# An Improved Geometric Theoretical Model and Throughflow Prediction Method for a CO2 Scroll Compressor of Automotive Air Conditioning System

Authors: Panpan Song; Ding Wu; Zhenbo Lu; Siyu Zheng; Mingshan Wei; Weilin Zhuge; Yangjun Zhang
Journal: International Journal of Energy Research
 (2023)

---

## Abstract

The scroll-type compressor is the core component of the refrigeration system using the natural refrigerant carbon dioxide. An accurate scroll geometric theoretical model is essential for evaluating and enhancing compressor performance. The present paper proposed and validated an improved geometric theoretical model of a scroll compressor. A vector triangle method-based general piecewise function describes precisely the volume variation of the working chambers, based on which the transient throughflow modeling and performance evaluation of a carbon dioxide scroll compressor were conducted. The mechanisms and influencing factors of the suction precompression and the asymmetry discharging were analyzed. The results indicate that the strength of the suction precompression is mainly influenced by the suction vacuum under different rotating speeds, partly associated with the tangential leakage under low rotating speeds. The increasing initial suction pressure contributes to the linear raising of the pressure difference between suction initial and ending pressures, decreasing the precompression extension slightly. The variation of the discharge throughflow areas of two symmetric discharge chambers dominates the pressure asymmetry, which is gradually eliminated by the overlapping throughflow area. Compared with the circular discharge structure, the waist-shape port reduces the pressure asymmetry degree and shortens its duration.


---

## Body

## 1. Introduction

Carbon dioxide, an alternative natural refrigerant for the next-generation refrigeration system, has been attracting considerable attention due to the advantages of environmental protection and high energy efficiency [1]. CO2 refrigeration systems have excellent development potential in the integrated thermal management of automobiles, especially electric vehicles. It should be emphasized that the popularization and application of automotive CO2 air conditioning technology mainly depend on the technical maturity of the compressor. A CO2 scroll compressor with the advantages of compact structure, high efficiency, high reliability, and low noise would be an ideal selection for automotive air conditioning systems [2]. In the design process, scroll geometric modeling plays a decisive role in evaluating and enhancing the quasisteady and fully unsteady performances. Many studies on scroll geometric modeling have been carried out in the past. The literature review shows that the mathematical methods of scroll geometry mainly include sector integration method [3], graphic method [4], midline method [5], and analytical method [6].Sector integration method, based on differential sector integration, is the initial and traditional method. The geometric model of scroll profiles without wrap modification was first studied by Morishita et al. [7]. Further, the geometric model under one-arc wrap modification was discussed Yanagisawa et al. [3] and Chen et al. [8]. Wang et al. [9] simplified the discharge chamber calculation in Ref. [8] and derived a general subsection function for the volume variation. Moreover, Liu et al. [10] solved the compression chamber volume constructed by the involute of circle with variable radii. The graphic method, a traditional and intuitive method, has the shortcoming of calculating complexity primarily for the discharge chamber. Wang and Liu [4] proposed a detailed geometric model for the scroll profiles with dual-arc tip modification. Mojiri et al. [11] calculated the discharge chamber volume of a corotating scroll compressor [12], referring to the same graphic method used in Ref. [13].The midline method, an ingenious mathematical method, could support the shortcut of the design period [14]. Liu and Wu [5] derived the chamber volumes of a constant wall thickness scroll compressor. And Peng et al. [15] conducted the chamber volume calculation of a variable wall thickness scroll compressor. In the forward design of profiles, Qiang generates the orbiting and fixed scrolls by offsetting the scroll midline [16]. Based on this work, the geometric theory of different scroll tip-modified profiles was developed, including dual-arc, arc-line-arc, and asymmetric arcs [17], and multiple segments of circular arcs [18]. In addition, this design method was also applied in a new type of vacuum pump with scroll tip modified by asymmetric circular arcs [19]. The analytical method, a general mathematical method, has better geometrical adaptability for different shapes of working chambers. Bush and Beagle [6] conducted the geometrical solution for a single compression chamber. Gravesen and Henriksen [20] proposed the analytical geometric model for the whole scroll compressor. Blunier et al. [21] utilized the analytical method into the geometry theory of a scroll compressor with one-arc tip modification. Bell [22] developed a comprehensive analytical model and especially discussed the arc-line-arc discharge geometry. Further, based on the vector triangle method taking the coordinate origin as the common starting point, Bell et al. [23] proposed a general analytical model for the chamber volume calculation. Considering the radial clearance, Shaffer and Groll [24] adopted the analytical method to describe the geometry theory of a variable wall thickness scroll compressor.The sector integration method, as well as the graphic method, increases the calculation complexity of the discharge chamber volume. The accuracy of the geometric model plays an essential role in the thermodynamic modeling [25]. The midline method has the error of volume calculation up to 0.015% [17], while that of the analytical method is within 0.003% [26]. Moreover, the midline method has difficulty deriving a general and completed geometric model due to the complex integration of the midline length. Compared with other methods, the vector triangle-based analytical method could concisely construct the geometric model. Based on Green’s formula, the analytical method focuses on the derivation of equivalent deformation, while the vector triangle method focuses on the derivation of area calculation from the geometric graphic aspect by combination of number and form. Although they are different in expression, they both come to the same conclusion by different ways.The development of geometry theory contributes to the scroll machine predesign accuracy and efficiency [27]. Further, to evaluate the performance of the predesigned prototype, several types of research approaches have been adapted including experiments [28, 29], thermodynamic modeling [30, 31], and CFD simulation modeling of dimensions 1D [32, 33], 2D [34, 35], and 3D [36, 37]. Due to the nonignorable consumption of time and experimental materials, researches of scroll machine experiments were very few. Moreover, the precision of thermodynamic modeling is limited by the simplified assumptions. On the other hand, CFD simulations are a promising tool to illustrate the scroll machine nonuniform and asymmetric inner flow characteristics [38]. The 1D CFD simulation modeling especially has the advantages of high efficiency and flexibility, which permits its potential for the analysis of off-design performance. Most mathematical models assumed that the suction pressure keeps no change in the suction process [39–41]. Actually, a remarkable physic phenomenon named precompression exists in the suction chamber [42], resulting in a higher pressure than the initial suction pressure. Previous studies considered that the precompression is mainly induced by the decrease of chamber volume at the later suction period [8, 43], but the mechanism and the influencing factors of precompression were not discussed deeply. Equally, asymmetric flow phenomenon existing in the symmetric working chambers has been mentioned and investigated in previous studies on the aerodynamic behaviors of scroll compressors [44–46] or expanders [47–49]. Asymmetric discharging has an essential impact on the aerodynamic performance of the scroll compressor. For example, the asymmetric pressure distribution in the discharge chamber could cause prominent discharging pulsation and irreversible losses. But few studies on the flow control and optimization of asymmetric discharging in the scroll compressor were conducted.This paper proposed an improved geometric model of a CO2 scroll compressor for automotive air conditioning system. A vector triangle method-based general piecewise function could precisely describe the working chamber geometry. An efficient transient 1D throughflow modeling and performance evaluation were conducted on this base. The mechanisms and influencing factors of two special flow behaviors, the suction precompression and the pressure asymmetry, were analyzed in detail. Effects of the rotating speed and initial suction pressure on suction precompression and discharge structure on asymmetric flow were revealed.


## 2. Vector Triangle Method for Scroll Geometry

As a widely used scroll profile [38], the circle involute was used to design the CO2 scroll compressor. The coordinates of scroll profiles and the application of the vector triangle method are introduced here. An accurate description of the scroll profile coordinates contributes to applying the vector triangle method in calculating the axial projected area of working chambers.


### 2.1. Coordinate Description of Scroll Profiles

The coordinates of the inner and outer involute profiles of the fixed scroll can be expressed as(1)xfi,yfi=rbcosϕ+ϕ−φi0sinϕ,rbsinϕ−ϕ−φi0cosϕ,(2)xfo,yfo=rbcosϕ+ϕ−φo0sinϕ,rbsinϕ−ϕ−φo0cosϕ,where ϕ ranges from φis to φie and φos to φoe for the inner and outer involutes, respectively. Initial angles φi0 and φo0 satisfy φi0=−φo0. In addition, the coordinate description of the dual-arc modified profiles is shown in Figure 1.Figure 1
Dual-arc modified wrap profile.The center coordinates of two modified circular arcs are expressed as(3)xa1=xfis−ra1sinφis,ya1=yfis+ra1cosφis,xa2=xfos−ra2sinφos,ya2=yfos+ra2cosφos,(4)xβ=xa1+ra1cosβ,yβ=ya1+ra1sinβ,βa1,1≤β≤βa1,2,(5)xβ=xa2+ra2cosβ,yβ=ya2+ra2sinβ,βa2,1≤β≤βa2,2,where radii ra1 and ra2 are obtained after choosing a proper modified position and satisfying the PMP condition. βa1,1, βa1,2, βa2,1, and βa2,2 represent the vector angles of the modified circular arcs, respectively.
(6)βa1,1=atan2ya2−ya1,xa2−xa1,βa1,2=atan2yfis−ya1,xfis−xa1,βa2,1=atan2ya1−ya2,xa1−xa2,βa2,2=atan2yfos−ya2,xfos−xa2,where the function atan2y,x returns the four-quadrant arctangent angle in the range -π to π.From Equation (1) to Equation (6), all coordinates of the fixed scroll can be obtained accurately. Concurrently, the coordinates of the orbiting scroll could also be gained according to the conjugate relation as defined in
(7)xo,yo=−xf+rocosθm,−yf+rosinθm,where the orbiting radius ro=rbπ−φi0+φo0 and the orbiting scroll offset angle can be expressed as θm=φie−θ−π/2. Obviously, θm means the beginning of the suction process at θ=0. It is noticeable that the orbiting angle ranges from the beginning of suction to the end of discharge, which is different from traditional expression of the orbiting angle [26].


### 2.2. Vector Triangle Method

Based on the coordinate description, the vector triangle method then could be applied. Detailed projected area calculations of two geometric type regions, triangle and sector type regions, are introduced.As shown in Figure2, the triangle area At can be calculated through one-half the cross product of two vectors formed by vertices xoos,yoos, xt1,yt1, and xt2,yt2, expressed in Equation (8). As for the sector type region displayed in Figure 2, the area Aoa,dd1 is calculated by integrating the cross product of two vectors formed by vertices xoos,yoos, x,y, and x+x′dβ,y+y′dβ expressed in Equation (10).
(8)At=12xt1−xoosyt2−yoos−xt2−xoosyt1−yoos.Figure 2
The area of the triangle regionAt and sector region Aoa,dd1.Through multiplying the areaAt by the scroll height hs, Vt can be expressed as
(9)Vt=hs2xt1−xoosyt2−yoos−xt2−xoosyt1−yoos,where xt2,yt2=xa1+ra1cosβa1,1,ya1+ra1sinβa1,1, xt1,yt1=−xt2+rocosθm,−yt2+rosinθm, and xoos,yoos=−rbcosφos−rbφos−φo0sinφos+rocosθm,−rbsinφos+rbφos−φo0cosφos+rosinθm.
(10)AOa,dd1=∫βa1,1βa1,212x−xoosy+y′βdβ−yoos−y−yoosx+x′βdβ−xoos=12∫βa1,1βa1,2x−xoosdydβ−y−yoosdxdβdβ.Through multiplying the areaAOa,dd1 by the scroll height hs, VOa,dd1 can be expressed as
(11)VOa,dd1=ra1hs2yoos−ya1cosβa1,2−cosβa1,1−xoos−xa1sinβa1,2−sinβa1,1+ra1βa1,2−βa1,1.


## 2.1. Coordinate Description of Scroll Profiles

The coordinates of the inner and outer involute profiles of the fixed scroll can be expressed as(1)xfi,yfi=rbcosϕ+ϕ−φi0sinϕ,rbsinϕ−ϕ−φi0cosϕ,(2)xfo,yfo=rbcosϕ+ϕ−φo0sinϕ,rbsinϕ−ϕ−φo0cosϕ,where ϕ ranges from φis to φie and φos to φoe for the inner and outer involutes, respectively. Initial angles φi0 and φo0 satisfy φi0=−φo0. In addition, the coordinate description of the dual-arc modified profiles is shown in Figure 1.Figure 1
Dual-arc modified wrap profile.The center coordinates of two modified circular arcs are expressed as(3)xa1=xfis−ra1sinφis,ya1=yfis+ra1cosφis,xa2=xfos−ra2sinφos,ya2=yfos+ra2cosφos,(4)xβ=xa1+ra1cosβ,yβ=ya1+ra1sinβ,βa1,1≤β≤βa1,2,(5)xβ=xa2+ra2cosβ,yβ=ya2+ra2sinβ,βa2,1≤β≤βa2,2,where radii ra1 and ra2 are obtained after choosing a proper modified position and satisfying the PMP condition. βa1,1, βa1,2, βa2,1, and βa2,2 represent the vector angles of the modified circular arcs, respectively.
(6)βa1,1=atan2ya2−ya1,xa2−xa1,βa1,2=atan2yfis−ya1,xfis−xa1,βa2,1=atan2ya1−ya2,xa1−xa2,βa2,2=atan2yfos−ya2,xfos−xa2,where the function atan2y,x returns the four-quadrant arctangent angle in the range -π to π.From Equation (1) to Equation (6), all coordinates of the fixed scroll can be obtained accurately. Concurrently, the coordinates of the orbiting scroll could also be gained according to the conjugate relation as defined in
(7)xo,yo=−xf+rocosθm,−yf+rosinθm,where the orbiting radius ro=rbπ−φi0+φo0 and the orbiting scroll offset angle can be expressed as θm=φie−θ−π/2. Obviously, θm means the beginning of the suction process at θ=0. It is noticeable that the orbiting angle ranges from the beginning of suction to the end of discharge, which is different from traditional expression of the orbiting angle [26].


## 2.2. Vector Triangle Method

Based on the coordinate description, the vector triangle method then could be applied. Detailed projected area calculations of two geometric type regions, triangle and sector type regions, are introduced.As shown in Figure2, the triangle area At can be calculated through one-half the cross product of two vectors formed by vertices xoos,yoos, xt1,yt1, and xt2,yt2, expressed in Equation (8). As for the sector type region displayed in Figure 2, the area Aoa,dd1 is calculated by integrating the cross product of two vectors formed by vertices xoos,yoos, x,y, and x+x′dβ,y+y′dβ expressed in Equation (10).
(8)At=12xt1−xoosyt2−yoos−xt2−xoosyt1−yoos.Figure 2
The area of the triangle regionAt and sector region Aoa,dd1.Through multiplying the areaAt by the scroll height hs, Vt can be expressed as
(9)Vt=hs2xt1−xoosyt2−yoos−xt2−xoosyt1−yoos,where xt2,yt2=xa1+ra1cosβa1,1,ya1+ra1sinβa1,1, xt1,yt1=−xt2+rocosθm,−yt2+rosinθm, and xoos,yoos=−rbcosφos−rbφos−φo0sinφos+rocosθm,−rbsinφos+rbφos−φo0cosφos+rosinθm.
(10)AOa,dd1=∫βa1,1βa1,212x−xoosy+y′βdβ−yoos−y−yoosx+x′βdβ−xoos=12∫βa1,1βa1,2x−xoosdydβ−y−yoosdxdβdβ.Through multiplying the areaAOa,dd1 by the scroll height hs, VOa,dd1 can be expressed as
(11)VOa,dd1=ra1hs2yoos−ya1cosβa1,2−cosβa1,1−xoos−xa1sinβa1,2−sinβa1,1+ra1βa1,2−βa1,1.


## 3. Geometric Model and Evaluation

### 3.1. Working Process Division

The scroll profiles usually consist of different geometric curves, such as circle involute, circular arc, and high order curve. Different division principles of working processes have been proposed in previous studies [4, 15]. In the present study, the working processes were divided into five working stages according to the geometric elements, as displayed in Table 1.Table 1
Geometric element composition in each stage.
StageWorking processGeometric element0SuctionInvolute of circle, segment1CompressionInvolute of circle2CompressionInvolute of the circle, circular arc3DischargeInvolute of the circle, circular arc4DischargeCircular arcThe division of the chamber projected area is the basic before applying the vector method. Figure3 displays the division principle of the chamber projected area from stage 2 to stage 4. In stage 2, the projected area of the compression chamber is divided into two parts by a segment between two demarcation points of the circle involute and circular arc. In stage 3, another segment connected with xt1,yt1 and xt2,yt2 separates the projected area of the central discharge chamber into two symmetrical parts. The geometric element of the central discharge chamber in stage 4 only includes the modified circular arc. Similarly, the projected area is divided into two symmetrical parts by the segment connected with xt1,yt1 and xt2,yt2.Figure 3
Area divisions.


### 3.2. Chamber Volume Calculation

#### 3.2.1. Suction Process

No matter the volume calculation stage, the orbiting angle range corresponding to the working chambers’ geometric elements should be determined first. The orbiting angle in stage 0 ranges fromθ=0 to 2π clearly. The orbiting angle value θd0 at the end of stage 1 is defined in Equation (12). According to meshing characteristics of PMP profile, the lasting angle of stage 2 is defined as Δθ expressed in Equation (13).
(12)θd0=φie−φos−π,(13)Δθ=βa1,2−βa1,1=βa2,2−βa2,1.The volume calculation of the working chamber in stage 0 and stage 1 had been analyzed accurately by former researchers [7, 20, 24]. The suction chamber Vs0 in stage 0 is expressed in Equation (14) quoted from Ref. [24].
(14)Vs0=hsrbrbφe−φi0−roθπ−2φe+θ−rbφe−π−φo02B−rbφe−π−φo0B2−rb3B3φe−φi0rbcosB+rbsinBφe−π+B−φo0−rosinθ+rbsinB−rbcosBφe−π+B−φo0−rocosθ−rosinB+θφe−π+B−φo0−rocosB+θ+ro0≤θ≤2π,where B=1/2φo0−φie+π+φo0−φie+π2−4D and D=ro/rbφi0−φiesinθ−cosθ+1/φie−φi0.


#### 3.2.2. Compression Process

(1) Chamber in Stage 1. The compression chamber Vs1 in stage 1 is expressed in Equation (15) quoted from Ref. [20].
(15)Vs1=2πhsrbro2φe+π−2θ2π≤θ≤θd0.(2) Chamber Section c1 in Stage 2. Before the compression chamber section c1 volume calculation in stage 2, as shown in Figure 3, the discharge chamber section d1 in stage 3 is firstly analyzed. Because chamber section d1 originates from section c1 with the only change in the range of the orbiting angle θ, the division of the projected area of section d1 especially can be expressed in Equation (16) more clearly, as shown in Figure 4.
(16)Ad1=AO,d1−AI,d1.Figure 4
Division of chamber section d1 and decomposition ofAI,d1.AO,d1 formed by the coordinate origin and inner wall curve of fixed scroll wrap is calculated in Equation (17). The volume VO,d1 of the corresponding chamber section is obtained in Equation (18). As displayed in Figure 4, AI,d1 is divided into three triangles and one sector type region. It is noticeable that the geometric symmetry is considered here and makes the analysis simpler, as expressed in Equation (19).
(17)AO,d1=12∫φos+πφie−θ+2πxdydϕ−ydxdϕdϕ,(18)VO,d1=hsrb2φie−θ+2π−φi03−φos+π−φi036,(19)AI,d1=AIa,d1+AIb,d1+AIc,d1+AId,d1.AIa,d1 formed by the coordinate origin and inner wall curve of fixed scroll wrap is calculated with the upper and lower limits of integration φie−θ+π and φos, as expressed in Equation (20). VIb,d1, VIc,d1, and VId,d1 are calculated by multiplying the area of the triangle by the scroll height hs, respectively. Each projected triangle area formed by three geometric vertices is listed in Table 2 for every chamber section.
(20)AIa,d1=rb2φie−θ+π−φo03−φos−φo036.Table 2
Vertices making up of projected triangle areas.
Projected areaVertices forming axial projected triangle areaAIb,d10,0; rocosθm,rosinθm; rbcosφos+rbφos−φo0sinφos,rbsinφos−rbφos−φo0cosφosAIc,d10,0; rocosθm,rosinθm; rbcosφie−θ+π+rbφie−θ+π−φo0sinφie−θ+π,rbsinφie−θ+π−rbφie−θ+π−φo0cosφie−θ+πAId,d1rocosθm,rosinθm;−rbcosφis−rbφis−φi0sinφis+rocosθm,−rbsinφis+rbφis−φi0cosφis+rosinθm; rbcosφos+rbφos−φo0sinφos,rbsinφos−rbφos−φo0cosφosVIb,d1, VIc,d1, and VId,d1 are given by Equation (21), Equation (22), and Equation (23), respectively. From Equation (16) to Equation (23), the volume expression of chamber section d1 in stage 3 θd0+Δθ≤θ≤θd0+2π is obtained in Equation (24). Finally, chamber section c1 in stage 2 has the consistent expression of the volume calculation as chamber section d1, just with a different range of orbiting angles θd0≤θ≤θd0+Δθ.
(21)VIb,d1=hsrbro2φos−φo0sinθ+φos−φie+cosθ+φos−φie,(22)VIc,d1=hsrbro2,(23)VId,d1=hsrbro2φos−φi0+πsinθ+φos−φie+cosθ+φos−φie+1,(24)Vd1=hsrb6rbφie−θ+2π−φi03−φos+π−φi03−rbφie−θ+π−φo03−φos−φo03−3ro2φos+πsinθ+φos−φie+2cosθ+φos−φie+2.(3) Calculation of Chamber Section cc1 in Stage 2. The projected area of chamber section cc1, Acc1, equals to the difference between AO,cc1 and AI,cc1, as shown in Figure 5. AO,cc1 and AI,cc1 can be calculated with two different upper and lower limits of integration [βa1,2+θd0−θ, βa1,2] and [βa2,2+θd0−θ, βa2,2], respectively. Then, the volumes of VO,cc1 and VI,cc1 are simplified as Equations (26) and (29), where Equation (28) describes the coordinates of the orbiting scroll modified arc. Finally, the volume of two compression chambers in stage 2, Vs2, is obtained from Equation (24), Equation (26), and Equation (29).
(25)AO,cc1=12∫βa1,2+θd0−θβa1,2x−xoosdydβ−y−yoosdxdβdβ,(26)VO,cc1=ra1hs2yoos−ya1cosβa1,2−cosβa1,2+θd0−θ−xoos−xa1sinβa1,2−sinβa1,2+θd0−θ+ra1θ−θd0,(27)AI,cc1=12∫βa2,2+θd0−θβa2,2xβ−xoosdyβdβ−yβ−yoosdxβdβdβ,where
(28)xβ=−xa2−ra2cosβ+rocosθm,yβ=−ya2−ra2sinβ+rosinθm,βa2,2+θd0−θ≤β≤βa2,2,(29)VI,cc1=ra2hs2xa2sinβa2,2−sinβa2,2+θd0−θ−ya2cosβa2,2−cosβa2,2+θd0−θ−rbsinβa2,2−φos−sinβa2,2+θd0−θ−φos−rbφos−φo0cosβa2,2−φos−cosβa2,2+θd0−θ−φos+ra2θ−θd0,(30)Vs2=2Vc1+VO,cc1−VI,cc1θd0≤θ≤Δθ+θd0.Figure 5
Divisions of chamber sections in stages 2 to 4.


#### 3.2.3. Discharge Process

The discharge process covers stage 3 and stage 4 continuously. In stage 3, the discharge chamber shown in Figure3 includes chamber sections d1, d2, dd1, and dd2. Considering the geometric symmetry, chamber sections d1 and dd1 were analyzed. The chamber section d1 had been described in Equation (24). As for the chamber section dd1, according to Figure 5, its projected area Add1 can be obtained via
(31)Add1=AOa,dd1+At−AIa,dd1.At and AOa,dd1 were also derived in Equation (8) and Equation (10). Hence, only AIa,dd1 needs to be derived in Equation (32). Finally, the total volume of the discharge chamber in stage 3 is obtained in Equation (34).
(32)AIa,dd1=12∫βa2,1βa2,2x−xoosdydβ−y−yoosdxdβdβ,(33)VIa,dd1=ra2hs2xa2sinβa2,2−sinβa2,1−ya2cosβa2,2−cosβa2,1−rbsinβa2,2−φos−sinβa2,1−φos−rbφos−φo0,cosβa2,2−φos−cosβa2,1−φos+ra2βa2,2−βa2,1,(34)Vs3=2Vd1+VOa,dd1+Vt−VIa,dd1θd0+Δθ≤θ≤θd0+2π.In stage 4, the projected area of discharge chamber section ddd1 can be obtained according toAddd1=AO,ddd1−AI,ddd1 shown in Figure 5. AO,ddd1, the sector type region, is expressed in Equation (35). The volume of VO,ddd1 was calculated according to Equation (37).
(35)AO,ddd1=12∫βa1,1βa1,2+θd0+2π−θx−xt1dydβ−y−yt1dxdβdβ,where the modified arc coordinates are expressed in
(36)xβ=xa1+ra1cosβ,yβ=ya1+ra1sinβ,βa1,1≤β≤βa1,2+θd0+2π−θ,(37)VO,ddd1=ra1hs2xa1+xt2sinβa1,2+θd0+2π−θ−sinβa1,1+ya1+yt2cosβa1,1−cosβa1,2+θd0+2π−θ+ra1βa1,2+θd0+2π−θ−βa1,1+rosinφie−θd0−2.5π−βa1,2−sinθm−βa1,1.AI,ddd1 is expressed in Equation (38), and Equation (40) derives VI,ddd1. Finally, the total volume of the central chamber in stage 4, Vs4, is obtained.
(38)AI,ddd1=12∫βa2,1βa2,2+θd0+2π−θx−xt1dydβ−y−yt1dxdβdβ,where the modified arc coordinates are expressed in
(39)xβ=−xa2−ra2cosβ+rocosθm,yβ=−ya2−ra2sinβ+rosinθm,βa2,1≤β≤βa2,2+θd0+2π−θ,(40)VI,ddd1=ra2hs2xa2−xt2sinβa2,2+θd0+2π−θ−sinβa2,1+ra2βa2,2+θd0+2π−θ−βa2,1+ya2−yt2cosβa2,1−cosβa2,2+θd0+2π−θ,(41)Vs4=2VO,ddd1−VI,ddd1θd0+2π≤θ≤θd0+2π+Δθ.Combining Equation (30), Equation (34), and Equation (41), a general piecewise function-based geometric model is finally established as described in Equation (42). In the following subsection, the precision and accuracy evaluation are conducted for the geometric model.
(42)V=Vs00≤θ≤2π,Vs12π≤θ≤θd0,Vs2θd0≤θ≤θd0+Δθ,Vs3θd0+Δθ≤θ≤θd0+2π,Vs4θd0+2π≤θ≤θd0+2π+Δθ.


### 3.3. Geometric Model Evaluation

According to a reference operation condition (case M10 in Yang’s work [50]) of a CO2 compressor in a vehicle air conditioning system, the geometric profiles of the scroll compressor were designed by an in-house code. The essential parameters of the scroll profiles are listed in Table 3.Table 3
Geometric parameters of the CO2 scroll compressor.
SymbolUnitValuehsmm6.530rbmm3.000romm4.210φo0rad-0.869φi0rad0.869φosrad1.571φisrad4.712φerad12.566ra1mm7.295ra2mm3.085For evaluation, the polygon-based method has been applied to the evaluation of projection area calculation accuracy in Ref. [26]. Hence, the polygon-based equivalent method to the discharge chamber projected area calculation in stage 3 was utilized. The independence verification of the polygon edge number was firstly conducted. Polygon edge numbers 1000, 2000, 4000, and 8000 were adapted, respectively, to calculate equivalently the discharge chamber projected area at θ=θd0=φie−φos−π=2.5π. Table 4 presents the relative error of the polygon area among different polygon edge numbers. As the number of edges increases, the accuracy becomes higher. Relative error decreases significantly from 0.0195‰ to 0.0052‰ after choosing 4000 edges. Hence, the polygon edge number 4000 was selected to calculate the projected area.Table 4
Independence verification of the polygon edge number.
Polygon edge numberProjected area (mm2)Relative error (‰)1000498.5678—2000498.57750.01954000498.58010.00528000498.58070.0012Based on a sufficient accuracy approximation by a polygon with 4000 edges, Figure6 shows the precision difference between the geometric model in the present paper and that in Bell’s study [22]. The relative errors of 0.001% at the beginning and 0.16% at the end of stage 3 indicate that both models agree well with the polygon approximated model. However, a significant relative error reaching up to 12.95% exists in the other model at the orbiting angle of 4π. The geometric model in the work of Liu et al. [51] and Wang et al. [52] was adapted to verify the present theoretical model. Figure 6 indicates that the present model has a sufficient prediction accuracy. As the scroll compressor 1D throughflow modeling prerequisite, the volume variation of the scroll working chamber could be simulated accurately.Figure 6
Accuracy evaluation and validation of scroll geometric model.


## 3.1. Working Process Division

The scroll profiles usually consist of different geometric curves, such as circle involute, circular arc, and high order curve. Different division principles of working processes have been proposed in previous studies [4, 15]. In the present study, the working processes were divided into five working stages according to the geometric elements, as displayed in Table 1.Table 1
Geometric element composition in each stage.
StageWorking processGeometric element0SuctionInvolute of circle, segment1CompressionInvolute of circle2CompressionInvolute of the circle, circular arc3DischargeInvolute of the circle, circular arc4DischargeCircular arcThe division of the chamber projected area is the basic before applying the vector method. Figure3 displays the division principle of the chamber projected area from stage 2 to stage 4. In stage 2, the projected area of the compression chamber is divided into two parts by a segment between two demarcation points of the circle involute and circular arc. In stage 3, another segment connected with xt1,yt1 and xt2,yt2 separates the projected area of the central discharge chamber into two symmetrical parts. The geometric element of the central discharge chamber in stage 4 only includes the modified circular arc. Similarly, the projected area is divided into two symmetrical parts by the segment connected with xt1,yt1 and xt2,yt2.Figure 3
Area divisions.


## 3.2. Chamber Volume Calculation

### 3.2.1. Suction Process

No matter the volume calculation stage, the orbiting angle range corresponding to the working chambers’ geometric elements should be determined first. The orbiting angle in stage 0 ranges fromθ=0 to 2π clearly. The orbiting angle value θd0 at the end of stage 1 is defined in Equation (12). According to meshing characteristics of PMP profile, the lasting angle of stage 2 is defined as Δθ expressed in Equation (13).
(12)θd0=φie−φos−π,(13)Δθ=βa1,2−βa1,1=βa2,2−βa2,1.The volume calculation of the working chamber in stage 0 and stage 1 had been analyzed accurately by former researchers [7, 20, 24]. The suction chamber Vs0 in stage 0 is expressed in Equation (14) quoted from Ref. [24].
(14)Vs0=hsrbrbφe−φi0−roθπ−2φe+θ−rbφe−π−φo02B−rbφe−π−φo0B2−rb3B3φe−φi0rbcosB+rbsinBφe−π+B−φo0−rosinθ+rbsinB−rbcosBφe−π+B−φo0−rocosθ−rosinB+θφe−π+B−φo0−rocosB+θ+ro0≤θ≤2π,where B=1/2φo0−φie+π+φo0−φie+π2−4D and D=ro/rbφi0−φiesinθ−cosθ+1/φie−φi0.


### 3.2.2. Compression Process

(1) Chamber in Stage 1. The compression chamber Vs1 in stage 1 is expressed in Equation (15) quoted from Ref. [20].
(15)Vs1=2πhsrbro2φe+π−2θ2π≤θ≤θd0.(2) Chamber Section c1 in Stage 2. Before the compression chamber section c1 volume calculation in stage 2, as shown in Figure 3, the discharge chamber section d1 in stage 3 is firstly analyzed. Because chamber section d1 originates from section c1 with the only change in the range of the orbiting angle θ, the division of the projected area of section d1 especially can be expressed in Equation (16) more clearly, as shown in Figure 4.
(16)Ad1=AO,d1−AI,d1.Figure 4
Division of chamber section d1 and decomposition ofAI,d1.AO,d1 formed by the coordinate origin and inner wall curve of fixed scroll wrap is calculated in Equation (17). The volume VO,d1 of the corresponding chamber section is obtained in Equation (18). As displayed in Figure 4, AI,d1 is divided into three triangles and one sector type region. It is noticeable that the geometric symmetry is considered here and makes the analysis simpler, as expressed in Equation (19).
(17)AO,d1=12∫φos+πφie−θ+2πxdydϕ−ydxdϕdϕ,(18)VO,d1=hsrb2φie−θ+2π−φi03−φos+π−φi036,(19)AI,d1=AIa,d1+AIb,d1+AIc,d1+AId,d1.AIa,d1 formed by the coordinate origin and inner wall curve of fixed scroll wrap is calculated with the upper and lower limits of integration φie−θ+π and φos, as expressed in Equation (20). VIb,d1, VIc,d1, and VId,d1 are calculated by multiplying the area of the triangle by the scroll height hs, respectively. Each projected triangle area formed by three geometric vertices is listed in Table 2 for every chamber section.
(20)AIa,d1=rb2φie−θ+π−φo03−φos−φo036.Table 2
Vertices making up of projected triangle areas.
Projected areaVertices forming axial projected triangle areaAIb,d10,0; rocosθm,rosinθm; rbcosφos+rbφos−φo0sinφos,rbsinφos−rbφos−φo0cosφosAIc,d10,0; rocosθm,rosinθm; rbcosφie−θ+π+rbφie−θ+π−φo0sinφie−θ+π,rbsinφie−θ+π−rbφie−θ+π−φo0cosφie−θ+πAId,d1rocosθm,rosinθm;−rbcosφis−rbφis−φi0sinφis+rocosθm,−rbsinφis+rbφis−φi0cosφis+rosinθm; rbcosφos+rbφos−φo0sinφos,rbsinφos−rbφos−φo0cosφosVIb,d1, VIc,d1, and VId,d1 are given by Equation (21), Equation (22), and Equation (23), respectively. From Equation (16) to Equation (23), the volume expression of chamber section d1 in stage 3 θd0+Δθ≤θ≤θd0+2π is obtained in Equation (24). Finally, chamber section c1 in stage 2 has the consistent expression of the volume calculation as chamber section d1, just with a different range of orbiting angles θd0≤θ≤θd0+Δθ.
(21)VIb,d1=hsrbro2φos−φo0sinθ+φos−φie+cosθ+φos−φie,(22)VIc,d1=hsrbro2,(23)VId,d1=hsrbro2φos−φi0+πsinθ+φos−φie+cosθ+φos−φie+1,(24)Vd1=hsrb6rbφie−θ+2π−φi03−φos+π−φi03−rbφie−θ+π−φo03−φos−φo03−3ro2φos+πsinθ+φos−φie+2cosθ+φos−φie+2.(3) Calculation of Chamber Section cc1 in Stage 2. The projected area of chamber section cc1, Acc1, equals to the difference between AO,cc1 and AI,cc1, as shown in Figure 5. AO,cc1 and AI,cc1 can be calculated with two different upper and lower limits of integration [βa1,2+θd0−θ, βa1,2] and [βa2,2+θd0−θ, βa2,2], respectively. Then, the volumes of VO,cc1 and VI,cc1 are simplified as Equations (26) and (29), where Equation (28) describes the coordinates of the orbiting scroll modified arc. Finally, the volume of two compression chambers in stage 2, Vs2, is obtained from Equation (24), Equation (26), and Equation (29).
(25)AO,cc1=12∫βa1,2+θd0−θβa1,2x−xoosdydβ−y−yoosdxdβdβ,(26)VO,cc1=ra1hs2yoos−ya1cosβa1,2−cosβa1,2+θd0−θ−xoos−xa1sinβa1,2−sinβa1,2+θd0−θ+ra1θ−θd0,(27)AI,cc1=12∫βa2,2+θd0−θβa2,2xβ−xoosdyβdβ−yβ−yoosdxβdβdβ,where
(28)xβ=−xa2−ra2cosβ+rocosθm,yβ=−ya2−ra2sinβ+rosinθm,βa2,2+θd0−θ≤β≤βa2,2,(29)VI,cc1=ra2hs2xa2sinβa2,2−sinβa2,2+θd0−θ−ya2cosβa2,2−cosβa2,2+θd0−θ−rbsinβa2,2−φos−sinβa2,2+θd0−θ−φos−rbφos−φo0cosβa2,2−φos−cosβa2,2+θd0−θ−φos+ra2θ−θd0,(30)Vs2=2Vc1+VO,cc1−VI,cc1θd0≤θ≤Δθ+θd0.Figure 5
Divisions of chamber sections in stages 2 to 4.


### 3.2.3. Discharge Process

The discharge process covers stage 3 and stage 4 continuously. In stage 3, the discharge chamber shown in Figure3 includes chamber sections d1, d2, dd1, and dd2. Considering the geometric symmetry, chamber sections d1 and dd1 were analyzed. The chamber section d1 had been described in Equation (24). As for the chamber section dd1, according to Figure 5, its projected area Add1 can be obtained via
(31)Add1=AOa,dd1+At−AIa,dd1.At and AOa,dd1 were also derived in Equation (8) and Equation (10). Hence, only AIa,dd1 needs to be derived in Equation (32). Finally, the total volume of the discharge chamber in stage 3 is obtained in Equation (34).
(32)AIa,dd1=12∫βa2,1βa2,2x−xoosdydβ−y−yoosdxdβdβ,(33)VIa,dd1=ra2hs2xa2sinβa2,2−sinβa2,1−ya2cosβa2,2−cosβa2,1−rbsinβa2,2−φos−sinβa2,1−φos−rbφos−φo0,cosβa2,2−φos−cosβa2,1−φos+ra2βa2,2−βa2,1,(34)Vs3=2Vd1+VOa,dd1+Vt−VIa,dd1θd0+Δθ≤θ≤θd0+2π.In stage 4, the projected area of discharge chamber section ddd1 can be obtained according toAddd1=AO,ddd1−AI,ddd1 shown in Figure 5. AO,ddd1, the sector type region, is expressed in Equation (35). The volume of VO,ddd1 was calculated according to Equation (37).
(35)AO,ddd1=12∫βa1,1βa1,2+θd0+2π−θx−xt1dydβ−y−yt1dxdβdβ,where the modified arc coordinates are expressed in
(36)xβ=xa1+ra1cosβ,yβ=ya1+ra1sinβ,βa1,1≤β≤βa1,2+θd0+2π−θ,(37)VO,ddd1=ra1hs2xa1+xt2sinβa1,2+θd0+2π−θ−sinβa1,1+ya1+yt2cosβa1,1−cosβa1,2+θd0+2π−θ+ra1βa1,2+θd0+2π−θ−βa1,1+rosinφie−θd0−2.5π−βa1,2−sinθm−βa1,1.AI,ddd1 is expressed in Equation (38), and Equation (40) derives VI,ddd1. Finally, the total volume of the central chamber in stage 4, Vs4, is obtained.
(38)AI,ddd1=12∫βa2,1βa2,2+θd0+2π−θx−xt1dydβ−y−yt1dxdβdβ,where the modified arc coordinates are expressed in
(39)xβ=−xa2−ra2cosβ+rocosθm,yβ=−ya2−ra2sinβ+rosinθm,βa2,1≤β≤βa2,2+θd0+2π−θ,(40)VI,ddd1=ra2hs2xa2−xt2sinβa2,2+θd0+2π−θ−sinβa2,1+ra2βa2,2+θd0+2π−θ−βa2,1+ya2−yt2cosβa2,1−cosβa2,2+θd0+2π−θ,(41)Vs4=2VO,ddd1−VI,ddd1θd0+2π≤θ≤θd0+2π+Δθ.Combining Equation (30), Equation (34), and Equation (41), a general piecewise function-based geometric model is finally established as described in Equation (42). In the following subsection, the precision and accuracy evaluation are conducted for the geometric model.
(42)V=Vs00≤θ≤2π,Vs12π≤θ≤θd0,Vs2θd0≤θ≤θd0+Δθ,Vs3θd0+Δθ≤θ≤θd0+2π,Vs4θd0+2π≤θ≤θd0+2π+Δθ.


## 3.2.1. Suction Process

No matter the volume calculation stage, the orbiting angle range corresponding to the working chambers’ geometric elements should be determined first. The orbiting angle in stage 0 ranges fromθ=0 to 2π clearly. The orbiting angle value θd0 at the end of stage 1 is defined in Equation (12). According to meshing characteristics of PMP profile, the lasting angle of stage 2 is defined as Δθ expressed in Equation (13).
(12)θd0=φie−φos−π,(13)Δθ=βa1,2−βa1,1=βa2,2−βa2,1.The volume calculation of the working chamber in stage 0 and stage 1 had been analyzed accurately by former researchers [7, 20, 24]. The suction chamber Vs0 in stage 0 is expressed in Equation (14) quoted from Ref. [24].
(14)Vs0=hsrbrbφe−φi0−roθπ−2φe+θ−rbφe−π−φo02B−rbφe−π−φo0B2−rb3B3φe−φi0rbcosB+rbsinBφe−π+B−φo0−rosinθ+rbsinB−rbcosBφe−π+B−φo0−rocosθ−rosinB+θφe−π+B−φo0−rocosB+θ+ro0≤θ≤2π,where B=1/2φo0−φie+π+φo0−φie+π2−4D and D=ro/rbφi0−φiesinθ−cosθ+1/φie−φi0.


## 3.2.2. Compression Process

(1) Chamber in Stage 1. The compression chamber Vs1 in stage 1 is expressed in Equation (15) quoted from Ref. [20].
(15)Vs1=2πhsrbro2φe+π−2θ2π≤θ≤θd0.(2) Chamber Section c1 in Stage 2. Before the compression chamber section c1 volume calculation in stage 2, as shown in Figure 3, the discharge chamber section d1 in stage 3 is firstly analyzed. Because chamber section d1 originates from section c1 with the only change in the range of the orbiting angle θ, the division of the projected area of section d1 especially can be expressed in Equation (16) more clearly, as shown in Figure 4.
(16)Ad1=AO,d1−AI,d1.Figure 4
Division of chamber section d1 and decomposition ofAI,d1.AO,d1 formed by the coordinate origin and inner wall curve of fixed scroll wrap is calculated in Equation (17). The volume VO,d1 of the corresponding chamber section is obtained in Equation (18). As displayed in Figure 4, AI,d1 is divided into three triangles and one sector type region. It is noticeable that the geometric symmetry is considered here and makes the analysis simpler, as expressed in Equation (19).
(17)AO,d1=12∫φos+πφie−θ+2πxdydϕ−ydxdϕdϕ,(18)VO,d1=hsrb2φie−θ+2π−φi03−φos+π−φi036,(19)AI,d1=AIa,d1+AIb,d1+AIc,d1+AId,d1.AIa,d1 formed by the coordinate origin and inner wall curve of fixed scroll wrap is calculated with the upper and lower limits of integration φie−θ+π and φos, as expressed in Equation (20). VIb,d1, VIc,d1, and VId,d1 are calculated by multiplying the area of the triangle by the scroll height hs, respectively. Each projected triangle area formed by three geometric vertices is listed in Table 2 for every chamber section.
(20)AIa,d1=rb2φie−θ+π−φo03−φos−φo036.Table 2
Vertices making up of projected triangle areas.
Projected areaVertices forming axial projected triangle areaAIb,d10,0; rocosθm,rosinθm; rbcosφos+rbφos−φo0sinφos,rbsinφos−rbφos−φo0cosφosAIc,d10,0; rocosθm,rosinθm; rbcosφie−θ+π+rbφie−θ+π−φo0sinφie−θ+π,rbsinφie−θ+π−rbφie−θ+π−φo0cosφie−θ+πAId,d1rocosθm,rosinθm;−rbcosφis−rbφis−φi0sinφis+rocosθm,−rbsinφis+rbφis−φi0cosφis+rosinθm; rbcosφos+rbφos−φo0sinφos,rbsinφos−rbφos−φo0cosφosVIb,d1, VIc,d1, and VId,d1 are given by Equation (21), Equation (22), and Equation (23), respectively. From Equation (16) to Equation (23), the volume expression of chamber section d1 in stage 3 θd0+Δθ≤θ≤θd0+2π is obtained in Equation (24). Finally, chamber section c1 in stage 2 has the consistent expression of the volume calculation as chamber section d1, just with a different range of orbiting angles θd0≤θ≤θd0+Δθ.
(21)VIb,d1=hsrbro2φos−φo0sinθ+φos−φie+cosθ+φos−φie,(22)VIc,d1=hsrbro2,(23)VId,d1=hsrbro2φos−φi0+πsinθ+φos−φie+cosθ+φos−φie+1,(24)Vd1=hsrb6rbφie−θ+2π−φi03−φos+π−φi03−rbφie−θ+π−φo03−φos−φo03−3ro2φos+πsinθ+φos−φie+2cosθ+φos−φie+2.(3) Calculation of Chamber Section cc1 in Stage 2. The projected area of chamber section cc1, Acc1, equals to the difference between AO,cc1 and AI,cc1, as shown in Figure 5. AO,cc1 and AI,cc1 can be calculated with two different upper and lower limits of integration [βa1,2+θd0−θ, βa1,2] and [βa2,2+θd0−θ, βa2,2], respectively. Then, the volumes of VO,cc1 and VI,cc1 are simplified as Equations (26) and (29), where Equation (28) describes the coordinates of the orbiting scroll modified arc. Finally, the volume of two compression chambers in stage 2, Vs2, is obtained from Equation (24), Equation (26), and Equation (29).
(25)AO,cc1=12∫βa1,2+θd0−θβa1,2x−xoosdydβ−y−yoosdxdβdβ,(26)VO,cc1=ra1hs2yoos−ya1cosβa1,2−cosβa1,2+θd0−θ−xoos−xa1sinβa1,2−sinβa1,2+θd0−θ+ra1θ−θd0,(27)AI,cc1=12∫βa2,2+θd0−θβa2,2xβ−xoosdyβdβ−yβ−yoosdxβdβdβ,where
(28)xβ=−xa2−ra2cosβ+rocosθm,yβ=−ya2−ra2sinβ+rosinθm,βa2,2+θd0−θ≤β≤βa2,2,(29)VI,cc1=ra2hs2xa2sinβa2,2−sinβa2,2+θd0−θ−ya2cosβa2,2−cosβa2,2+θd0−θ−rbsinβa2,2−φos−sinβa2,2+θd0−θ−φos−rbφos−φo0cosβa2,2−φos−cosβa2,2+θd0−θ−φos+ra2θ−θd0,(30)Vs2=2Vc1+VO,cc1−VI,cc1θd0≤θ≤Δθ+θd0.Figure 5
Divisions of chamber sections in stages 2 to 4.


## 3.2.3. Discharge Process

The discharge process covers stage 3 and stage 4 continuously. In stage 3, the discharge chamber shown in Figure3 includes chamber sections d1, d2, dd1, and dd2. Considering the geometric symmetry, chamber sections d1 and dd1 were analyzed. The chamber section d1 had been described in Equation (24). As for the chamber section dd1, according to Figure 5, its projected area Add1 can be obtained via
(31)Add1=AOa,dd1+At−AIa,dd1.At and AOa,dd1 were also derived in Equation (8) and Equation (10). Hence, only AIa,dd1 needs to be derived in Equation (32). Finally, the total volume of the discharge chamber in stage 3 is obtained in Equation (34).
(32)AIa,dd1=12∫βa2,1βa2,2x−xoosdydβ−y−yoosdxdβdβ,(33)VIa,dd1=ra2hs2xa2sinβa2,2−sinβa2,1−ya2cosβa2,2−cosβa2,1−rbsinβa2,2−φos−sinβa2,1−φos−rbφos−φo0,cosβa2,2−φos−cosβa2,1−φos+ra2βa2,2−βa2,1,(34)Vs3=2Vd1+VOa,dd1+Vt−VIa,dd1θd0+Δθ≤θ≤θd0+2π.In stage 4, the projected area of discharge chamber section ddd1 can be obtained according toAddd1=AO,ddd1−AI,ddd1 shown in Figure 5. AO,ddd1, the sector type region, is expressed in Equation (35). The volume of VO,ddd1 was calculated according to Equation (37).
(35)AO,ddd1=12∫βa1,1βa1,2+θd0+2π−θx−xt1dydβ−y−yt1dxdβdβ,where the modified arc coordinates are expressed in
(36)xβ=xa1+ra1cosβ,yβ=ya1+ra1sinβ,βa1,1≤β≤βa1,2+θd0+2π−θ,(37)VO,ddd1=ra1hs2xa1+xt2sinβa1,2+θd0+2π−θ−sinβa1,1+ya1+yt2cosβa1,1−cosβa1,2+θd0+2π−θ+ra1βa1,2+θd0+2π−θ−βa1,1+rosinφie−θd0−2.5π−βa1,2−sinθm−βa1,1.AI,ddd1 is expressed in Equation (38), and Equation (40) derives VI,ddd1. Finally, the total volume of the central chamber in stage 4, Vs4, is obtained.
(38)AI,ddd1=12∫βa2,1βa2,2+θd0+2π−θx−xt1dydβ−y−yt1dxdβdβ,where the modified arc coordinates are expressed in
(39)xβ=−xa2−ra2cosβ+rocosθm,yβ=−ya2−ra2sinβ+rosinθm,βa2,1≤β≤βa2,2+θd0+2π−θ,(40)VI,ddd1=ra2hs2xa2−xt2sinβa2,2+θd0+2π−θ−sinβa2,1+ra2βa2,2+θd0+2π−θ−βa2,1+ya2−yt2cosβa2,1−cosβa2,2+θd0+2π−θ,(41)Vs4=2VO,ddd1−VI,ddd1θd0+2π≤θ≤θd0+2π+Δθ.Combining Equation (30), Equation (34), and Equation (41), a general piecewise function-based geometric model is finally established as described in Equation (42). In the following subsection, the precision and accuracy evaluation are conducted for the geometric model.
(42)V=Vs00≤θ≤2π,Vs12π≤θ≤θd0,Vs2θd0≤θ≤θd0+Δθ,Vs3θd0+Δθ≤θ≤θd0+2π,Vs4θd0+2π≤θ≤θd0+2π+Δθ.


## 3.3. Geometric Model Evaluation

According to a reference operation condition (case M10 in Yang’s work [50]) of a CO2 compressor in a vehicle air conditioning system, the geometric profiles of the scroll compressor were designed by an in-house code. The essential parameters of the scroll profiles are listed in Table 3.Table 3
Geometric parameters of the CO2 scroll compressor.
SymbolUnitValuehsmm6.530rbmm3.000romm4.210φo0rad-0.869φi0rad0.869φosrad1.571φisrad4.712φerad12.566ra1mm7.295ra2mm3.085For evaluation, the polygon-based method has been applied to the evaluation of projection area calculation accuracy in Ref. [26]. Hence, the polygon-based equivalent method to the discharge chamber projected area calculation in stage 3 was utilized. The independence verification of the polygon edge number was firstly conducted. Polygon edge numbers 1000, 2000, 4000, and 8000 were adapted, respectively, to calculate equivalently the discharge chamber projected area at θ=θd0=φie−φos−π=2.5π. Table 4 presents the relative error of the polygon area among different polygon edge numbers. As the number of edges increases, the accuracy becomes higher. Relative error decreases significantly from 0.0195‰ to 0.0052‰ after choosing 4000 edges. Hence, the polygon edge number 4000 was selected to calculate the projected area.Table 4
Independence verification of the polygon edge number.
Polygon edge numberProjected area (mm2)Relative error (‰)1000498.5678—2000498.57750.01954000498.58010.00528000498.58070.0012Based on a sufficient accuracy approximation by a polygon with 4000 edges, Figure6 shows the precision difference between the geometric model in the present paper and that in Bell’s study [22]. The relative errors of 0.001% at the beginning and 0.16% at the end of stage 3 indicate that both models agree well with the polygon approximated model. However, a significant relative error reaching up to 12.95% exists in the other model at the orbiting angle of 4π. The geometric model in the work of Liu et al. [51] and Wang et al. [52] was adapted to verify the present theoretical model. Figure 6 indicates that the present model has a sufficient prediction accuracy. As the scroll compressor 1D throughflow modeling prerequisite, the volume variation of the scroll working chamber could be simulated accurately.Figure 6
Accuracy evaluation and validation of scroll geometric model.


## 4. Throughflow Modeling of CO2 Scroll Compressor

Figure7 shows three pairs of scroll symmetric chambers 1A/1B, 2A/2B, and 3A/3B. The suction chamber 1A and chamber 1B include two virtual inlet planes (VIPs) marked as inlet A and inlet B. The compression chamber 2A and chamber 2B connect with the upstream and downstream working chambers through the radial clearances. Similar to the description in Ref. [32], the central discharge chamber was divided into two symmetric chambers 3A and 3B by a virtual overlapping plane (VOLP), marked by a red dotted line. Moreover, the outlet was also divided into two virtual outlet planes (VOPs) marked as outlet A and outlet B. As a result, the central discharge chamber can be considered as two separated symmetric working chambers in the whole discharging process. The variations of the chamber volume and the areas of virtual planes with the orbiting angle are also displayed in Figure 7, which was utilized in the CO2 scroll compressor 1D throughflow model developed in the GT-SUITE simulation environment.Figure 7
Chamber illustration and the variation of volume and virtual areas in 1D throughflow model.The 1D throughflow model consists of different equivalent modules such as the suction and discharge pipe system, working chamber, crankshaft, clearance leakage channel, and the overlapping surface of the discharge chamber. The most important module working chamber could simulate the whole volume variation process from suction to discharge. Three pairs of symmetric working chambers represent different working processes at the same orbiting angle via the crankshaft module. Moreover, a rotating phase difference of 360 degrees ensures the continuity of the working process. An equivalent Orifice module describes the flank leakage through the radial clearance, but the radial leakage flow through the axial clearance is not included here. The overlapping surface module can communicate the mass flow exchange between two discharge chambers.The pressure and temperature at the inlet boundary are 33.9 bar and 302.65 K. The outlet pressure is 95.6 bar. The CO2 properties were obtained from NIST REFPROP 9.1 database. The design rotation speed of the compressor is 5000 rpm. The energy loss of the heat transfer between the working fluid and the scroll wrap/plate wall was also neglected by the adiabatic wall condition.


## 5. Results and Discussion

### 5.1. The Precompression Effect

#### 5.1.1. Influence of Rotating Speed

First, the present paper investigated the impacts of the rotating speed and the initial suction pressure on the precompression. The influence of the rotating speed is clearly displayed in Figures8 and 9. Figure 8 shows two common phenomena of pressure variation among different rotating speeds. It can be seen that the minimum pressure in the suction chamber decreases significantly with the increase of the rotation speed, which means that the degree of the suction vacuum gets larger, leading to the increase of the pressure difference between the inlet and the suction chamber. The mass flow driven by pressure difference therefore will increase after more working fluids enter the finite volume chamber, which leads to the enhancement of the precompression. This will be further discussed in Figure 9. The suction vacuum exists approximately covering the initial two-thirds of the suction process, and the precompression appears in the remaining one-third period. With the throughflow areas of inlet A and inlet B decreasing continuously, less reverse flow through the suction ports is beneficial for the precompression in the suction chamber. The suction vacuum and the precompression become more severe as the rotating speed increases. The level of suction vacuum increases with the rotating speed due to the increment of the volume variation rate.Figure 8
Variation of the suction pressure and precompression extension under different rotating speeds.Figure 9
Flow paths of refrigerant in chamber 1A and mass flow rate through inlet A during the suction process.Figure8 quantifies the precompression effect. First, the precompression extension was defined as Epc in Equation (43). pactual is the actual pressure at the end of the suction process, and pin represents the initial suction pressure.
(43)Epc=pactual−pinpin×100%.pactual as well as Epc increases with the rotating speed from 2000 rpm to 8000 rpm. CO2 is sucked into the suction chamber by a certain vacuum degree. As shown in Figure 9, the inflow or outflow through the inlet A and RC1A of the control volume 1A determines the net refrigerant mass in the suction chamber, which mainly affects the final suction pressure and the precompression extension. Figure 9 also shows the mass flow rate through inlet A during the suction process under different rotating speeds. Before the end of the suction process, the pressure difference between the chamber and inlet pipe causes the outflow through inlet A in a short period. In the inflow period, much more refrigerant enters the chamber 1A at the same orbiting angle position due to the higher vacuum degree as the rotating speed increases. Meanwhile, the outflow increases slightly with the rotating speed. As a result, the increase of rotating speed benefits the refrigerant flow entering the suction chamber.Figure10 exhibits the integrated absolute and relative refrigerant leakage mass entering into chamber 1A through radial clearance RC1A during the suction process under different rotating speeds. To quantify the influence of tangential leakage on the suction precompression, the tangential leakage flow rate qm and the absolute and relative leakage masses, Mlc,abs and Mlc,r, were obtained by Equation (44), Equation (45), and Equation (46), respectively. As shown in Figure 10, the absolute leakage mass goes down significantly from 0.026 g to 0.0069 g as the rotating speed increases. The relative leakage mass also decreases from 4.4% to 1.17%, which means the tangential leakage of refrigerant through RC1A makes less contribution to the suction precompression.
(44)qm=CdρπD242ph−plρ,(45)Mlc,abs=16N∫0360qmdθ,(46)Mlc,r=Mlc,absMideal×100%,where qm refers to the transient mass flow rate of refrigerant during the suction process, Cd represents the effective discharge coefficient, ρ refers to the refrigerant density, D means the equivalent hydraulic diameter, ph and pl purport the pressures of the upstream chamber and downstream suction chamber, respectively, Mlc,abs is the total leakage mass through the racial clearance, N represents the rotating speed of the crankshaft, and Mideal refers to the ideal mass through a quasistatic suction process.Figure 10
Total and relative leakage mass flow under different rotating speeds.As abovementioned, the suction vacuum dominates the extension of the precompression under different rotating speeds. At the low rotating speed, besides the suction vacuum, the tangential leakage partially contributes to the precompression. With the rotating speed increase, the tangential leakage decreases significantly, and its influence on the precompression effect becomes weaker gradually.


#### 5.1.2. Influence of Initial Suction Pressure

Figure11 shows the impact of initial suction pressure on the precompression extension. In the initial two-thirds period of the suction process, the pressure curves generally reach to the initial suction pressures with a similar variation pattern. Nevertheless, the initial suction pressure variation leads to a slight difference of the precompression in the remaining one-third period, which is quantified in Figure 11. According to the definition, the increasing initial suction pressure could cause a noticeable reduction of the precompression extension. However, the increasing initial suction pressure weakens the suction outflow through the inlet port; hence, the pressure difference between the end pressure and the initial suction pressures raises linearly in reality. As a result, the precompression extension just slightly decreases from 8.196% to 8.16%.Figure 11
Variation of the suction pressure and precompression extension under different initial suction pressures.


### 5.2. Asymmetric Discharging Effect

Based on revealing the unsteady behaviors of the compressor discharging flow, the present paper compared the effects of two different structures of the discharge port on the asymmetric discharging flow. The most popular circular type discharge port was firstly arranged tangent with the discharge arc a1 at the disengaged pointxt2,yt2 displayed in Figure 1. The predesigned circular discharge port is shown in Figure 12. The pressure variations in the discharge chambers 3A and 3B and corresponding throughflow areas are displayed in Figure 13. The discharge process of the predesigned scroll compressor with geometric parameters listed in Table 3 lasts from orbiting angle 594.69 degrees to 954.69 degrees. The discharge process was separated into three substages in Figure 13 according to the asymmetric pressure distribution between two discharge chambers 3A and 3B.Figure 12
Positions of the circular and waist-shape discharge ports.Figure 13
The pressure and throughflow areas of discharge chambers and throughflow areas at representative positions.At the beginning of the substage ①, although the discharge volume decreases, the pressure in the chambers 3A and 3B rises slightly when the throughflow areas are still too small. For instance, Figure13 shows the small areas of the discharge port and overlapping surface at a representative position of substage ①. In the remaining process, as the discharge volume decreases continuously, the pressure in chamber 3B still increases rapidly while chamber 3A pressure drops rapidly. This special pressure variation behavior can be explained by the variation of throughflow areas. Compared with the discharge throughflow area of chamber 3A, that of chamber 3B is much smaller, resulting in a higher discharge flow resistance of chamber 3B. On the other hand, an overlapping surface area nearby zero limits the mass flow transfer between two symmetric discharge chambers in the substage ①. As a result, asymmetric pressure distribution becomes more and more evident with the increase of the orbiting angle.In substage ②, the asymmetric pressure distribution gets weaker. The pressure decrease of chamber 3B mainly contributes to the pressure balance process in this substage. The quick pressure drop of chamber 3B is mainly associated with the apparent increase of the overlapping area, as shown in Figure13. In this case, the pressure difference between chambers 3A and 3B drives a relatively high mass flow transferring from chambers 3B to 3A, as displayed in Figure 14. The mass flow rate through the overlapping area reaches its maximum, resulting in an acceleration of the pressure equilibrium in substage ②. In substage ③, even though the discharge throughflow areas of chambers 3A and 3B vary with the different pattern as the orbiting angle increases, the pressure equilibrium is still dominated by the overlapping area, as displayed in Figure 13. During the later period of the substage ③, the direction of mass flow transfer gets reversed. Because the discharge throughflow area of chamber 3B is greater than that of chamber 3A shown in Figure 13, the refrigerant in chamber 3A has to “leak” into chamber 3B to keep the pressure equilibrium.Figure 14
Mass flow rate through overlapping area.As abovementioned, the asymmetric pressure distribution in the discharge process is mainly induced by the variation of the discharge throughflow areas and is weakened by the variation of the overlapping surface area. And also, the pressure equilibrium in discharge chambers is maintained by the mass flow transfer through the overlapping surface. Hence, variations of two types of throughflow areas control the internal flow pattern in the discharge chambers. The discharging controlling methods on the asymmetric flow involve the discharge port location optimization and structure design adopted in several previous studies. Gelke et al. [53] reduced the maximum pressure in the discharge chamber with appropriate positioning of the discharge ports. Zhao et al. [54] weakened the asymmetric flow via the optimization of the position and the shape of the discharge port. The present paper compared the circular discharge port and the waist-shape discharge port investigated in Ref. [55], as displayed in Figure 12.Figure15 illustrates the effect of the circular and waist-shape discharge ports on the variation of discharge pressures. The pressure in chamber 3A has no apparent change in substage ① for both the circular and waist-shape discharge ports. However, compared with the circular discharge port, the waist-shape one brings about a lower pressure in chamber 3B at the end of the substage ①, which means a weaker asymmetric pressure distribution. The maximum pressure in chamber 3B drops 1.2 bar from 117.01 bar to 115.81 bar. These pressure distributions for both discharge ports last until the end of the substage ②. However, the waist-shape one reduced the duration of the asymmetric pressure distribution by about 6 degrees of orbiting angle from θc=639° to θw=633°. The structural adjustment of the discharge port has no significant impact on the pressure value and the pressure equilibrium between two discharge chambers in substage ③. As mentioned above, the pressure equilibrium in the substage ③ was dominated by the overlapping throughflow between two discharge chambers.Figure 15
Discharge pressures of chambers 3A and 3B under two discharge ports.


## 5.1. The Precompression Effect

### 5.1.1. Influence of Rotating Speed

First, the present paper investigated the impacts of the rotating speed and the initial suction pressure on the precompression. The influence of the rotating speed is clearly displayed in Figures8 and 9. Figure 8 shows two common phenomena of pressure variation among different rotating speeds. It can be seen that the minimum pressure in the suction chamber decreases significantly with the increase of the rotation speed, which means that the degree of the suction vacuum gets larger, leading to the increase of the pressure difference between the inlet and the suction chamber. The mass flow driven by pressure difference therefore will increase after more working fluids enter the finite volume chamber, which leads to the enhancement of the precompression. This will be further discussed in Figure 9. The suction vacuum exists approximately covering the initial two-thirds of the suction process, and the precompression appears in the remaining one-third period. With the throughflow areas of inlet A and inlet B decreasing continuously, less reverse flow through the suction ports is beneficial for the precompression in the suction chamber. The suction vacuum and the precompression become more severe as the rotating speed increases. The level of suction vacuum increases with the rotating speed due to the increment of the volume variation rate.Figure 8
Variation of the suction pressure and precompression extension under different rotating speeds.Figure 9
Flow paths of refrigerant in chamber 1A and mass flow rate through inlet A during the suction process.Figure8 quantifies the precompression effect. First, the precompression extension was defined as Epc in Equation (43). pactual is the actual pressure at the end of the suction process, and pin represents the initial suction pressure.
(43)Epc=pactual−pinpin×100%.pactual as well as Epc increases with the rotating speed from 2000 rpm to 8000 rpm. CO2 is sucked into the suction chamber by a certain vacuum degree. As shown in Figure 9, the inflow or outflow through the inlet A and RC1A of the control volume 1A determines the net refrigerant mass in the suction chamber, which mainly affects the final suction pressure and the precompression extension. Figure 9 also shows the mass flow rate through inlet A during the suction process under different rotating speeds. Before the end of the suction process, the pressure difference between the chamber and inlet pipe causes the outflow through inlet A in a short period. In the inflow period, much more refrigerant enters the chamber 1A at the same orbiting angle position due to the higher vacuum degree as the rotating speed increases. Meanwhile, the outflow increases slightly with the rotating speed. As a result, the increase of rotating speed benefits the refrigerant flow entering the suction chamber.Figure10 exhibits the integrated absolute and relative refrigerant leakage mass entering into chamber 1A through radial clearance RC1A during the suction process under different rotating speeds. To quantify the influence of tangential leakage on the suction precompression, the tangential leakage flow rate qm and the absolute and relative leakage masses, Mlc,abs and Mlc,r, were obtained by Equation (44), Equation (45), and Equation (46), respectively. As shown in Figure 10, the absolute leakage mass goes down significantly from 0.026 g to 0.0069 g as the rotating speed increases. The relative leakage mass also decreases from 4.4% to 1.17%, which means the tangential leakage of refrigerant through RC1A makes less contribution to the suction precompression.
(44)qm=CdρπD242ph−plρ,(45)Mlc,abs=16N∫0360qmdθ,(46)Mlc,r=Mlc,absMideal×100%,where qm refers to the transient mass flow rate of refrigerant during the suction process, Cd represents the effective discharge coefficient, ρ refers to the refrigerant density, D means the equivalent hydraulic diameter, ph and pl purport the pressures of the upstream chamber and downstream suction chamber, respectively, Mlc,abs is the total leakage mass through the racial clearance, N represents the rotating speed of the crankshaft, and Mideal refers to the ideal mass through a quasistatic suction process.Figure 10
Total and relative leakage mass flow under different rotating speeds.As abovementioned, the suction vacuum dominates the extension of the precompression under different rotating speeds. At the low rotating speed, besides the suction vacuum, the tangential leakage partially contributes to the precompression. With the rotating speed increase, the tangential leakage decreases significantly, and its influence on the precompression effect becomes weaker gradually.


### 5.1.2. Influence of Initial Suction Pressure

Figure11 shows the impact of initial suction pressure on the precompression extension. In the initial two-thirds period of the suction process, the pressure curves generally reach to the initial suction pressures with a similar variation pattern. Nevertheless, the initial suction pressure variation leads to a slight difference of the precompression in the remaining one-third period, which is quantified in Figure 11. According to the definition, the increasing initial suction pressure could cause a noticeable reduction of the precompression extension. However, the increasing initial suction pressure weakens the suction outflow through the inlet port; hence, the pressure difference between the end pressure and the initial suction pressures raises linearly in reality. As a result, the precompression extension just slightly decreases from 8.196% to 8.16%.Figure 11
Variation of the suction pressure and precompression extension under different initial suction pressures.


## 5.1.1. Influence of Rotating Speed

First, the present paper investigated the impacts of the rotating speed and the initial suction pressure on the precompression. The influence of the rotating speed is clearly displayed in Figures8 and 9. Figure 8 shows two common phenomena of pressure variation among different rotating speeds. It can be seen that the minimum pressure in the suction chamber decreases significantly with the increase of the rotation speed, which means that the degree of the suction vacuum gets larger, leading to the increase of the pressure difference between the inlet and the suction chamber. The mass flow driven by pressure difference therefore will increase after more working fluids enter the finite volume chamber, which leads to the enhancement of the precompression. This will be further discussed in Figure 9. The suction vacuum exists approximately covering the initial two-thirds of the suction process, and the precompression appears in the remaining one-third period. With the throughflow areas of inlet A and inlet B decreasing continuously, less reverse flow through the suction ports is beneficial for the precompression in the suction chamber. The suction vacuum and the precompression become more severe as the rotating speed increases. The level of suction vacuum increases with the rotating speed due to the increment of the volume variation rate.Figure 8
Variation of the suction pressure and precompression extension under different rotating speeds.Figure 9
Flow paths of refrigerant in chamber 1A and mass flow rate through inlet A during the suction process.Figure8 quantifies the precompression effect. First, the precompression extension was defined as Epc in Equation (43). pactual is the actual pressure at the end of the suction process, and pin represents the initial suction pressure.
(43)Epc=pactual−pinpin×100%.pactual as well as Epc increases with the rotating speed from 2000 rpm to 8000 rpm. CO2 is sucked into the suction chamber by a certain vacuum degree. As shown in Figure 9, the inflow or outflow through the inlet A and RC1A of the control volume 1A determines the net refrigerant mass in the suction chamber, which mainly affects the final suction pressure and the precompression extension. Figure 9 also shows the mass flow rate through inlet A during the suction process under different rotating speeds. Before the end of the suction process, the pressure difference between the chamber and inlet pipe causes the outflow through inlet A in a short period. In the inflow period, much more refrigerant enters the chamber 1A at the same orbiting angle position due to the higher vacuum degree as the rotating speed increases. Meanwhile, the outflow increases slightly with the rotating speed. As a result, the increase of rotating speed benefits the refrigerant flow entering the suction chamber.Figure10 exhibits the integrated absolute and relative refrigerant leakage mass entering into chamber 1A through radial clearance RC1A during the suction process under different rotating speeds. To quantify the influence of tangential leakage on the suction precompression, the tangential leakage flow rate qm and the absolute and relative leakage masses, Mlc,abs and Mlc,r, were obtained by Equation (44), Equation (45), and Equation (46), respectively. As shown in Figure 10, the absolute leakage mass goes down significantly from 0.026 g to 0.0069 g as the rotating speed increases. The relative leakage mass also decreases from 4.4% to 1.17%, which means the tangential leakage of refrigerant through RC1A makes less contribution to the suction precompression.
(44)qm=CdρπD242ph−plρ,(45)Mlc,abs=16N∫0360qmdθ,(46)Mlc,r=Mlc,absMideal×100%,where qm refers to the transient mass flow rate of refrigerant during the suction process, Cd represents the effective discharge coefficient, ρ refers to the refrigerant density, D means the equivalent hydraulic diameter, ph and pl purport the pressures of the upstream chamber and downstream suction chamber, respectively, Mlc,abs is the total leakage mass through the racial clearance, N represents the rotating speed of the crankshaft, and Mideal refers to the ideal mass through a quasistatic suction process.Figure 10
Total and relative leakage mass flow under different rotating speeds.As abovementioned, the suction vacuum dominates the extension of the precompression under different rotating speeds. At the low rotating speed, besides the suction vacuum, the tangential leakage partially contributes to the precompression. With the rotating speed increase, the tangential leakage decreases significantly, and its influence on the precompression effect becomes weaker gradually.


## 5.1.2. Influence of Initial Suction Pressure

Figure11 shows the impact of initial suction pressure on the precompression extension. In the initial two-thirds period of the suction process, the pressure curves generally reach to the initial suction pressures with a similar variation pattern. Nevertheless, the initial suction pressure variation leads to a slight difference of the precompression in the remaining one-third period, which is quantified in Figure 11. According to the definition, the increasing initial suction pressure could cause a noticeable reduction of the precompression extension. However, the increasing initial suction pressure weakens the suction outflow through the inlet port; hence, the pressure difference between the end pressure and the initial suction pressures raises linearly in reality. As a result, the precompression extension just slightly decreases from 8.196% to 8.16%.Figure 11
Variation of the suction pressure and precompression extension under different initial suction pressures.


## 5.2. Asymmetric Discharging Effect

Based on revealing the unsteady behaviors of the compressor discharging flow, the present paper compared the effects of two different structures of the discharge port on the asymmetric discharging flow. The most popular circular type discharge port was firstly arranged tangent with the discharge arc a1 at the disengaged pointxt2,yt2 displayed in Figure 1. The predesigned circular discharge port is shown in Figure 12. The pressure variations in the discharge chambers 3A and 3B and corresponding throughflow areas are displayed in Figure 13. The discharge process of the predesigned scroll compressor with geometric parameters listed in Table 3 lasts from orbiting angle 594.69 degrees to 954.69 degrees. The discharge process was separated into three substages in Figure 13 according to the asymmetric pressure distribution between two discharge chambers 3A and 3B.Figure 12
Positions of the circular and waist-shape discharge ports.Figure 13
The pressure and throughflow areas of discharge chambers and throughflow areas at representative positions.At the beginning of the substage ①, although the discharge volume decreases, the pressure in the chambers 3A and 3B rises slightly when the throughflow areas are still too small. For instance, Figure13 shows the small areas of the discharge port and overlapping surface at a representative position of substage ①. In the remaining process, as the discharge volume decreases continuously, the pressure in chamber 3B still increases rapidly while chamber 3A pressure drops rapidly. This special pressure variation behavior can be explained by the variation of throughflow areas. Compared with the discharge throughflow area of chamber 3A, that of chamber 3B is much smaller, resulting in a higher discharge flow resistance of chamber 3B. On the other hand, an overlapping surface area nearby zero limits the mass flow transfer between two symmetric discharge chambers in the substage ①. As a result, asymmetric pressure distribution becomes more and more evident with the increase of the orbiting angle.In substage ②, the asymmetric pressure distribution gets weaker. The pressure decrease of chamber 3B mainly contributes to the pressure balance process in this substage. The quick pressure drop of chamber 3B is mainly associated with the apparent increase of the overlapping area, as shown in Figure13. In this case, the pressure difference between chambers 3A and 3B drives a relatively high mass flow transferring from chambers 3B to 3A, as displayed in Figure 14. The mass flow rate through the overlapping area reaches its maximum, resulting in an acceleration of the pressure equilibrium in substage ②. In substage ③, even though the discharge throughflow areas of chambers 3A and 3B vary with the different pattern as the orbiting angle increases, the pressure equilibrium is still dominated by the overlapping area, as displayed in Figure 13. During the later period of the substage ③, the direction of mass flow transfer gets reversed. Because the discharge throughflow area of chamber 3B is greater than that of chamber 3A shown in Figure 13, the refrigerant in chamber 3A has to “leak” into chamber 3B to keep the pressure equilibrium.Figure 14
Mass flow rate through overlapping area.As abovementioned, the asymmetric pressure distribution in the discharge process is mainly induced by the variation of the discharge throughflow areas and is weakened by the variation of the overlapping surface area. And also, the pressure equilibrium in discharge chambers is maintained by the mass flow transfer through the overlapping surface. Hence, variations of two types of throughflow areas control the internal flow pattern in the discharge chambers. The discharging controlling methods on the asymmetric flow involve the discharge port location optimization and structure design adopted in several previous studies. Gelke et al. [53] reduced the maximum pressure in the discharge chamber with appropriate positioning of the discharge ports. Zhao et al. [54] weakened the asymmetric flow via the optimization of the position and the shape of the discharge port. The present paper compared the circular discharge port and the waist-shape discharge port investigated in Ref. [55], as displayed in Figure 12.Figure15 illustrates the effect of the circular and waist-shape discharge ports on the variation of discharge pressures. The pressure in chamber 3A has no apparent change in substage ① for both the circular and waist-shape discharge ports. However, compared with the circular discharge port, the waist-shape one brings about a lower pressure in chamber 3B at the end of the substage ①, which means a weaker asymmetric pressure distribution. The maximum pressure in chamber 3B drops 1.2 bar from 117.01 bar to 115.81 bar. These pressure distributions for both discharge ports last until the end of the substage ②. However, the waist-shape one reduced the duration of the asymmetric pressure distribution by about 6 degrees of orbiting angle from θc=639° to θw=633°. The structural adjustment of the discharge port has no significant impact on the pressure value and the pressure equilibrium between two discharge chambers in substage ③. As mentioned above, the pressure equilibrium in the substage ③ was dominated by the overlapping throughflow between two discharge chambers.Figure 15
Discharge pressures of chambers 3A and 3B under two discharge ports.


## 6. Conclusions

This study focused on developing the scroll geometric model based on the vector triangle method. Furthermore, one-dimensional flow modeling of a carbon dioxide scroll compressor for automotive air conditioning system was established to investigate the compressor’s precompression and asymmetric discharge effects. The main conclusions are summarized as follows:(1)
An improved geometry model of scroll compressor with dual-arc tip modification was proposed and validated. Based on an original division of the working processes, a general piecewise function covering the suction, compression, and discharge processes continuously and accurately describes working chambers’ volume variation. One-dimensional throughflow model considering tangential leakage of a carbon dioxide scroll compressor was established based on the proposed geometric model(2)
The strength of the suction precompression is mainly affected by the suction vacuum under different rotating speeds. The pressure rise at the end of the suction process is partly associated with the tangential leakage besides the suction vacuum at the low rotating speed. As the rotating speed increases, the effect of tangential leakage becomes weaker gradually due to the less leakage flow(3)
The increasing initial suction pressure results in a noticeable reduction of the precompression at the end of the suction process. Because the suction outflow through the inlet port is weakened by the increasing initial suction pressure, the precompression extension just slightly decreases from 8.196% to 8.16%(4)
The variation of the discharge throughflow areas of two symmetric discharge chambers dominates the asymmetric pressure distribution, which is weakened by the variation of the overlapping surface area. The mass flow transfer maintains the pressure equilibrium between two discharge chambers through the overlapping surface. Compared with the circular discharge structure, the waist-shape one not only reduces the pressure asymmetry degree but shortens its duration by six degrees of the orbiting angle



---

*Source: 9382690-2023-02-06.xml*