polytropic process, expressed as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2)

The cylinder displacement of a Wankel compressor can be expressed as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3)

where y is the compression ratio, [gamma] = [P.sub.2]/[P.sub.1].

A Wankel compressor consists of three working chambers, and each volume is bounded by the internal surface of the cylinder, rotor, apex seals, and two endplates. The volume of the working chambers can be described as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4)

The Wankel compressor must have clearance between the rotor and internal surface of the cylinder because of the limitation in machining accuracy and the need for friction reduction. Considering clearance (also called offset a, as shown in Figure 2), the volume of the working chamber of a Wankcl compressor can be approximately calculated using the following equation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (5)

[FIGURE 2 OMITTED]

The maximum ([V.sub.1]) and minimum volumes ([V.sub.3]) of the working chamber are expressed as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], (6)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (7)

Volumes [V.sub.1] and [V.sub.3] are substituted into Equation 3. The displacement of the Wankel compressor can then be obtained by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. (8)

The rotor and cylinder profiles are vital to the shape and volume of the working chamber. The corresponding cross-sectional area of the working chamber plays a decisive role in cylinder displacement and in the performance of the Wankel compressor. The profile equations of the Wankel compressor cylinder are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (9)

The cross-sectional area of the cylinder can be calculated by the column coordinate equations of a cylinder, expressed as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (10)

where [beta] = [pi]/3

The cylinder displacement per unit cross-sectional area of a Wankel compressor is obtained by the ratio of cylinder displacement [V.su.d] to the cross-sectional area of cylinder A:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (11)

According to Equations 8 and 11, the factors causing cylinder displacement in a Wankel compressor include the compression ratio ([lambda]), shape factor (K), eccentric distance (e), offset (a), and cylinder height (a).

Results and discussion of the cylinder displacement of a Wankel compressor

According to theoretical Equations 8 and 11, the cylinder displacement and cylinder displacement per unit cross-sectional area of a Wankel compressor can be obtained using different parameters. Figures 3-5 show the results at different compression ratios, shape factors, and eccentric distance (the offset is assumed to be 0.5 mm).

At shape factor, eccentric distance, and cylinder height values of 6, 3 (0.118 in.) and 11.5 mm (0.453 in.), respectively, the cylinder displacement and cylinder displacement per unit cross-sectional area gradually decrease with increasing compression ratio--the rate at which the displacement decreases continuously drops, as shown in Figure 3. The other parameters are unchanged, except for the shape factor; the compression ratio is 3.5. The cylinder displacement constantly increases with increasing shape factor, whereas the cylinder displacement per unit cross-sectional area shows an opposite trend (Figure 4). The common range of the shape factor is from 6 to 7 (Zhang and Wang 2011) in the design of a Wankel compressor. To make the compressor more compact, the cylinder displacement per unit cross-sectional area should be as large as possible. Therefore, the shape factor of a Wankel compressor should have a value of around 6, especially for the meso- and micro-Wankel-type units.

[FIGURE 3 OMITTED]

Figure 5 shows that the cylinder displacement and cylinder displacement per unit cross-sectional area continuously increase with increasing eccentric distance under the same parameters (K = 6, B = 11.5 mm [0.453 in.], and [lambdda] = 3.5). The growing rate of the cylinder displacement continuously increases with the increase in eccentric distance, but the cylinder displacement per unit cross-sectional area exhibits a converse growth rate.

According to the analysis above, the cylinder displacement of a Wankel compressor is primarily affected by the shape factor, eccentric distance, and cylinder height on the basis that the compression ratio is determined by the design parameters. Among these design parameters, the shape factor and eccentric distance are the most important factors that influence the dimensions and performance of a Wankel compressor. Therefore, the effect of the shape factor and eccentric distance on the performance of Wankel compressors is analyzed using computational fluid dynamics.

Numerical simulation of a Wankel compressor

Initial conditions of a Wankel compressor

In general, the flow field in a Wankel compressor is three dimensional. The degree of importance of the three-dimensional effects is related to the position of the inlet and outlet, as well as their heights. In the present investigation, the inlet and outlet are assumed to be uniformly spread along the height of a Wankel compressor (Figure 6). Therefore, the three-dimensional effects can be disregarded, and two-dimensional models can be used for the simulations in this study.

[FIGURE 4 OMITTED]

For a simplified simulation of a Wankel compressor, the following assumptions are made:

1) the three-dimensional effects on the flow field are disregarded, and the two-dimensional model is used for the simulation;

2) the seals apex comes into contact with the internal surface of the cylinder, indicating the leakage between the three chambers is disregarded and no fluid exchange occurs between the three chambers;

3) the friction between moving parts of the compressor parts (that is, not considering friction heat) is disregarded; and

4) all solid boundaries are assumed adiabatic.

[FIGURE 5 OMITTED]

The design of a Wankel compressor is based on its cooling capacity and the state parameters of intake and exhaust gases. In this study, the simulation object is a meso-Wankel compressor, and the operational parameters are shown in Table 1.

[FIGURE 6 OMITTED]

Simulation method for a Wankel compressor

The simulation of the internal flow field of the compressor is carried out using the fluid dynamic software FLUENT (2D). The governing equations for the simulation include the conservation equations of mass, momentum, and energy. A finite-volume approach and the implicit-difference method based on pressure are used to solve the governing equations. A second-order upwind scheme is selected to discretize the governing equations, and the SIMPLE algorithm is used for pressure-velocity coupling.

In general, Wankel compressors have high rotation speed, and the internal flow field is a complex turbulent flow. Aside from the governing equations, turbulence models for the calculation of Reynolds stresses are also added to the solution equations for the simulation. For turbulence modeling, the RNG [kappa]-[epsilon] model is used, while the turbulent viscosity is computed from a differential equation. This model is a modified representation of the turbulent flows with Reynolds number in a large range and a case with curved streamlines in circulation regions. The modified terms in Equation e of the RNG [kappa]-[epsilon] model enhance the dissipation rate in the vicinity of the stagnation region and prevent the augmentation of turbulent kinetic energy in this region.

The model of the Wankei compressor is meshed with an unstructured grid. The meshes near the apex seal are refined to ensure that the minimum clearance has a three-layer mesh, which is the minimum number of mesh layers required in simulating fluid flow. The time step is 0.000005 s, while the rotating speed of the compressor is 1800 rpm in the simulation processes. The operation of a Wankel compressor is an unsteady process, and the shape and volume of the working chambers constantly change as the rotor rotates. Therefore, a dynamic mesh should be used in simulating Wankel compressors; in the current study, a user-defined function is created to define the movement principle and control rotor movement.

In accordance with the analysis above, the following boundary conditions are used in the simulation.

Inlet and outlet boundary condition: the pressure boundary condition is used for the inlet and outlet, and the total and static pressures are determined based on initial conditions.

Wall boundary condition: the effect of heat transfer is disregarded, and a no slip boundary condition is used for the internal surface of the cylinder and external surface of the rotor. The rotor has a rotating wall, and the rotation speed is about 600 rpm.

Boundary conditions in relation to turbulence: Standard wall functions are used for the turbulence model. Turbulent kinetic energy and the turbulent kinetic energy dissipation rate can be obtained by the following empirical formula:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (12)

This article focuses on the influence of the shape factor and eccentric distance on the performance of a Wankel compressor.

First, shape factor K, which controls the shape of the cylinder and the rotor, is considered, using the same eccentric distance and the ideal design of the inlet and outlet (the intake and exhaust are controlled by the valve on the inlet and outlet) as a basis. A reasonable shape factor is obtained in designing the Wankel compressor.

Second, the effect of eccentric distance is taken into account on the basis of the ideal design of the inlet and outlet, as well as the shape factor obtained in the previous step. The reasonable eccentric distance is determined using the design parameters in Table 1.

Finally, a meso-Wankel compressor prototype is created on the basis of the results from the above-mentioned steps and the simulation carried out in this article.

During the simulation, variations in some parameters of the working chamber should be monitored to ensure the reliability of the simulation results. These parameters include gas pressure, gas temperature, and gas velocity.

Effect of shape factor on performance under the same cylinder heights and eccentric distances

To analyze the influence of the shape factor on the performance of a Wankel compressor, the following initial conditions should be set:

1) an eccentric distance of 3 mm (0.118 in.);

2) a cylinder height of 11.5 mm (0.453 in.);

3) the ideal design of the inlet and outlet is used in the simulation, and the intake and exhaust gases are controlled by the valves set on the inlet and outlet; and

4) the shape factor is usually within the range of 6 to 7 for a Wankel machine. To increase the universality and applicability of the simulation, the shape factor value chosen is 5.5 to 7.5.

[FIGURE 7 OMITTED]

Figures 7-9 show that the opening time of the exhaust valve gradually advances with increasing shape factor, and the corresponding exhaust time continuously increases. The cylinder displacement, exhaust mass flow, and cooling capacity exhibit the same growth trend as the shape factor increases. The cooling capacity per unit cross-sectional area of the cylinder decreases with an increasing shape factor, and better performance is achieved when the shape factor of a Wankel machine is in the range of 6 to 7. In accordance with the decreasing trend of the cooling capacity per unit cross-sectional area of the cylinder with increasing shape factor, the value of ~6 is taken as tile optimum shape factor to make the Wankel compressor more compact, especially for meso- and micro-Wankel compressors.

Effect of eccentric distance on performance under the same shape factors and cylinder heights

On the basis of the analysis above, ~6 is chosen as the value of the shape factor of the Wankel compressor. In addition to the shape factor, the following initial conditions should be established to analyze the influence of the eccentric distance on the performance of the compressor:

1) a cylinder height of 11.5 mm (0.453 in.);

2) the ideal design of the inlet and outlet is used in the simulation, and the intake and exhaust gases are controlled by the valves set on the inlet and outlet; and

3) in accordance with the geometrical relationship and strength requirement of a Wankel compressor, the eccentric distance is restricted in the range of 2 to 5 mm (0.0787 to 0.197 in.) (Zhang and Wang 2011).

[FIGURE 8 OMITTED]

[FIGURE 9 OMITTED]

As shown in Figures 10-12, the exhaust time, cylinder displacement, exhaust mass flow, cooling capacity, and cooling capacity per unit cylinder cross-sectional area continually increase with an increasing eccentric distance. However, their growth rates are different from one another. Among the growth rates, those of the exhaust time and cooling capacity per unit cross-sectional area gradually decrease with the increase in eccentric distance. The growth rates of the other parameters continuously increase.

The simulation results for the effect of the shape factor and eccentric distance on the performance of the Wankel compressor agree well with the theoretical results. The reasonable shape factor is approximately 6. In accordance with the design cooling capacity and considering a certain surplus in cooling capacity (because leakage is not considered in this article), a value of 3 mm (0.118 in.) is taken as the eccentric distance.

[FIGURE 10 OMITTED]

Simulation of the Wankel compressor prototype

A meso-Wankel compressor prototype (Figure 13) is created based on the analysis, in which shape factor and eccentric distance are 6 and 3 mm (0.118 in.), respectively. The prototype is composed of a main shaft, gear pairs, seals, rotor, cylinder, endplates, and valve plates. The cylinder and rotor are both 11.5 mm (0.453 in.) high, and the compressor has a footprint of 40 x 50 mm (1.575 x 1.969 in.) and a thickness of 24 mm (0.945 in.).

[FIGURE 11 OMITTED]

The simulation of the meso-Wankel compressor prototype is conducted, and a variation in gas pressure in the working chambers ([V.sub.1], [V.sub.2], and [V.sub.3]) is shown in Table 2. Working chamber [V.sub.3] is taken as an example in comprehensively analyzing the pressure variation.

The exhaust process of working chamber [V.sub.3] ends when the time is 0 s, at which the volume reaches its minimum and the operation turns into the expansion process. With increasing [V.sub.3] volume, the gas pressure gradually decreases, and the process turns into the intake process until the gas pressure in [V.sub.3] reaches the point of intake pressure. The intake time is 0.00855 to 0.025 s, and the volume of [V.sub.3] reaches its maximum at the end of the intake process. The compression process then begins. The gas pressure in [V.sub.3] continuously increases with the decrease in the volume of [V.sub.3] until the gas pressure reaches the point of back pressure (1.2 MPa in this article). The process of the Wankel compressor then turns into the exhaust process, which ends when the volume of [V.sub.3] reaches its minimum. The exhaust time is approximately 0.00804 s (from 0.04196 to 0.05 s) for the Wankel compressor with valves on the inlet and outlet. The corresponding cooling capacity is about 352 W.

[FIGURE 12 OMITTED]

[FIGURE 13 OMITTED]

According to the dimensions of the mesoWankel compressor prototype, the maximum and minimum volumes can be measured and substituted into Equation 3. The cylinder displacement and corresponding cooling capacity are approximately 900 [mm.sup.3] (0.055 in. (3)) and 418 W The relative deviation is about 16% compared with the simulation results.

Conclusion

Through simulation and theoretical analyses, it was determined that the compression ratio and dimensions (shape factor, eccentric distance, and cylinder height) considerably affect the performance of a Wankel compressor.

According to the consistency of results between the simulation and theoretical discussions, the cylinder displacement constantly increases with an increasing shape factor, whereas the cylinder displacement per unit cross-sectional area gradually decreases. Both the cylinder displacement and cylinder displacement per unit cross-sectional area of the cylinder increase with increasing eccentric distance and have a linear increase with cylinder height. The same changing trends apply to the cooling capacity and cooling capacity per unit cross-sectional area.

The optimum shape Factor is approximately 6 for the Wankel compressor, and the value of 3 mm (0.118 in.) is taken as the eccentric distance based on the design cooling capacity and considering a certain surplus of cooling capacity. The cooling capacity obtained from the theoretical and simulation results are 418 and 352 W, respectively, and the relative deviation is about 16%.

DOI: 10.1080/10789669.2012.635948

Acknowledgment

This work was supported by the National Natural Science Foundation of China (grant 50976067).

Nomenclature

a = offset, mm (in.)

A = cross-sectional area of cylinder, [mm.sup.2] ([in.sup..2])

B = height of the cylinder, mm (in.) [C.sub.[micro]] = constants of the turbulence model, the default value [c.sub.[mu]]= 0.09 is used in this simulation

e = eccentric distance, mm (in.)

I = turbulence intensity

K = shape factor of the Wankel compressor, equal to the ratio of generation radius and eccentric distance

l = turbulence length, m (in.)

n = polytropic exponent

[p.sub.1] and [p.sub.2] = gas pressures before and after compression process, Pa (psi)

R = generation radius of the rotor, mm (in.)

u = mean velocity, m * [s.sup.-1] fit * [s.sup.-1])

[V.sub.1], [V.sub.3], and [V.sub.2] = maximum chamber volume, minimum chamber volume, and volume of the compression chamber at the end of the compression process, [mm.sup.3] ([in..sup.3])

[V.sub.d] and [V.sub.ds] = cylinder displacement of cylinder, mm 3, and cylinder displacement per unit cross-sectional area, mm (in.)

Greek symbols

[gamma] = compression ratio

[epsilon] = turbulent kinetic energy dissipation rate, [m.sup.2] * [s.sup.-3] ([ft.sup.2] * [s.sup.-3])

[[theta].sub.max] = maximum swinging angle, rad

[kappa] = the turbulent kinetic energy, [m.sup.2] * [s.sup.-2] ([ft.sup.2] * [s.sup.-2])

[phi] = eccentric angle, rad

Received April 20, 2011; accepted September 8, 2011

References

Fu, K., A. Knobloch, F. Martinez, D. Walther, A.C. FernandezPello, A. Pisano, and D. Liepmann. 2001. Design and fabrication of a silicon-based MEMS rotary engine. ASME International Mechanical Engineering Congress and Exposition, New York, November 11-16, pp. 875-80.

Heppner, J.D., D.C. Walther, and A.P. Pisano. 2007. The design of ARCTIC: A rotary compressor thermally insulated [mu]cooler. Sensors and Actuators A 134(1):47-56.

Lee, C.H., K.C. Jiang, P. Jin, and RD. Prewett. 2004. Design and fabrication of a micro Wankel engine using MEMS technology. Microelectronic Engineering 73(4):529-34.

Martinez, F., A. Knobloch, K. Fu, and A.P. Pisano. 2003. Apex seal design for the MEMS rotary engine power system. Proceedings of ASME 2003 International Mechanical Engineering Congress and Exposition (IMECE), Washington DC, November 16-21, IMECE2003-42071.

Swanger, M., D.C. Walther, A.C. Fernandez-pello, and A.P. Pisano. 2004. Small-scale rotary engine power system development status. WSS-04S-9 for Western States Section/Combustion Institute, Davis, CA.

Zhang, Y.L., and W. Wang. 2011. Effects of leakage and friction on miniaturization of Wankel compressor. Frontiers of Energy and Power Engineering in China 5(1):83-92.

Yilin Zhang is PhD Student. Wen Wang, PhD, is Professor.

Yilin Zhang and Wen Wang *

Institute of Refrigeration and Cryogenics, Shanghai Jiao Tong University 800 Dongehuan Road, Shanghai 200240, China

* Corresponding author e-mail: wenwang@sjtu.edu.cn

Introduction

With the increasing requirement for temperature control of meso- and micro-facilities, the demand for micro-cooling systems has become increasingly urgent in recent years. Meso- and micro-compressors act as the core parts of a cooling system, and Wankel-type units have become prime choices for micro-cooling systems because of their unique advantages, including simplicity in structure, high efficiency, and low noise. Some achievements in research on the micro-Wankel engine and micro-Wankel compressor include prototype design and manufacturing, as well as corresponding experimental research (Heppner et al. 2007; Lee et al. 2004; Fu et al. 2001). Martinez et al. (2003) and Swanger et al. (2004) improved the sealing mode of the Wankel engine, thereby increasing the sealing efficiency by 10% to 50%. A numerical simulation of the performance of a meso-scale Wankel compressor was conducted, and the factors affecting the miniaturization of the compressor (leakage, friction loss, and design limit) were discussed (Zhang and Wang 2011).

In decreasing the dimensions of a Wankel compressor to satisfy the miniaturization requirements of a micro-cooling system, the surface-to-volume ratio becomes larger, and the surfaces in relative motion cannot be adequately lubricated as in the normal scale. Because of these effects, the performance of meso- and micro-Wankel compressors tremendously decreases, and high efficiency becomes difficult to achieve. Therefore, strategies for improving the performance of a Wankel compressor have become an important issue, especially for micro--and meso-Wankel compressors.

Many factors affect the performance of a Wankel compressor. Such factors mainly include the compression ratio and the dimensions affecting the profile and volume of the working chamber. The dimensions are the shape factor and eccentric distance, which affect the profile of the chamber, as well as the height of the cylinder.

In the current work, the effects of these factors on the performance of a Wankel compressor are analyzed in detail. Related equations are also obtained. The theoretical analysis results are supported by the simulation. The variations in flow field, cylinder displacement, exhaust time, cooling capacity, and cooling capacity per unit cross-sectional area of the cylinder are analyzed under the corresponding operational conditions. Finally, a Wankel compressor prototype is designed based on the optimum dimensions obtained from the theoretical and simulation analyses.

Performance of a Wankel compresssor

The P-V diagram of the thermal process of a Wankel compressor is shown in Figure 1. The sequence 1-2-3-4-1 represents the ideal thermal process of a Wankel compressor. Sequence 1-2 illustrates the compression process, in which the gas pressure in the compression chamber gradually increases with decreasing volume. The gas pressure continues to increase until it reaches the point of back pressure, after which the operation turns into an exhaust process, as shown in sequence 2-3. The gas pressure is kept stable during the exhaust process until the minimum volume of the exhaust chamber is achieved. At this time, the exhaust valve is shut, and the expansion process begins. The gas pressure declines with increasing clearance volume until the pressure reaches the intake process, as illustrated in sequence 3-4. The operation process transforms into the intake process (sequence 4-1), and the pressure is kept unchanged until the volume of the intake chamber reaches its maximum.

[FIGURE 1 OMITTED]

Cylinder displacement of a Wankel compressor

The cylinder displacement of a Wankel compressor is calculated by the difference between the compression chamber volume at the end of the compression process ([V.sub.2]) and the exhaust chamber volume at the end of the exhaust process ([V.sub.3]) (Figure 1):

[V.sub.d] = [V.sub.2]- [V.sub.3]. (1)

The compression process in the compression chamber is assumed to be a polytropic process, expressed as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2)

The cylinder displacement of a Wankel compressor can be expressed as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3)

where y is the compression ratio, [gamma] = [P.sub.2]/[P.sub.1].

A Wankel compressor consists of three working chambers, and each volume is bounded by the internal surface of the cylinder, rotor, apex seals, and two endplates. The volume of the working chambers can be described as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4)

The Wankel compressor must have clearance between the rotor and internal surface of the cylinder because of the limitation in machining accuracy and the need for friction reduction. Considering clearance (also called offset a, as shown in Figure 2), the volume of the working chamber of a Wankcl compressor can be approximately calculated using the following equation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (5)

[FIGURE 2 OMITTED]

The maximum ([V.sub.1]) and minimum volumes ([V.sub.3]) of the working chamber are expressed as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], (6)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (7)

Volumes [V.sub.1] and [V.sub.3] are substituted into Equation 3. The displacement of the Wankel compressor can then be obtained by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. (8)

The rotor and cylinder profiles are vital to the shape and volume of the working chamber. The corresponding cross-sectional area of the working chamber plays a decisive role in cylinder displacement and in the performance of the Wankel compressor. The profile equations of the Wankel compressor cylinder are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (9)

The cross-sectional area of the cylinder can be calculated by the column coordinate equations of a cylinder, expressed as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (10)

where [beta] = [pi]/3

The cylinder displacement per unit cross-sectional area of a Wankel compressor is obtained by the ratio of cylinder displacement [V.su.d] to the cross-sectional area of cylinder A:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (11)

According to Equations 8 and 11, the factors causing cylinder displacement in a Wankel compressor include the compression ratio ([lambda]), shape factor (K), eccentric distance (e), offset (a), and cylinder height (a).

Results and discussion of the cylinder displacement of a Wankel compressor

According to theoretical Equations 8 and 11, the cylinder displacement and cylinder displacement per unit cross-sectional area of a Wankel compressor can be obtained using different parameters. Figures 3-5 show the results at different compression ratios, shape factors, and eccentric distance (the offset is assumed to be 0.5 mm).

At shape factor, eccentric distance, and cylinder height values of 6, 3 (0.118 in.) and 11.5 mm (0.453 in.), respectively, the cylinder displacement and cylinder displacement per unit cross-sectional area gradually decrease with increasing compression ratio--the rate at which the displacement decreases continuously drops, as shown in Figure 3. The other parameters are unchanged, except for the shape factor; the compression ratio is 3.5. The cylinder displacement constantly increases with increasing shape factor, whereas the cylinder displacement per unit cross-sectional area shows an opposite trend (Figure 4). The common range of the shape factor is from 6 to 7 (Zhang and Wang 2011) in the design of a Wankel compressor. To make the compressor more compact, the cylinder displacement per unit cross-sectional area should be as large as possible. Therefore, the shape factor of a Wankel compressor should have a value of around 6, especially for the meso- and micro-Wankel-type units.

[FIGURE 3 OMITTED]

Figure 5 shows that the cylinder displacement and cylinder displacement per unit cross-sectional area continuously increase with increasing eccentric distance under the same parameters (K = 6, B = 11.5 mm [0.453 in.], and [lambdda] = 3.5). The growing rate of the cylinder displacement continuously increases with the increase in eccentric distance, but the cylinder displacement per unit cross-sectional area exhibits a converse growth rate.

According to the analysis above, the cylinder displacement of a Wankel compressor is primarily affected by the shape factor, eccentric distance, and cylinder height on the basis that the compression ratio is determined by the design parameters. Among these design parameters, the shape factor and eccentric distance are the most important factors that influence the dimensions and performance of a Wankel compressor. Therefore, the effect of the shape factor and eccentric distance on the performance of Wankel compressors is analyzed using computational fluid dynamics.

Numerical simulation of a Wankel compressor

Initial conditions of a Wankel compressor

In general, the flow field in a Wankel compressor is three dimensional. The degree of importance of the three-dimensional effects is related to the position of the inlet and outlet, as well as their heights. In the present investigation, the inlet and outlet are assumed to be uniformly spread along the height of a Wankel compressor (Figure 6). Therefore, the three-dimensional effects can be disregarded, and two-dimensional models can be used for the simulations in this study.

[FIGURE 4 OMITTED]

For a simplified simulation of a Wankel compressor, the following assumptions are made:

1) the three-dimensional effects on the flow field are disregarded, and the two-dimensional model is used for the simulation;

2) the seals apex comes into contact with the internal surface of the cylinder, indicating the leakage between the three chambers is disregarded and no fluid exchange occurs between the three chambers;

3) the friction between moving parts of the compressor parts (that is, not considering friction heat) is disregarded; and

4) all solid boundaries are assumed adiabatic.

[FIGURE 5 OMITTED]

The design of a Wankel compressor is based on its cooling capacity and the state parameters of intake and exhaust gases. In this study, the simulation object is a meso-Wankel compressor, and the operational parameters are shown in Table 1.

[FIGURE 6 OMITTED]

Simulation method for a Wankel compressor

The simulation of the internal flow field of the compressor is carried out using the fluid dynamic software FLUENT (2D). The governing equations for the simulation include the conservation equations of mass, momentum, and energy. A finite-volume approach and the implicit-difference method based on pressure are used to solve the governing equations. A second-order upwind scheme is selected to discretize the governing equations, and the SIMPLE algorithm is used for pressure-velocity coupling.

In general, Wankel compressors have high rotation speed, and the internal flow field is a complex turbulent flow. Aside from the governing equations, turbulence models for the calculation of Reynolds stresses are also added to the solution equations for the simulation. For turbulence modeling, the RNG [kappa]-[epsilon] model is used, while the turbulent viscosity is computed from a differential equation. This model is a modified representation of the turbulent flows with Reynolds number in a large range and a case with curved streamlines in circulation regions. The modified terms in Equation e of the RNG [kappa]-[epsilon] model enhance the dissipation rate in the vicinity of the stagnation region and prevent the augmentation of turbulent kinetic energy in this region.

The model of the Wankei compressor is meshed with an unstructured grid. The meshes near the apex seal are refined to ensure that the minimum clearance has a three-layer mesh, which is the minimum number of mesh layers required in simulating fluid flow. The time step is 0.000005 s, while the rotating speed of the compressor is 1800 rpm in the simulation processes. The operation of a Wankel compressor is an unsteady process, and the shape and volume of the working chambers constantly change as the rotor rotates. Therefore, a dynamic mesh should be used in simulating Wankel compressors; in the current study, a user-defined function is created to define the movement principle and control rotor movement.

In accordance with the analysis above, the following boundary conditions are used in the simulation.

Inlet and outlet boundary condition: the pressure boundary condition is used for the inlet and outlet, and the total and static pressures are determined based on initial conditions.

Wall boundary condition: the effect of heat transfer is disregarded, and a no slip boundary condition is used for the internal surface of the cylinder and external surface of the rotor. The rotor has a rotating wall, and the rotation speed is about 600 rpm.

Boundary conditions in relation to turbulence: Standard wall functions are used for the turbulence model. Turbulent kinetic energy and the turbulent kinetic energy dissipation rate can be obtained by the following empirical formula:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (12)

This article focuses on the influence of the shape factor and eccentric distance on the performance of a Wankel compressor.

First, shape factor K, which controls the shape of the cylinder and the rotor, is considered, using the same eccentric distance and the ideal design of the inlet and outlet (the intake and exhaust are controlled by the valve on the inlet and outlet) as a basis. A reasonable shape factor is obtained in designing the Wankel compressor.

Second, the effect of eccentric distance is taken into account on the basis of the ideal design of the inlet and outlet, as well as the shape factor obtained in the previous step. The reasonable eccentric distance is determined using the design parameters in Table 1.

Finally, a meso-Wankel compressor prototype is created on the basis of the results from the above-mentioned steps and the simulation carried out in this article.

During the simulation, variations in some parameters of the working chamber should be monitored to ensure the reliability of the simulation results. These parameters include gas pressure, gas temperature, and gas velocity.

Effect of shape factor on performance under the same cylinder heights and eccentric distances

To analyze the influence of the shape factor on the performance of a Wankel compressor, the following initial conditions should be set:

1) an eccentric distance of 3 mm (0.118 in.);

2) a cylinder height of 11.5 mm (0.453 in.);

3) the ideal design of the inlet and outlet is used in the simulation, and the intake and exhaust gases are controlled by the valves set on the inlet and outlet; and

4) the shape factor is usually within the range of 6 to 7 for a Wankel machine. To increase the universality and applicability of the simulation, the shape factor value chosen is 5.5 to 7.5.

[FIGURE 7 OMITTED]

Figures 7-9 show that the opening time of the exhaust valve gradually advances with increasing shape factor, and the corresponding exhaust time continuously increases. The cylinder displacement, exhaust mass flow, and cooling capacity exhibit the same growth trend as the shape factor increases. The cooling capacity per unit cross-sectional area of the cylinder decreases with an increasing shape factor, and better performance is achieved when the shape factor of a Wankel machine is in the range of 6 to 7. In accordance with the decreasing trend of the cooling capacity per unit cross-sectional area of the cylinder with increasing shape factor, the value of ~6 is taken as tile optimum shape factor to make the Wankel compressor more compact, especially for meso- and micro-Wankel compressors.

Effect of eccentric distance on performance under the same shape factors and cylinder heights

On the basis of the analysis above, ~6 is chosen as the value of the shape factor of the Wankel compressor. In addition to the shape factor, the following initial conditions should be established to analyze the influence of the eccentric distance on the performance of the compressor:

1) a cylinder height of 11.5 mm (0.453 in.);

2) the ideal design of the inlet and outlet is used in the simulation, and the intake and exhaust gases are controlled by the valves set on the inlet and outlet; and

3) in accordance with the geometrical relationship and strength requirement of a Wankel compressor, the eccentric distance is restricted in the range of 2 to 5 mm (0.0787 to 0.197 in.) (Zhang and Wang 2011).

[FIGURE 8 OMITTED]

[FIGURE 9 OMITTED]

As shown in Figures 10-12, the exhaust time, cylinder displacement, exhaust mass flow, cooling capacity, and cooling capacity per unit cylinder cross-sectional area continually increase with an increasing eccentric distance. However, their growth rates are different from one another. Among the growth rates, those of the exhaust time and cooling capacity per unit cross-sectional area gradually decrease with the increase in eccentric distance. The growth rates of the other parameters continuously increase.

The simulation results for the effect of the shape factor and eccentric distance on the performance of the Wankel compressor agree well with the theoretical results. The reasonable shape factor is approximately 6. In accordance with the design cooling capacity and considering a certain surplus in cooling capacity (because leakage is not considered in this article), a value of 3 mm (0.118 in.) is taken as the eccentric distance.

[FIGURE 10 OMITTED]

Simulation of the Wankel compressor prototype

A meso-Wankel compressor prototype (Figure 13) is created based on the analysis, in which shape factor and eccentric distance are 6 and 3 mm (0.118 in.), respectively. The prototype is composed of a main shaft, gear pairs, seals, rotor, cylinder, endplates, and valve plates. The cylinder and rotor are both 11.5 mm (0.453 in.) high, and the compressor has a footprint of 40 x 50 mm (1.575 x 1.969 in.) and a thickness of 24 mm (0.945 in.).

[FIGURE 11 OMITTED]

The simulation of the meso-Wankel compressor prototype is conducted, and a variation in gas pressure in the working chambers ([V.sub.1], [V.sub.2], and [V.sub.3]) is shown in Table 2. Working chamber [V.sub.3] is taken as an example in comprehensively analyzing the pressure variation.

The exhaust process of working chamber [V.sub.3] ends when the time is 0 s, at which the volume reaches its minimum and the operation turns into the expansion process. With increasing [V.sub.3] volume, the gas pressure gradually decreases, and the process turns into the intake process until the gas pressure in [V.sub.3] reaches the point of intake pressure. The intake time is 0.00855 to 0.025 s, and the volume of [V.sub.3] reaches its maximum at the end of the intake process. The compression process then begins. The gas pressure in [V.sub.3] continuously increases with the decrease in the volume of [V.sub.3] until the gas pressure reaches the point of back pressure (1.2 MPa in this article). The process of the Wankel compressor then turns into the exhaust process, which ends when the volume of [V.sub.3] reaches its minimum. The exhaust time is approximately 0.00804 s (from 0.04196 to 0.05 s) for the Wankel compressor with valves on the inlet and outlet. The corresponding cooling capacity is about 352 W.

[FIGURE 12 OMITTED]

[FIGURE 13 OMITTED]

According to the dimensions of the mesoWankel compressor prototype, the maximum and minimum volumes can be measured and substituted into Equation 3. The cylinder displacement and corresponding cooling capacity are approximately 900 [mm.sup.3] (0.055 in. (3)) and 418 W The relative deviation is about 16% compared with the simulation results.

Conclusion

Through simulation and theoretical analyses, it was determined that the compression ratio and dimensions (shape factor, eccentric distance, and cylinder height) considerably affect the performance of a Wankel compressor.

According to the consistency of results between the simulation and theoretical discussions, the cylinder displacement constantly increases with an increasing shape factor, whereas the cylinder displacement per unit cross-sectional area gradually decreases. Both the cylinder displacement and cylinder displacement per unit cross-sectional area of the cylinder increase with increasing eccentric distance and have a linear increase with cylinder height. The same changing trends apply to the cooling capacity and cooling capacity per unit cross-sectional area.

The optimum shape Factor is approximately 6 for the Wankel compressor, and the value of 3 mm (0.118 in.) is taken as the eccentric distance based on the design cooling capacity and considering a certain surplus of cooling capacity. The cooling capacity obtained from the theoretical and simulation results are 418 and 352 W, respectively, and the relative deviation is about 16%.

DOI: 10.1080/10789669.2012.635948

Acknowledgment

This work was supported by the National Natural Science Foundation of China (grant 50976067).

Nomenclature

a = offset, mm (in.)

A = cross-sectional area of cylinder, [mm.sup.2] ([in.sup..2])

B = height of the cylinder, mm (in.) [C.sub.[micro]] = constants of the turbulence model, the default value [c.sub.[mu]]= 0.09 is used in this simulation

e = eccentric distance, mm (in.)

I = turbulence intensity

K = shape factor of the Wankel compressor, equal to the ratio of generation radius and eccentric distance

l = turbulence length, m (in.)

n = polytropic exponent

[p.sub.1] and [p.sub.2] = gas pressures before and after compression process, Pa (psi)

R = generation radius of the rotor, mm (in.)

u = mean velocity, m * [s.sup.-1] fit * [s.sup.-1])

[V.sub.1], [V.sub.3], and [V.sub.2] = maximum chamber volume, minimum chamber volume, and volume of the compression chamber at the end of the compression process, [mm.sup.3] ([in..sup.3])

[V.sub.d] and [V.sub.ds] = cylinder displacement of cylinder, mm 3, and cylinder displacement per unit cross-sectional area, mm (in.)

Greek symbols

[gamma] = compression ratio

[epsilon] = turbulent kinetic energy dissipation rate, [m.sup.2] * [s.sup.-3] ([ft.sup.2] * [s.sup.-3])

[[theta].sub.max] = maximum swinging angle, rad

[kappa] = the turbulent kinetic energy, [m.sup.2] * [s.sup.-2] ([ft.sup.2] * [s.sup.-2])

[phi] = eccentric angle, rad

Received April 20, 2011; accepted September 8, 2011

References

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Heppner, J.D., D.C. Walther, and A.P. Pisano. 2007. The design of ARCTIC: A rotary compressor thermally insulated [mu]cooler. Sensors and Actuators A 134(1):47-56.

Lee, C.H., K.C. Jiang, P. Jin, and RD. Prewett. 2004. Design and fabrication of a micro Wankel engine using MEMS technology. Microelectronic Engineering 73(4):529-34.

Martinez, F., A. Knobloch, K. Fu, and A.P. Pisano. 2003. Apex seal design for the MEMS rotary engine power system. Proceedings of ASME 2003 International Mechanical Engineering Congress and Exposition (IMECE), Washington DC, November 16-21, IMECE2003-42071.

Swanger, M., D.C. Walther, A.C. Fernandez-pello, and A.P. Pisano. 2004. Small-scale rotary engine power system development status. WSS-04S-9 for Western States Section/Combustion Institute, Davis, CA.

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Yilin Zhang is PhD Student. Wen Wang, PhD, is Professor.

Yilin Zhang and Wen Wang *

Institute of Refrigeration and Cryogenics, Shanghai Jiao Tong University 800 Dongehuan Road, Shanghai 200240, China

* Corresponding author e-mail: wenwang@sjtu.edu.cn

Table 1. Operational conditions of a Wankel compressor with refrigerant R134a. No. Parameter Description Value 1 Q Cooling capacity 300 W (1023.6 Btu * h-1) 2 n Motor speed 1800 rpm 3 [T.sub.eva] Evaporation 5[degrees]C (41[degrees]F) temperature 4 [T.sub.con] Condensation 46[degrees]C temperature (114.8[degrees]F) 5 [P.sub.in] Intake pressure 0.35 MPa (50,763 psi) 6 [P.sub.out] Back pressure 1.20 MPa (174,045 psi) Table 2. Variation in pressure and the state of the Wankel compressor prototype. [V.sub.1] [V.sub.2] Time, s Pressure, MPa State Pressure, MPa State 0 0.457 Compression 0.35 Intake process process 0.00855 1.18 Compression 0.353 Compression process process 0.025 0.362 Expansion 1.14 Compression process process 0.04196 0.351 Compression 0.35 Intake process process 0.05 0.457 Compression 0.35 Intake process process [V.sub.3] Time, s Pressure, MPa State 0 1.2 End of exhaust process 0.00855 0.35 Beginning of intake process 0.025 0.35 End of intake process 0.04196 1.2 Beginning of exhaust process 0.05 1.2 End of exhaust process

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Author: | Zhang, Yilin; Wang, Wen |
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Publication: | HVAC & R Research |

Date: | Jun 1, 2012 |

Words: | 4132 |

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