The scroll expander studied is modified from the scroll compressor of a certain type of vehicle air conditioner, and its main geometric parameters are shown in Table 1. The scroll expander is mainly composed of orbiting scroll and fixed scroll. The two scrolls are installed with a phase difference of 180° and mesh with each other. The orbiting scroll rotates and translates around the fixed scroll. The geometric profiles of orbiting scroll and fixed scroll are the same and are generated by circular involute profiles. The involute equation of a circle can be expressed by Eq. 1. Then, combined with the initial involute angle, the inner and outer wall profile equation of the scroll expander can be obtained, as shown in Eqs 2, 3. In SolidWorks, combined with the geometric parameters of the model and the profile equation, the geometric model of the expander is established in the form of equation driven curve, as shown in Figure 1. x = r b cos ⁡ φ + φ ⁡ sin ⁡ φ y = r b sin ⁡ φ − φ ⁡ cos ⁡ φ (1)

Inner wall involute profile equation: x i = r b cos ⁡ φ + φ + α sin ⁡ φ y i = r b sin ⁡ φ − φ + α cos ⁡ φ (2)

Outer wall involute profile equation: x o = r b cos ⁡ φ + φ − α sin ⁡ φ y o = r b sin ⁡ φ − φ − α cos ⁡ φ (3) Where φ is the involute angle, r b is the radius of the base circle, α is the initial angle of the inner and outer wall involute; i , o are subscripts, representing the inner and outer wall lines.

The structure design of the scroll profile is the key to the meshing of the scroll, and the processing of the profile has a greater impact on the performance and efficiency of the scroll expander. Especially for the tooth head part, if reasonable tooth head correction is not performed to avoid interference in the center part of scroll profile, the tooth head will be deformed more under high temperature and pressure, which will affect the meshing effect. In severe cases, it may even cause the tooth head to break and affect the operation of the whole machine. Nowadays, common methods of tooth head correction include dual arc correction, multiple pairs of arc correction, and arc plus straight-line correction, etc. In this paper, the influence of different tooth head corrections on the flow field distribution and deformation characteristics of the scroll is further explored by comparing the dual-arc correction and dual-arc plus straight-line correction.

The fluid calculation domain of scroll expander is obtained by cavity extraction in ANSYS Workbench, as shown in Figure 4A. Then, the computational domain consisting of inlet domain, outlet domain and working domain is meshed in ICEM, as shown in Figure 4B. In order to ensure the quality of the grid, the working domain grid is stretched from triangular surface grid to triangular prism grid. The inlet domain and outlet domain are meshed by hexahedral structure and they are connected to the working domain by interface. The whole calculation domain has 782,784 nodes and 1,478,268 cells.

It is meshed and then loaded into FLUENT to simulate flow and heat transfer. The whole working process obeys the continuity equation, the momentum conservation equation and the energy conservation equation. K-epsilon model is selected as the turbulence model. R123 real gas in NIST database is selected as the working fluid. Coupled algorithm is selected as the pressure-velocity coupling scheme which solves the pressure-based and momentum continuity equations together. The standard wall function method is used in the area near the wall. The inlet and outlet are set as the pressure boundary, where the total pressure and temperature are 1,000 kPa and 405 K respectively. In particular, the interfaces between gas and surfaces of scroll parts are defined as coupled walls for data transmission.

In order to verify the reliability of the numerical model, experimental measurements were carried out under four different test conditions. The test rig has been introduced in the previous work (Wei et al., 2015; Liu et al., 2018), and the test conditions are also shown in Table 3. By controlling and adjusting exhaust flow/temperature, working fluid flow/temperature, metering pump stroke, inlet/outlet gate valve opening, bypass valve opening and load, the performance parameters of expander under four working conditions are obtained. They include the flow rate, speed, effective output power, inlet/outlet pressure and temperature of the scroll expander. The output power is obtained by measuring the voltage and current of the electrical load, namely the generator electric power, as shown in Eq. 4. The mass flow rate is measured by turbine flowmeter. Meanwhile, the numerical simulation of the four working conditions is carried out, and the corresponding numerical calculation results are obtained, where the simulated shaft power can be obtained by Eq. 5. Ultimately, a comparison of the experimental results and numerical calculations results under four conditions is shown in Figures 5A, B. W   exp = U ⋅ I η (4) W s i m = 2 π n 60 T ⋅ ∫ t t + T M d t d t (5) Where U and I represent measured voltage and current, respectively; internal friction losses and external electro-mechanical losses are lumped into one single efficiency η; n is the rotating speed of the orbiting scroll; M d represents instantaneous driving gas moment exerted on the orbiting scroll.

It seems that the experimental results are in good agreement with the calculation results, and the maximum deviation is less than 5%. Therefore, the reliability of the calculation results is verified in a reasonable error range, and the numerical method can be used for subsequent research.

The static analysis module is established in ANSYS after importing and the scroll plate model. The global mesh size is set up and the independent mesh generation is adopted. By means of a one-way coupling method, the temperature and pressure results obtained by CFD simulation are transmitted to the finite element model as boundary conditions for the stress and deformation analysis of the scroll plate. According to the actual working conditions of the scroll expander, the corresponding displacement constraints are set for the orbiting scroll:

(1) The displacement in the Z direction of the side wall around the end plate is constrained;

The scroll plate material in this study is aluminum alloy 4032, and its material parameters are shown in Table 4.

The flow field distribution of the working chamber is affected differently by the modified profile of the tooth head with different combinations. The positions with spindle angles of 60°, 180° and 240° are selected to study the flow field distribution characteristics in the working chamber of the scroll plate after the tooth head is modified by PMP and EA-SAL.

Since the correction of the tooth head has a greater influence on the flow field distribution at the center of the tooth head, this paper mainly studies the flow field distribution of the central suction chamber of the scroll expander. In order to better observe the spatial distribution of the flow field after PMP correction and EA-SAL correction, the circular planes with z = 8 mm, 18 mm and 28 mm are established in CFD-POST with the center of the fixed scroll as the center. When the shaft rotation angle is 60°, the temperature distribution of the central suction chamber under PMP and EA-SAL correction is shown in Figure 6.

It can be seen from the cloud diagram that when the shaft rotates to the position of 60°, the temperature at the center of the scroll plate under the two tooth head corrections is lower than that at other positions. This is because at this position, the connecting area between the inlet and the central suction chamber is nearly blocked by the orbiting scroll, which causes mall connected area and serious throttling. In addition, it shows that the temperature at the center of the scroll under the EA-SAL correction is greater than that under PMP correction by comparing the temperature scales. The average difference is about 2 K and the reason is that different tooth head modifications will cause different tooth head shapes. The modified geometric structure of EA-SAL makes the larger volume of the central suction chamber formed by meshing, which leads to more high temperature working fluid being inhaled. As shown in the frame line in the figure, the positions of the maximum temperature in the suction chamber under the two corrections are different. They are distributed at both ends of the suction chamber. This is caused by the position of the inlet port and the correction profile. In the case of PMP correction, the radius of the correction arc is larger than that of the connection arc, so the working fluid is more likely to flow to the A side along the fixed scroll wall. In the case of EA-SAL correction, the inlet port is located at the correction straight line position and close to side B, so the temperature of side B is higher.

With the rotation of the spindle, the connected area between the inlet port and the suction chamber increases, the throttling weakens, and the temperature of the working chamber increases. At the 180° position, the opening of the inlet port is the largest, and the temperature of the central suction chamber is the highest. However, due to different tooth head profile and different curvature changes, there are differences in the temperature distribution of the suction chamber. As shown in Figures 7A, B, the temperature is the highest in the vertical direction of the inlet port, and is slightly lower on both sides. The temperature near the surface of the orbiting scroll and the meshing point has obvious stratification and is relatively small. In addition, along the downward direction of the Z axis, the area of high temperature becomes larger and larger, and the temperature distribution is uneven. As shown in Figure 7C, the high temperature working fluid enters the central suction chamber and flows rapidly along the direction of the tooth height. When the airflow hits the end plate of the orbiting scroll, the velocity decrease and the direction changes, so that a local high temperature is obtained in the lower part of the suction chamber. In comparison, it can be found that this phenomenon is weaker under EA-SAL correction. The reason is that the combination of arc and straight line makes the flow fluctuation smaller. It is more favorable for the working fluid to the flow and transfer heat and easier to disperse to the surrounding during the flow process.

As shown in Figure 8, at the 240° spindle rotation angle, the temperature distribution of the central suction chamber of the two modified scroll plates shows polarization. Which means taking the suction center as the boundary, the temperature distribution on both sides is quite different. The temperature on the left is higher than that on the right. In addition, the temperature on the right side is unevenly distributed with local high temperature. The specific reason is that the working fluid flows more smoothly towards the left side along the inner wall of the fixed scroll under the action of kinetic energy after entering the central suction chamber. On the contrary, the working fluid flowing to the right side is blocked by a part of the inner wall of the orbiting scroll. Therefore, the working fluid gas on the left side accumulates quickly, resulting in temperature rise and more uniform distribution. On the right side, the average temperature under the PMP correction is greater than that under EA-SAL correction. The reason is that the curvature of the connecting arc at the beginning of the tooth head of the former is smaller than that of the latter, so it is more conducive to the flow of working fluid. As shown by the circle in the figure, local high temperature appears under both corrections, and this phenomenon is more obvious under EA-SAL correction. The reason for this phenomenon is that the flow of working fluid at the circle is significantly blocked by the wall, and the curvature is large. This leads to the decrease of working fluid flow velocity, the formation of local low-speed vortex and the increase of temperature.

At different spindle rotation angles, the pressure distribution of the central suction chamber of the scroll after PMP correction and EA-SAI correction is shown in Figures 9A–C and Figures 9D–F. When the spindle angle is 60°, the pressure value at the center of the scroll under EA-SAL correction is significantly greater than that under PMP correction. The reason is the same as the temperature distribution. In this position, the former suction chamber volume is larger and more high-pressure gas is inhaled. With the rotation of the orbiting scroll, the volume changing rate of the suction chamber in the center of the scroll under the PMP correction is greater, so the gap between the two corrections is gradually reduced. At the angle of 180°, the average pressure under the PMP correction is greater than that under EA-SAL correction. At the angle of 240°, the pressure value under EA-SAL correction is higher than that under PMP correction by about 5 kPa. In addition, due to the combination of circular arc and straight line at this position, the pressure distribution is uneven caused by the more complex flow of the working fluid in the central suction chamber and the larger flow resistance is larger.

The numerical simulation results of flow field calculated by CFD are imported into the static analysis module for stress and deformation analysis of scroll tooth. In order to observe the deformation law of the scroll tooth under the two tooth head correction methods more directly, the deformation curve of the top of the scroll tooth is drawn. The deformation curve is drawn by taking the modified starting point of the scroll tooth head as the origin, the inner arc length as x coordinate and the deformation amount as y coordinate. In this paper, the arc length range of 1–50 mm is selected to make a comparative analysis of tooth head deformation after PMP correction and EA-SAL correction.

In Figures 10A–F show the top deformation curves of the orbiting scroll tooth head modified by PMP and EA-SAL under the action of temperature load at 60°, 180° and 240° angles position respectively. It can be seen intuitively from the curve distribution that the temperature load has a greater influence on the axial deformation of the tooth head. The axial deformation trend of the tooth head under the two corrections are basically the same, which gradually decrease outward along the scroll profile. The axial deformation of the tooth head under EA-SAL correction is slightly larger than that under PMP correction within a certain range of spindle rotation angle. However, the gap between them is reduced with the increase of spindle rotation angle and is eventually reversed. At the position of 60° angle, the average difference of axial deformation is 8um. The reason is the same as the flow field analysis. At this position, the average temperature at the center of the tooth head under EA-SAL correction is higher than that under PMP correction, so the thermal stress deformation is slightly larger than that under PMP correction. Since the modified tooth is thicker and the degrees of freedom in the X and Y directions of the bearing holes are limited, the figure shows that the radial deformation is smaller than the axial deformation. In addition, the former part of the radial deformation curve shows a downward trend, because this stage is in the connecting arc section. The curvature of the position is larger, the flow rate of the working fluid is faster, and the internal energy conversion is high, which makes the deformation in a contraction state. The large fluctuation of radial deformation at the 60° position is caused by the large change of temperature gradient and the change of direction. Moreover, the average difference of radial deformation between the tooth heads is less than 5 um, and all of them tend to increase outward.

It is not difficult to find that the overall total deformation of the tooth head corrected by EA-SAL is slightly larger than that by PMP based on the total deformation curve under temperature load. Moreover, their difference gradually decreases with the increase of the angle, which is related to the temperature distribution at different rotation angles. In the range of 0–12 mm arc length, the tooth thickness gradually increases from the beginning of the tooth head, so the larger wall thickness reduces the bending stress caused by the temperature, resulting in rapid reduction of deformation.

The top deformation curves of tooth head modified by PMP and EA-SAL under internal pressure load at different spindle positions are shown in Figures 11A–F. From the deformation curves at different spindle positions, it can be seen that the trends of radial deformation distribution of the tooth head modified by the two methods are roughly the same. However, due to the different correction forms and different tooth thickness, the deformation is different in the local range. In general, the average deformation of the tooth after PMP correction is greater than that after EA-SAL correction. At the angle of 60°, the deformation curves under the two modifications both show steep inflection points. The EA-SAL shows expansion effect, whilst the PMP shows contraction effect. This can be attributed to the deformation fluctuation near the inlet port caused by the sudden large pressure difference between the inside and outside of scroll tooth.

As can be observed from the total pressure deformation curve, the total pressure deformation of the tooth head after EA-SAL correction is greater than that of the tooth head after PMP correction at a small spindle angles because of the difference in suction chamber volume. However, with the increase of rotation angle, under the influence of expansion volume and tooth head wall thickness, the deformation effect of the two corrections is reversed. The total deformation of the tooth head under PMP correction is gradually larger than that under EA-SAL. Therefore, it can be seen that the deformation of the tooth head is complex and changeable, which is affected by temperature, pressure, correction method and spindle rotation angle.

In Figures 12, 13 show deformation curves of scroll tooth heads corrected by different correction angles under temperature load and pressure load at the same rotation angle. From the deformation curve under temperature load, it can be seen that the deformation trend of the tooth head after correction with different correction angles is similar, the PMP correction has a smoother trend and less fluctuation than the EA-SAL correction. Combined with the data in Table 5, it can be found that the deformation of tooth head under the action of temperature load decreases with the decrease of correction angle. The deformation difference between the EA-SAL correction of γ = 43° and the PMP correction of γ = 52° under the action of temperature load is larger. The reason is that the thickness of the tooth head corrected by EA-SAL is larger and less affected by thermal stress. So appropriately reducing the correction Angle is conducive to reducing the deformation caused by thermal stress.

From the deformation curve under internal pressure load, it can be seen that the overall deformation of the tooth head under PMP correction is greater than that under EA-SAL correction, and the difference is significant at the front end of the tooth head, which is about 20%. In addition, as the EA-SAL correction increases the tooth thickness, the deformation of the tooth head under different correction angles is relatively close. Under the action of internal pressure, the deformation of tooth head also decreases with the decrease of the correction angle. But after 30 mm, the deformation gradually increases, and the smaller the correction angle, the greater the deformation. The reason is that the smaller the correction angle, the greater the rate of change of the tooth thickness, resulting in greater changes in the amount of deformation.