modeling of top scroll profile using equidistant-curve approach for

Research ArticleModeling of Top Scroll Profile Using Equidistant-CurveApproach for a Scroll Compressor

Tao Liu and Zaixin Wu

School ofMechanical and Electronical Engineering LanzhouUniversity of Technology 287 Langongping Road Lanzhou 730050 China

Correspondence should be addressed to Tao Liu cathyliu1999126com

Received 6 February 2015 Revised 22 May 2015 Accepted 31 May 2015

Academic Editor Filippo Ubertini

Copyright copy 2015 T Liu and Z Wu This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Scroll profile plays a key role in determining the performance of a scroll compressor In this study geometric and dynamiccharacteristics of the scroll profile are analyzed to investigate the influence of its geometric continuity on property of a scrollcompressor Firstly scroll profiles are created to redesign the geometry of scroll wrap by using the equidistant-curve approachon the basis of a generation line consisting of involute of circle and circular arc Subsequently the geometric and dynamic modelsof the scroll compressor are established These models are related to parameters of the generation line of scroll profile and rotationangle of a moving scroll Lastly some simulation examples of second-order continuity (SOC) scroll profile are compared with first-order continuity (FOC) scroll profiles and some important conclusions are obtained Results show that SOC scroll profile is superiorto FOC profile in terms of volume ratio stability of gas force and possible leakage loss in a scroll compressor

1 Introduction

Scroll compressors are widely used for compressing airor refrigerants in refrigerators and air conditioners Scrollcompressors have the advantage of higher efficiency andlower noise and vibration levels and they are more reliablebecause of the smaller number of moving parts required fortheir operation comparedwith other types of compressors [1]

One problem when developing a scroll compressor is thedesign of scroll profile which is an important factor affectingthe performance of the compressor Many researchers havefocused on this subject and done a lot of work [2ndash7] based oneither dynamic analysis or optimum design considerationsLiu et al [8] presented a graphic and mathematical methodfor the modification of the start part of scroll profile After-wards Lee and Wu [9] gave an analytical method to designscroll profile with first-order continuity (FOC) These fore-going researches had focused on interpolating scroll profileby using circular arcs with geometry condition to zero orderor FOC for a scroll compressor Notably Bush and Beagle [10]put forward a general relationship which governs conjugacyof scroll profiles and proved that the general conditions forconjugacy could assure first- and second-order continuity(SOC)More recently Gravesen andHenriksen [11] redefined

the scroll geometry by radius of curvature as a function oftangent direction In line with the previous work Blunier etal [12] proposed a new way of describing the geometry of thescroll wrap by using an orthogonal reference frame Howeverthere is little information available in literature about SOCscroll profiles for scroll fluid machines Furthermore theapproach to generate two meshed profiles with inherent con-jugate relationship has not been given publicly Neverthelessfurther research may still be needed

This paper presents a new design of SOC top scrollprofile with circular arcs curves based on the curvatureradius function proposed by Gravesen and Henriksen [11]The generation of scroll wrap is illustrated in detail Theestimations of geometric and dynamic properties are analyt-ically described and implemented by computer simulationFor comparison purposes some simulation examples for SOCand FOC profiles are presented The results suggest manyadvantages of SOC scroll profiles over FOC ones

2 Generation Line of Scroll Profiles

The typical profile of scroll compressors is formed by aninvolute of circle Involute designs have several advantagessimple formulation uniform wall thickness and small force

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 403249 8 pageshttpdxdoiorg1011552015403249

2 Mathematical Problems in Engineering

vibration However there are some limitations Only profileoutside of base circle of the involute can be generated whileusing this kind of curve The top profile within base circle isusually formed by interference between a cutter and the scrollcomponent when it is machined However the thus formedtop profile of two scrolls cannot engage each other whichresults in a smaller volume ratio and weaker top scroll wrapof a scroll compressor In order to overcome these defects wereconstruct the starting part of the involute with a circular arcgoing through ordinate origin to form an engaged top profileAs a result a complete scroll profile including the portionwithin base circle is obtained and defined as the generationline of scroll profiles as shown in Figure 1 In the generationline the circular arc and the involute of circle are jointed atpoint119875The starting tangential angle of circular arc is definedas 1205931 and the ending tangential angle of circular arc is definedas 1205932 and the radius of circular arc is represented as 119877

119904 In

order to assure SOC geometric boundary condition of scrollprofiles the following requirements must be met

(1) Tangential angle 1205932 should be equal to the tangentialangle of involute of circle at point 119875

(2) The radius of curvature of point 119875 on circular curveshould be the same as that on involute of circle

If the above requirement (2) cannot be satisfied thenthe geometric boundary condition for scroll profile can onlymaintain FOC

In a scroll compressor the two identical principal com-ponents that is the fixed and moving scrolls are positionedso that they share a common center any arbitrary pair ofconjugation points is separated by a distance 119877or the rotationradius directed normally to the respective surfacesThus thegeneration line and its normal equidistant lines can be appliedas scroll profiles that have conjugate relation as illustrated inFigure 2

The mathematical model for the generation line of scrollprofile is given as

119909 = int

120593

1205931

120588 (120593) cos120593 d120593

119910 = int

120593

1205931

120588 (120593) sin120593 d120593

(120593 isin [1205931 120593max])

(1)

where radius of curvature of involute

120588 (120593) =

119877119904

(120593 isin [1205931 1205932])

119886120593 (120593 isin [1205932 120593max]) (2)

The equation for outer profile of moving scroll can berepresented as follows

119909mo = int

120593

1205931

[120588 (120593) minus119877or2] cos120593 d120593

119910mo = int

120593

1205931

[120588 (120593) minus119877or2] sin120593 d120593

(120593 isin [1205931 120593max])

(3)

P

Y

XO

Base circle

Circular arc

Involute of circle

2a

1205932

120593 1

2Rs

Figure 1 Generation line of scroll profile

O X

Y

Moving scroll

Fixed scrollGeneration line

Ror

2 Ror

2

Figure 2 Schematic of scroll profile

The equation for inner profile of fixed scroll can bewrittenas

119909fi = int

120593

1205931

[120588 (120593) +119877or2] cos120593 d120593

119910fi = int

120593

1205931

[120588 (120593) +119877or2] sin120593 d120593

(120593 isin [1205931 120593max])

(4)

The equation for inner profile of moving scroll can befollowed as

119909mi = minusint

120593

1205931

[120588 (120593) +119877or2] cos120593 d120593

119910mi = minusint

120593

1205931

[120588 (120593) +119877or2] sin120593 d120593

(120593 isin [1205931 120593max])

(5)

The equation for outer profile of fixed scroll can bewrittenas

119909fo = minusint

120593

1205931

[120588 (120593) minus119877or2] cos120593 d120593

119910fo = minusint

120593

1205931

[120588 (120593) minus119877or2] sin120593 d120593

(120593 isin [1205931 120593max])

(6)

Mathematical Problems in Engineering 3

3 Geometrical Description ofthe Chamber Areas

The intermeshed scroll wraps form a series of sealed cham-bers namely suction chamber compression chamber anddischarge chamber as seen in Figure 3 The volumes ofthese chambers are of critical importance in determining thegeometrical property of a scroll compressor

31 Suction Chamber Figure 4(a) shows a pair of conjugationsurfaces of a suction chamber which are positioned so theyshare a common center In Figure 4(b) the moving scrollrotates around the fixed scroll at a radius of 119877or so thata suction chamber is formed at two extreme locations onthe periphery Assume the maximum tangential angle of thescroll profile is 120593max and the rotation angle of moving scrollis 120579 Therefore the suction area is given by the parameters ofthe generation line

1198601 (120579) = 119877or int120593maxminus120579

120593maxminus120579minus2120587119886120593 d120593 (7)

32 Compression Chamber The area calculation of a com-pression chamber is divided into two different stages

During the first stage the compression chamber isenclosed solely by involute of circle and the area of compres-sion chamber can be written as follows

1198602 (120579) = 119877or int120593maxminus120579minus2120587

120593maxminus120579minus4120587119886120593 d120593 (0 le 120579 le

31205872

minus 1205932) (8)

In the second stage the compression chamber is boundedby both involutes of circle and circular arcs as shown inFigure 5 The area of compression chamber is given by

1198602 (120579) = 119877or (int1205932

120593maxminus120579minus2120587119877119904d120593+int

120593maxminus120579

1205932

119886120593 d120593)

+Δ1198602 (120579) (31205872

minus 1205932 le 120579 le31205872

minus 1205931)

(9)

where

Δ1198602 (120579) = 119877or [119877119904 sin (120593max minus 120579minus 2120587minus1205931) minus 119886]

(31205872

minus 1205932 le 120579 le31205872

minus 1205931) (10)

When a compression chamber just opens into the dis-charge zone at the center a discharge angle of 120579lowast is definedas follows

120579lowast

=31205872

minus1205931 (11)

The volume ratio of a scroll compressor can be formulatedas

V =1198601 (0)1198602 (120579lowast) (12)

Compression chamber

Suction chamber

Discharge chamber

Fixed scroll

Moving scroll

Figure 3 Schematic view of sealed chambers in a scroll compressor

33 Discharge Chamber The determination of a dischargearea is divided into three stages in accordance with the rota-tion angle of the moving scroll

In the first stage the discharge chamber bounded by bothinvolutes and circular curves is shown in Figure 6 The areacomputation of discharge chamber follows

1198603 (120579) = 119877or (int31205872minus120579

1205932

119886120593 d120593+int1205932

1205931

119877119904d120593minus 119886)

(0 le 120579 le31205872

minus 1205932)

(13)

During the second stage the discharge pocket is solelyformed by circular curves as can be seen in Figure 7The areaof discharge chamber is computed as follows

1198603 (120579) = 119877or119877119904 [(120579lowast

minus 120579) minus sin (120579lowast minus 120579)]

(31205872

minus 1205932 le 120579 le 120579lowast

)

(14)

At the beginning of the last stage the largest dischargechamber formed when the rotation angle reaches dischargeangle 120579lowast Consequently the area function of discharge cham-ber since then could be expressed by

1198603 (120579) = 119877or (int(72)120587minus120579

1205932

119886120593 d120593+int1205932

1205931

119877119904d120593minus 119886)

(120579lowast

le 120579 le 2120587)

(15)

4 Dynamical Model of the Gas Forces

The gas forces developed within scroll chambers are wellknown to be axial tangential and radial [1 13] Each forcevaries in magnitude as the moving scroll rotates through onerevolution

41 Axial Force The axial gas load is produced by thecollection of gas pressures acting in the respective sealedchamber which can be calculated by

119865119886(120579) =

119873

sum

119894=1

119901119894119860119894(120579) (0 le 120579 le 2120587) (16)


1

2

Ror

(a)

1

2

A1

(b)

Figure 4 Schematic of a suction chamber

ΔA2

Ror

120579

120579lowast

(a)

2a

A2

Rs

(b)

Figure 5 Schematic of a compression chamber

2a

Involute of circle

Circular arc

Rs

1205791205932

1205931

Figure 6 Discharge chamber bounded by two kinds of curves

where

119901119894= 119901119904[1198601(0)

119860119894(120579)

]

119896

(0 le 120579 le 2120587) (17)

where 119901119904indicates the suction pressure of scroll compressor

and 119896 represents adiabatic index

A3

2Rs

Ror

Ror2

120579lowast minus120579

Figure 7 Discharge chamber bounded by circular curves

42 Tangential Force The tangential force acts normally tothe lines of flank contact points on both scrolls There arethree pairs of contacting points 1-11015840 2-21015840 and 3-31015840 in the scrollcompressor and the tangential distance between each pair ofthe points is defined as tangential action line which is 119871

1199051

1198711199052 and 119871

1199053 respectively as illustrated in Figure 8


1

2

3

3998400

2998400

1998400

Lr3

Lt3

Lr2

Lt2

Lr1

Lt1

Figure 8 Schematic of action lines of gas force

The tangential line 1198711199053between the innermost points 3

and 31015840 changes in accordance with the rotation angle and canbe written as

1198711199053

=

2119886 (31205872

minus 120579) (0 le 120579 le31205872

minus 1205932)

2119877119904[1 minus cos (120579lowast minus 120579)] (

31205872


)

2119886 (71205872

minus 120579) (120579lowast

le 120579 le 2120587)

(18)

The tangential line between other contact points can beexpressed as

119871119905119894= 2119886 [3120587

2sdot 2120587 (119873minus 119894) minus 120579]

(0 le 120579 le 2120587 119894 = 1 2) (19)

Therefore the tangential gas force can be obtained by

119865119905(120579) = ℎ

119873

sum

119894=1

119871119905119894(119901119894minus119901119894+1) (0 le 120579 le 2120587) (20)

43 Radial Force The radial gas force acts parallel to the linesof flank contact points on both scrolls Radial action line119871119903119894is the radial distance between the flank contact points

as indicated in Figure 8 The tangential line 1198711199033

betweencontact points 3 and 31015840 in discharge chamber changeswith therotation angle through one revolution The length of radialaction line 119871

1199033can be given by

1198711199033=

2119886 (0 le 120579 le31205872

minus 1205932)

2119877119904sin (120579lowast minus 120579) (

31205872


)

2119886 (120579lowast

le 120579 le 2120587)

(21)

On the contrary the radial action lines between contactpoints 2-21015840 and 1-11015840 are constants and can be expressed as

119871119903119894= 2119886 (0 le 120579 le 2120587 119894 = 1 2) (22)

So the radial gas force can be obtained by

119865119903(120579) = ℎ

119873

sum

119894=1

119871119903119894(119901119894minus119901119894+1) (0 le 120579 le 2120587 119894 = 1 2) (23)

5 Results and Discussions

To investigate the influence of scroll profile on performanceof a scroll compressor we constructed SOC and FOCmodelsof scroll profile for comparison Due to the constraint ofthickness of top scroll wrap FOC models with maximumending tangential angle 1205932 le 270∘ are considered inparticular Numerical examples are performed and someresults regarding geometric and dynamic characteristics ofthe scroll profiles will be shown herein In carrying outthe computer simulation the following basic parameters aregiven based on operating condition of a 375 kW refrigerationscroll compressor 119886 = 2069mm 119877or = 4mm 119873 = 3ℎ = 40mm 119896 = 14 119901

119904= 05MPa and 120593max = 55120587

(1) Top profile has a very significant effect on the shape ofscroll wrap and volume ration As indicated in Figure 9 thecentral portion of the scroll wrap is thicker in FOC designsthan that in SOC design A thicker scroll wrap in general willincrease the resistance to stress and deformation developedon the tipwrapHowever thicker topwrap results in a smallervolume ratio as listed in Table 1

(2) The axial gas force acting on moving scroll variesgreatly with different top profile designs As shown inFigure 10 the largest value of axial force occurs when rotationangle takes value of 0∘ and at thatmoment the suction processis just finished Compared to FOC designs the axial force forSOC is increased at least by 1054 as indicated in Table 2Greater vibrations in axial force curves result from the FOCscroll wrap Special attention should be paid to bearing designof scroll compressors

(3) Tangential gas forces exerted on moving scroll withFOC and SOC profiles are compared in Figure 11 and Table 2Like axial force the tangential force in SOC design is largerin FOC designs The largest value appears when dischargeprocess begins The maximum tangential force in SOC is atleast 1795 larger than that in FOCTiny vibration is found intangential force curve in FOC

4design while this occurrence

that takes place in other FOC designs is imperceptibleThe discontinuous slope coefficient of tangential action linethat occurred in FOC designs is believed to cause thisphenomenon In contrast no vibrations occur in tangentialforce curve in SOC design because of the continuous changeof radius of curvature along the whole scroll profile

(4) Radial forces in SOC and FOC designs are comparedin Figure 12 and Table 2The radial gas force observed in SOCscroll has smaller variation than that in the FOC ones anddoes not reverse When rotation angle 31205872 le 120579 le 120579

lowast thatis the two scrolls mesh along scroll wraps with circular arcs


Table 1 Key geometric parameters for SOC and FOC designs

1205932(∘

) 1205931(∘

) 119877s(mm) 120588(1205932)(mm) 120579lowast

(∘

) 1198602(120579lowast

)(mm2) V ΔV ()

SOC 57 minus33 2069 2069 303 1354 5340 mdashFOC1 90 minus25 2284 3249 295 1413 5202 minus258FOC2 135 1 2877 4875 269 1599 4596 minus139FOC3 180 35 3579 6499 235 1838 3687 minus309FOC4 270 114 5094 9749 156 2385 2977 minus443

Table 2 Comparison of dynamic properties for SOC and FOC designs

119865119886max(kN) 119865

119886(120579lowast

)(kN) 119865119905(120579lowast

)(kN) 119865119903max(N) 119865

119903min(N) 119865119903st(N)

SOC 2653 2333 5054 8011 7056 8011FOC1 2588 2303 4845 7909 6691 7501FOC2 2406 2217 4285 7429 5649 6174FOC3 2215 2123 3723 6801 4602 4936FOC4 1893 1954 2807 5607 3018 3173

SOCFOC1

FOC2

FOC3

Figure 9 Wrap comparison of SOC and FOC profiles

the radial force is much smaller than other rotation ranges inSOC design Comparatively in FOC designs the radial forcedeveloped in the sealed chamber changes periodically

The SOCdesign yieldedmuch smaller radial gas variationin the simulated computation by at least 8 as compared tothe FOC ones considered The SOC is favorable with radialgas load fluctuation rate ranging from 0 to minus135 while amaximum of corresponding fluctuation rate varying from767 to minus488 is found when the FOC

4design is adopted

This finding helps minimizing the strength and backlashrequirements of the antirotation coupling which ensuresthe moving scroll rotating in a plane motion as well asreducing potential wear of mechanical components andother effects of load reversal in the mechanism of a SOCdesign Moreover with smaller radial gas force variationa smaller radial clearance between the moving and fixedscrolls can be achieved which leads to an optimal design ofscroll compressor in terms of leakage loss bearing life andreliability

6 Conclusion

This paper proposed a way to redesign the geometry ofscroll wrap in which the central and main part of the profile

0 40 80 120 160 200 240 280 320 360

1200

1400

1600

1800

2000

2200

2400

2600

2800

Rotation angle 120579 (∘)

Axi

al g

as fo

rceF

a(N

)

SOCFOC2

FOC3

FOC4

Figure 10 Axial gas force

can be expressed in a general form This work is essentialfor the profile design of scroll components to ensure goodperformance of scroll compressors Area estimations aregiven for the sealed chambers formed duringworking processof a scroll compressor Afterwards the gas force acting onthe moving scroll is calculated The geometric and dynamicmodels have been validated by numerical simulation of ascroll compressor For comparison purposes some simulationexamples for SOC and FOC profile designs are presented Asa result of the above derivation and simulation the followingconclusions can be drawn with regard to the proposed scrollprofile designs

(1) This study intends to identify the geometrical anddynamical characteristics of SOC and FOC scroll profiles


0 40 80 120 160 200 240

2

25

3

35

4

45

Small vibration


Tang

entia

l gas

forc

eFt

(kN

)

SOCFOC2

FOC3

FOC4

Figure 11 Tangential gas force

300

400

500

600

700

800

900

0 40 80 120 160 200 240 280 320 360Rotation angle 120579 (∘)

SOCFOC2

FOC3

FOC4

Redi

al g

as fo

rceFr

(N)

Figure 12 Radial gas force

and offers a way to achieve enabling high quality design ofthe scroll compressors covering top geometry volume ratioand gas forces aspects

(2) It is found that there are no vibration in tangential gasforce and no load reversal in radial gas force when the SOCscroll profile is applied which not only helps minimizing thestrength and backlash requirements of mechanical compo-nents but also is favorable in terms of leakage loss bearinglife and reliability in the design and performance of a scrollcompressor

Nomenclature

120593 Tangential angle1205931 Starting tangential angle of circular arc1205932 Ending tangential angle of circular arc120593max Maximum tangential angle of scroll profile119877119904 Radius of circular arc

119886 Radius of the base circle of involute119877or Rotation radius120588 Radius of curvature of scroll profile120579 Rotation angle of moving scroll1198601 Area of suction chamber1198602 Area of compression chamber1198603 Area of discharge chamber120579lowast Discharge angleV Volume ratioΔV Increment volume ratio119865119886 Axial gas load

119901119904 Suction pressure

120581 Adiabatic index119873 Number of pairs of chambers119865119905 Tangential gas load

119871119905119894 Length of tangential action line

ℎ Height of scroll wrap119865119903 Radial gas load

119871119903119894 Length of radial action line

119865119903max Maximum radial gas load119865119903min Minimum radial gas load119865119903st Stable radial gas load

Subscripts

119886 Axial119905 Tangential119903 Radial119904 Suctionst StableOr Orbitingmax Maximummin Minimum119894 Inner119900 Outer119898 Moving119891 Fixed

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors thank Professor Liu Zhenquan for valuablediscussions This study is supported by National NaturalScience Foundation of China (Grant no 51265027) and TheFundamental Research Funds for the Universities in GansuProvince (Grant no 1302ZTC034)


References

[1] E Morishita and M Sugihara ldquoSome design problems of scrollcompressorrdquo Bulletin of the JSME vol 29 no 258 pp 4139ndash4146 1986

[2] Q Jianguo and T Wen ldquoNumerical modelling of temperaturedistribution for orbiting scroll wrap in an air scroll compressorrdquoInternational Journal of Heat andMass Transfer vol 67 pp 678ndash689 2013

[3] M M Cui ldquoNumerical study of unsteady flows in a scrollcompressorrdquo ASME Journal of Fluids Engineering vol 128 no5 pp 947ndash955 2006

[4] Z Jiang D K Harrison and K Cheng ldquoComputer-aideddesign and manufacturing of scroll compressorsrdquo Journal ofMaterials Processing Technology vol 138 no 1ndash3 pp 145ndash1512003

[5] E Winandy C S Saavedra O and J Lebrun ldquoExperimentalanalysis and simplified modelling of a hermetic scroll refriger-ation compressorrdquo Applied Thermal Engineering vol 22 no 2pp 107ndash120 2002

[6] D A Cha O K Kwon and M D Oh ldquoAn experimentalstudy on semiconductor process chiller using the digital scrollcompressorrdquo Journal of Mechanical Science and Technology vol28 no 8 pp 3345ndash3352 2014

[7] Y Liu C Hung and Y Chang ldquoStudy on involute of circle withvariable radii in a scroll compressorrdquo Mechanism and MachineTheory vol 45 no 11 pp 1520ndash1536 2010

[8] Z Liu G Du Z Qi and J Gu ldquoThe conjugacy analysis ofmodified part of scroll profilesrdquo in Proceedings of the Interna-tional Compressor Engineering Conference at Purdue pp 479ndash484 West Lafayette Ind USA 1994

[9] Y-R Lee and W-F Wu ldquoOn the profile design of a scrollcompressorrdquo International Journal of Refrigeration vol 18 no5 pp 308ndash317 1995

[10] J W Bush and W P Beagle ldquoDerivation of a general relationgoverning the conjugacy of scroll profilesrdquo in Proceedings of theInternational Compressor Engineering Conference at Purdue pp1079ndash1087 1992

[11] J Gravesen and C Henriksen ldquoThe geometry of the scrollcompressorrdquo SIAM Review vol 43 no 1 pp 113ndash126 2001

[12] B Blunier G Cirrincione Y Herve and A Miraoui ldquoA newanalytical and dynamical model of a scroll compressor withexperimental validationrdquo International Journal of Refrigerationvol 32 no 5 pp 874ndash891 2009

[13] Y Chen N P Halm E A Groll and J E Braun ldquoMathematicalmodeling of scroll compressorsmdashpart I compression processmodelingrdquo International Journal of Refrigeration vol 25 no 6pp 731ndash750 2002

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vibration However there are some limitations Only profileoutside of base circle of the involute can be generated whileusing this kind of curve The top profile within base circle isusually formed by interference between a cutter and the scrollcomponent when it is machined However the thus formedtop profile of two scrolls cannot engage each other whichresults in a smaller volume ratio and weaker top scroll wrapof a scroll compressor In order to overcome these defects wereconstruct the starting part of the involute with a circular arcgoing through ordinate origin to form an engaged top profileAs a result a complete scroll profile including the portionwithin base circle is obtained and defined as the generationline of scroll profiles as shown in Figure 1 In the generationline the circular arc and the involute of circle are jointed atpoint119875The starting tangential angle of circular arc is definedas 1205931 and the ending tangential angle of circular arc is definedas 1205932 and the radius of circular arc is represented as 119877

119904 In

order to assure SOC geometric boundary condition of scrollprofiles the following requirements must be met

(1) Tangential angle 1205932 should be equal to the tangentialangle of involute of circle at point 119875

(2) The radius of curvature of point 119875 on circular curveshould be the same as that on involute of circle

If the above requirement (2) cannot be satisfied thenthe geometric boundary condition for scroll profile can onlymaintain FOC

In a scroll compressor the two identical principal com-ponents that is the fixed and moving scrolls are positionedso that they share a common center any arbitrary pair ofconjugation points is separated by a distance 119877or the rotationradius directed normally to the respective surfacesThus thegeneration line and its normal equidistant lines can be appliedas scroll profiles that have conjugate relation as illustrated inFigure 2

The mathematical model for the generation line of scrollprofile is given as

119909 = int

120593

1205931

120588 (120593) cos120593 d120593

119910 = int

120593

1205931

120588 (120593) sin120593 d120593

(120593 isin [1205931 120593max])

(1)

where radius of curvature of involute

120588 (120593) =

119877119904

(120593 isin [1205931 1205932])

119886120593 (120593 isin [1205932 120593max]) (2)

The equation for outer profile of moving scroll can berepresented as follows

119909mo = int

120593

1205931

[120588 (120593) minus119877or2] cos120593 d120593

119910mo = int

120593

1205931

[120588 (120593) minus119877or2] sin120593 d120593

(120593 isin [1205931 120593max])

(3)

P

Y

XO

Base circle

Circular arc

Involute of circle

2a

1205932

120593 1

2Rs

Figure 1 Generation line of scroll profile

O X

Y

Moving scroll

Fixed scrollGeneration line

Ror

2 Ror

2

Figure 2 Schematic of scroll profile

The equation for inner profile of fixed scroll can bewrittenas

119909fi = int

120593

1205931

[120588 (120593) +119877or2] cos120593 d120593

119910fi = int

120593

1205931

[120588 (120593) +119877or2] sin120593 d120593

(120593 isin [1205931 120593max])

(4)

The equation for inner profile of moving scroll can befollowed as

119909mi = minusint

120593

1205931

[120588 (120593) +119877or2] cos120593 d120593

119910mi = minusint

120593

1205931

[120588 (120593) +119877or2] sin120593 d120593

(120593 isin [1205931 120593max])

(5)

The equation for outer profile of fixed scroll can bewrittenas

119909fo = minusint

120593

1205931

[120588 (120593) minus119877or2] cos120593 d120593

119910fo = minusint

120593

1205931

[120588 (120593) minus119877or2] sin120593 d120593

(120593 isin [1205931 120593max])

(6)





1198601 (120579) = 119877or int120593maxminus120579

120593maxminus120579minus2120587119886120593 d120593 (7)





31205872

minus 1205932) (8)


1198602 (120579) = 119877or (int1205932



1205932

119886120593 d120593)

+Δ1198602 (120579) (31205872

minus 1205932 le 120579 le31205872

minus 1205931)

(9)

where


(31205872

minus 1205932 le 120579 le31205872

minus 1205931) (10)


120579lowast

=31205872

minus1205931 (11)


V =1198601 (0)1198602 (120579lowast) (12)

Compression chamber

Suction chamber

Discharge chamber

Fixed scroll

Moving scroll




1198603 (120579) = 119877or (int31205872minus120579

1205932

119886120593 d120593+int1205932

1205931

119877119904d120593minus 119886)

(0 le 120579 le31205872

minus 1205932)

(13)


1198603 (120579) = 119877or119877119904 [(120579lowast


(31205872


)

(14)


1198603 (120579) = 119877or (int(72)120587minus120579

1205932

119886120593 d120593+int1205932

1205931

119877119904d120593minus 119886)

(120579lowast

le 120579 le 2120587)

(15)




119865119886(120579) =

119873

sum

119894=1

119901119894119860119894(120579) (0 le 120579 le 2120587) (16)


1

2

Ror

(a)

1

2

A1

(b)


ΔA2

Ror

120579

120579lowast

(a)

2a

A2

Rs

(b)


2a

Involute of circle

Circular arc

Rs

1205791205932

1205931


where

119901119894= 119901119904[1198601(0)

119860119894(120579)

]

119896

(0 le 120579 le 2120587) (17)



A3

2Rs

Ror

Ror2




1199051

1198711199052 and 119871



1

2

3

3998400

2998400

1998400

Lr3

Lt3

Lr2

Lt2

Lr1

Lt1




1198711199053

=

2119886 (31205872

minus 120579) (0 le 120579 le31205872

minus 1205932)


31205872


)

2119886 (71205872


le 120579 le 2120587)

(18)


119871119905119894= 2119886 [3120587

2sdot 2120587 (119873minus 119894) minus 120579]

(0 le 120579 le 2120587 119894 = 1 2) (19)


119865119905(120579) = ℎ

119873

sum

119894=1

119871119905119894(119901119894minus119901119894+1) (0 le 120579 le 2120587) (20)





1198711199033=

2119886 (0 le 120579 le31205872

minus 1205932)

2119877119904sin (120579lowast minus 120579) (

31205872


)

2119886 (120579lowast

le 120579 le 2120587)

(21)


119871119903119894= 2119886 (0 le 120579 le 2120587 119894 = 1 2) (22)


119865119903(120579) = ℎ

119873

sum

119894=1

119871119903119894(119901119894minus119901119894+1) (0 le 120579 le 2120587 119894 = 1 2) (23)



119904= 05MPa and 120593max = 55120587










1205932(∘

) 1205931(∘

) 119877s(mm) 120588(1205932)(mm) 120579lowast

(∘

) 1198602(120579lowast

)(mm2) V ΔV ()



119865119886max(kN) 119865

119886(120579lowast

)(kN) 119865119905(120579lowast

)(kN) 119865119903max(N) 119865

119903min(N) 119865119903st(N)


SOCFOC1

FOC2

FOC3




4design is adopted


6 Conclusion


0 40 80 120 160 200 240 280 320 360

1200

1400

1600

1800

2000

2200

2400

2600

2800


Axi

al g

as fo

rceF

a(N

)

SOCFOC2

FOC3

FOC4





0 40 80 120 160 200 240

2

25

3

35

4

45

Small vibration


Tang

entia

l gas

forc

eFt

(kN

)

SOCFOC2

FOC3

FOC4


300

400

500

600

700

800

900

0 40 80 120 160 200 240 280 320 360Rotation angle 120579 (∘)

SOCFOC2

FOC3

FOC4

Redi

al g

as fo

rceFr

(N)




Nomenclature









Subscripts




Acknowledgments



References





















Volume 2014




Journal of











Journal of


Function Spaces






Algebra













1198601 (120579) = 119877or int120593maxminus120579

120593maxminus120579minus2120587119886120593 d120593 (7)





31205872

minus 1205932) (8)


1198602 (120579) = 119877or (int1205932



1205932

119886120593 d120593)

+Δ1198602 (120579) (31205872

minus 1205932 le 120579 le31205872

minus 1205931)

(9)

where


(31205872

minus 1205932 le 120579 le31205872

minus 1205931) (10)


120579lowast

=31205872

minus1205931 (11)


V =1198601 (0)1198602 (120579lowast) (12)

Compression chamber

Suction chamber

Discharge chamber

Fixed scroll

Moving scroll




1198603 (120579) = 119877or (int31205872minus120579

1205932

119886120593 d120593+int1205932

1205931

119877119904d120593minus 119886)

(0 le 120579 le31205872

minus 1205932)

(13)


1198603 (120579) = 119877or119877119904 [(120579lowast


(31205872


)

(14)


1198603 (120579) = 119877or (int(72)120587minus120579

1205932

119886120593 d120593+int1205932

1205931

119877119904d120593minus 119886)

(120579lowast

le 120579 le 2120587)

(15)




119865119886(120579) =

119873

sum

119894=1

119901119894119860119894(120579) (0 le 120579 le 2120587) (16)


1

2

Ror

(a)

1

2

A1

(b)


ΔA2

Ror

120579

120579lowast

(a)

2a

A2

Rs

(b)


2a

Involute of circle

Circular arc

Rs

1205791205932

1205931


where

119901119894= 119901119904[1198601(0)

119860119894(120579)

]

119896

(0 le 120579 le 2120587) (17)



A3

2Rs

Ror

Ror2




1199051

1198711199052 and 119871



1

2

3

3998400

2998400

1998400

Lr3

Lt3

Lr2

Lt2

Lr1

Lt1




1198711199053

=

2119886 (31205872

minus 120579) (0 le 120579 le31205872

minus 1205932)


31205872


)

2119886 (71205872


le 120579 le 2120587)

(18)


119871119905119894= 2119886 [3120587

2sdot 2120587 (119873minus 119894) minus 120579]

(0 le 120579 le 2120587 119894 = 1 2) (19)


119865119905(120579) = ℎ

119873

sum

119894=1

119871119905119894(119901119894minus119901119894+1) (0 le 120579 le 2120587) (20)





1198711199033=

2119886 (0 le 120579 le31205872

minus 1205932)

2119877119904sin (120579lowast minus 120579) (

31205872


)

2119886 (120579lowast

le 120579 le 2120587)

(21)


119871119903119894= 2119886 (0 le 120579 le 2120587 119894 = 1 2) (22)


119865119903(120579) = ℎ

119873

sum

119894=1

119871119903119894(119901119894minus119901119894+1) (0 le 120579 le 2120587 119894 = 1 2) (23)



119904= 05MPa and 120593max = 55120587










1205932(∘

) 1205931(∘

) 119877s(mm) 120588(1205932)(mm) 120579lowast

(∘

) 1198602(120579lowast

)(mm2) V ΔV ()



119865119886max(kN) 119865

119886(120579lowast

)(kN) 119865119905(120579lowast

)(kN) 119865119903max(N) 119865

119903min(N) 119865119903st(N)


SOCFOC1

FOC2

FOC3




4design is adopted


6 Conclusion


0 40 80 120 160 200 240 280 320 360

1200

1400

1600

1800

2000

2200

2400

2600

2800


Axi

al g

as fo

rceF

a(N

)

SOCFOC2

FOC3

FOC4





0 40 80 120 160 200 240

2

25

3

35

4

45

Small vibration


Tang

entia

l gas

forc

eFt

(kN

)

SOCFOC2

FOC3

FOC4


300

400

500

600

700

800

900

0 40 80 120 160 200 240 280 320 360Rotation angle 120579 (∘)

SOCFOC2

FOC3

FOC4

Redi

al g

as fo

rceFr

(N)




Nomenclature









Subscripts




Acknowledgments



References





















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1

2

Ror

(a)

1

2

A1

(b)


ΔA2

Ror

120579

120579lowast

(a)

2a

A2

Rs

(b)


2a

Involute of circle

Circular arc

Rs

1205791205932

1205931


where

119901119894= 119901119904[1198601(0)

119860119894(120579)

]

119896

(0 le 120579 le 2120587) (17)



A3

2Rs

Ror

Ror2




1199051

1198711199052 and 119871



1

2

3

3998400

2998400

1998400

Lr3

Lt3

Lr2

Lt2

Lr1

Lt1




1198711199053

=

2119886 (31205872

minus 120579) (0 le 120579 le31205872

minus 1205932)


31205872


)

2119886 (71205872


le 120579 le 2120587)

(18)


119871119905119894= 2119886 [3120587

2sdot 2120587 (119873minus 119894) minus 120579]

(0 le 120579 le 2120587 119894 = 1 2) (19)


119865119905(120579) = ℎ

119873

sum

119894=1

119871119905119894(119901119894minus119901119894+1) (0 le 120579 le 2120587) (20)





1198711199033=

2119886 (0 le 120579 le31205872

minus 1205932)

2119877119904sin (120579lowast minus 120579) (

31205872


)

2119886 (120579lowast

le 120579 le 2120587)

(21)


119871119903119894= 2119886 (0 le 120579 le 2120587 119894 = 1 2) (22)


119865119903(120579) = ℎ

119873

sum

119894=1

119871119903119894(119901119894minus119901119894+1) (0 le 120579 le 2120587 119894 = 1 2) (23)



119904= 05MPa and 120593max = 55120587










1205932(∘

) 1205931(∘

) 119877s(mm) 120588(1205932)(mm) 120579lowast

(∘

) 1198602(120579lowast

)(mm2) V ΔV ()



119865119886max(kN) 119865

119886(120579lowast

)(kN) 119865119905(120579lowast

)(kN) 119865119903max(N) 119865

119903min(N) 119865119903st(N)


SOCFOC1

FOC2

FOC3




4design is adopted


6 Conclusion


0 40 80 120 160 200 240 280 320 360

1200

1400

1600

1800

2000

2200

2400

2600

2800


Axi

al g

as fo

rceF

a(N

)

SOCFOC2

FOC3

FOC4





0 40 80 120 160 200 240

2

25

3

35

4

45

Small vibration


Tang

entia

l gas

forc

eFt

(kN

)

SOCFOC2

FOC3

FOC4


300

400

500

600

700

800

900

0 40 80 120 160 200 240 280 320 360Rotation angle 120579 (∘)

SOCFOC2

FOC3

FOC4

Redi

al g

as fo

rceFr

(N)




Nomenclature









Subscripts




Acknowledgments



References





















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1

2

3

3998400

2998400

1998400

Lr3

Lt3

Lr2

Lt2

Lr1

Lt1




1198711199053

=

2119886 (31205872

minus 120579) (0 le 120579 le31205872

minus 1205932)


31205872


)

2119886 (71205872


le 120579 le 2120587)

(18)


119871119905119894= 2119886 [3120587

2sdot 2120587 (119873minus 119894) minus 120579]

(0 le 120579 le 2120587 119894 = 1 2) (19)


119865119905(120579) = ℎ

119873

sum

119894=1

119871119905119894(119901119894minus119901119894+1) (0 le 120579 le 2120587) (20)





1198711199033=

2119886 (0 le 120579 le31205872

minus 1205932)

2119877119904sin (120579lowast minus 120579) (

31205872


)

2119886 (120579lowast

le 120579 le 2120587)

(21)


119871119903119894= 2119886 (0 le 120579 le 2120587 119894 = 1 2) (22)


119865119903(120579) = ℎ

119873

sum

119894=1

119871119903119894(119901119894minus119901119894+1) (0 le 120579 le 2120587 119894 = 1 2) (23)



119904= 05MPa and 120593max = 55120587










1205932(∘

) 1205931(∘

) 119877s(mm) 120588(1205932)(mm) 120579lowast

(∘

) 1198602(120579lowast

)(mm2) V ΔV ()



119865119886max(kN) 119865

119886(120579lowast

)(kN) 119865119905(120579lowast

)(kN) 119865119903max(N) 119865

119903min(N) 119865119903st(N)


SOCFOC1

FOC2

FOC3




4design is adopted


6 Conclusion


0 40 80 120 160 200 240 280 320 360

1200

1400

1600

1800

2000

2200

2400

2600

2800


Axi

al g

as fo

rceF

a(N

)

SOCFOC2

FOC3

FOC4





0 40 80 120 160 200 240

2

25

3

35

4

45

Small vibration


Tang

entia

l gas

forc

eFt

(kN

)

SOCFOC2

FOC3

FOC4


300

400

500

600

700

800

900

0 40 80 120 160 200 240 280 320 360Rotation angle 120579 (∘)

SOCFOC2

FOC3

FOC4

Redi

al g

as fo

rceFr

(N)




Nomenclature









Subscripts




Acknowledgments



References





















Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1205932(∘

) 1205931(∘

) 119877s(mm) 120588(1205932)(mm) 120579lowast

(∘

) 1198602(120579lowast

)(mm2) V ΔV ()



119865119886max(kN) 119865

119886(120579lowast

)(kN) 119865119905(120579lowast

)(kN) 119865119903max(N) 119865

119903min(N) 119865119903st(N)


SOCFOC1

FOC2

FOC3




4design is adopted


6 Conclusion


0 40 80 120 160 200 240 280 320 360

1200

1400

1600

1800

2000

2200

2400

2600

2800


Axi

al g

as fo

rceF

a(N

)

SOCFOC2

FOC3

FOC4





0 40 80 120 160 200 240

2

25

3

35

4

45

Small vibration


Tang

entia

l gas

forc

eFt

(kN

)

SOCFOC2

FOC3

FOC4


300

400

500

600

700

800

900

0 40 80 120 160 200 240 280 320 360Rotation angle 120579 (∘)

SOCFOC2

FOC3

FOC4

Redi

al g

as fo

rceFr

(N)




Nomenclature









Subscripts




Acknowledgments



References





















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 40 80 120 160 200 240

2

25

3

35

4

45

Small vibration


Tang

entia

l gas

forc

eFt

(kN

)

SOCFOC2

FOC3

FOC4


300

400

500

600

700

800

900

0 40 80 120 160 200 240 280 320 360Rotation angle 120579 (∘)

SOCFOC2

FOC3

FOC4

Redi

al g

as fo

rceFr

(N)




Nomenclature









Subscripts




Acknowledgments



References





















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










References





















Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









modeling of top scroll profile using equidistant-curve approach for

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