Purdue Universi Purdue e-pubs Inernaional Compressor Engineering Conference School of echanical Engineering 6 Calculaion on he Iner-Lobe Clearance Disribuion of Twin-Screw Compressor b Opimiaion ehod Wei iong i an Universi of Archiecure & Technolog Follow his and addiional works a: hps://docs.lib.purdue.edu/icec iong Wei "Calculaion on he Iner-Lobe Clearance Disribuion of Twin-Screw Compressor b Opimiaion ehod" 6. Inernaional Compressor Engineering Conference. Paper 89. hps://docs.lib.purdue.edu/icec/89 This documen has been made available hrough Purdue e-pubs a service of he Purdue Universi Libraries. Please conac epubs@purdue.edu for addiional informaion. Complee proceedings ma be acquired in prin and on CD-RO direcl from he Ra W. Herrick Laboraories a hps://engineering.purdue.edu/ Herrick/Evens/orderli.hml
C Page CALCULATION OF THE INTER-LOBE CLEARANCE DISTRIBUTION OF TWIN-SCREW COPRESSOR B OPTIIZATION ETHOD Wei IONG School of Environmenal & unicipal Engineering i an Universi of Archiecure &Technolog i an 755 shaani China Phone: 86-9-879 Fa: 86-9-879 E-mail: hangbin@mail.u.edu.cn ABSTRACT Small roor clearance is oda a vial prerequisie for an efficien screw compressor. The clearance will increase compressor leakage and hereb minimie boh he volumeric and adiabaic efficienc. Bu he iner-lobe clearance is necessar for compressor running. Therefore a mahemaical apparaus o quanif he iner-lobe clearance is precondiion of aking a reasonable iner-lobe clearance. An opimiaion mehod is presened in his paper in which he iner-lobe clearance is calculaed as he minimum disance beween he wo roors along he conac line. And he acual posiion of he roors in running sae was considered in which he roors conaced wih each oher in he drive surface. Therefore he calculaion of iner-lobe clearance is ransformed o he calculaion of he minimum disance disribuion beween he wo roors along he conac line in conac sae. The resuls show ha his algorihm obains full iner-lobe clearance beween he wo roor surfaces wih higher accurac.. INTRODUCTION Screw compressors have he advanage of simple srucure high reliabili and eas o operae. So he are used for he compression of a vas range of gases and vapors including refrigerans. The are widel used in he indusrial process indusries; he are also used as he compressors in facor based compressed air supplies. Iner-lobe clearance is oda a vial facor for an efficien screw compressor. anufacurers produce clearance beween roors o accommodae olerance build-up gas force deflecions and he hermal disorions ha occur during operaion. The clearance will increase compressor leakage and hereb minimie boh he volumeric and adiabaic efficienc. Bu he iner-lobe clearance is necessar for compressor running. The reliabili of screw compressor depends on he iner-lobe clearance disribuion and is varie while he compressor is running. Therefore a mahemaical apparaus o quanif he iner-lobe clearance is he precondiion of aking a reasonable iner-lobe clearance. In pracice roor abrades or roor seiure ofen occurs when he compressors are running his acciden damages he compressor greal and cause grea loss o he users. Roors abrade or roors seiure occurs because of inerference of he wo screw roors while he are running. So an accurae calculaes and analsis of he iner-lobe clearance of he screw roors in running sae will provide he foundaion for erasing he acciden. Tang e al. 994 presen wo was of seing clearances of screw compressors and discuss a design echnique for obaining an opimum clearance disribuion. Sosic and Smih presen a mahemaical apparaus which simplifies he analsis and allows he problem o be presened as a wo dimensional problem in he end plane of he roor and his apparaus was applied o quanif he saic and dnamic effecs of he roor inerference in screw compressors. Fong e al. propose a mahemaical procedure o calculae he iner-lobe clearance beween wo maing screw roors his mehod regards he proecion of aial difference of wo roor surfaces in he normal direcion as he iner-lobe clearance beween wo maing screw roors. Bu he sharps of screw compressors are ver complicaed here are difference beween he proecion of aial difference in he normal direcion and he iner-lobe clearance. The conac line of screw compressors is he line of inersecion of he wo maing roor surfaces. The conac line divides he cavi of he screw roor ino wo porions he low- pressure porion and high-pressure porion and seals he fluid in he high-pressure porion. In pracice he conac line ransforms o seal bel because of he eisence of he iner-lobe clearance. Because he poins on he conac line of he wo roors are in meshing sae a his momen
C Page he clearance beween he wo roors near he conac line is he minimum disance beween he wo roors. So in his paper he iner-lobe clearance is referred o wha is mean is he minimum disance normal o he wo local helical surfaces near he conac line. Therefore he calculaion of iner-lobe clearance is ransformed o he minimum disance calculaion beween he wo roors near he conac line. And in his paper a mahemaical procedure is proposed o calculae he iner-lobe clearance beween he wo maing screw roors in conac sae in which he roors conaced wih each oher in he drive surface. So he mahemaical procedure can be used o quanif evenual roor inerference in screw compressors due o he bearing clearance and he imperfecions in compressor housing manufacuring.. COORDINATE SSTES AND COORDINATE TRANSFORATION As shown in Fig. wo maing screw roors roae wih consan gear raio in he opposie direcions abou parallel aes. There are five coordinae ssems in Fig.. The coordinae ssem o fied o he male roor he coordinae ssem o fied o he female roor and he coordinae ssems O Z O Z and OZ fied o he compressor housing. The relaionship beween he five coordinae ssems can be epressed b coordinae ransformaion marices as follows: The coordinae ransformaion for he coordinae ssem o and O Z. i [ Z ] [ ] cosϕ sinϕ sinϕ cosϕ The coordinae ransformaion for he coordinae ssem o and O Z. [ Z ] [ ] cosiϕ sin iϕ sin iϕ cosiϕ Figure : Coordinae ssems
C Page 3 3 The coordinae ransformaion for he coordinae ssem O Z and OZ. [ ] [ ] Z Z A 3 4 The coordinae ransformaion for he coordinae ssem O Z and OZ. [ ] [ ] Z Z 4 The surfaces of screw roors can be represened in he coordinae ssem OZ b he operaion of coordinae ransformaion. ale roor: [ ] [ ] K Z k 5 Female roor: [ ] [ ] K Z K 6 3. SCREW SURFACES AND THE NORAL VECTORS OF SURFACES The surfaces of he maing roors are epressed b he following equaions: r r r r 7 8 As in Fig. he normal vecors of he surfaces are represened in coordinae ssem o and o b r r n
C Page 4 Figure : Normal vecors of he screw surface and he minimum disance beween he wo surfaces where i r r n k 9 Unie he mesh condiional epression and he funcion of screw roor surfaces he epression of conac line was gained ing. 4. CALCULATION ON THE INTER-LOBE CLEARANCE 4. Coordinae ransformaion The surfaces of screw roors are epressed in he coordinae ssems o o b equaions 7 8. Q is a poin on he surface of he male roor n is he normal vecors passing he poin. Q is a poin on he surface of he female roor. This paper ransforms he surfaces of screw roors and is normal vecors o he coordinae ssems OZ b coordinae ransformaion as below: Q Z [ ] [ ] K n n k [ Z ] [ ] K Q 4. Calculaion on he iner-lobe clearance The Opimiaion mehod is used o calculae he iner-lobe clearance beween he wo roors in his paper. The iner-lobe clearance is regard as he minimum disance beween he wo roors along he conac line. The mahemaical model of his problem is as follows: f min Q Q Q Q f 3
Q Q C Page 5 The smbol epresses coordinaes of he poin on he helical surface of he male roor or female roor in he coordinae ssem fied o he compressor housing which is gained b he formula and formula. The parameers which are ehibied as epress locaion parameers of he helical surface of he wo roors. The obecive funcion represened b he smbol f epresses square of he disance beween he wo roors. Because he funcional relaionship beween f and is nonlinear he direc search mehod is used o search he minimum value of he disance Simple mehod for insance. The seps of solving he iner-lobe clearance and is locaion parameers are as follows. A firs selec a poin on he conac line on he surface of he male rooror female roor and se one parameer of is locaion parameers as consan and se he res hree parameers as iniial value and subsiue hem in he formula 3. The value of he res hree parameers and corresponding minimum disance are gained b opimiaion calculaion. The minimum disance resolved is us he iner lobe clearance near he poin seleced. Then selec he ne poin on he conac line in proper sequence resolve he iner-lobe clearance b he same mehod. When we adus he densi of he poins seleced o a proper level his algorihm obain full iner-lobe clearance wih higher accurac. 5. CONTACT STATE DETERINATION OF SCREW ROTORS 5. Coordinae ransformaion marices of screw roor roaing abou is ais In he coordinae ssem OZ R represens he screw roor surface in he coordinae ssem faed o he compressor housing. R represens he screw roor surface roaed abou is shaf wih an angle θ in he coordinae ssem faed o he compressor housing. Then he roor surface afer roaing abou is shaf wih an angle θ can be gained hrough he following mehod. R Z RR θ Z R θ 4 [ ] [ ] R Z cos θ sin θ θ sin θ cos θ 5 5. Esimae of he conac sae beween he wo maing roors A clearance is needed beween he roors o allow for gas loading deflecions and hermal disorions of he roors bearings and housing and for manufacuring errors and oil-film hickness. There are in general five mehods for producing clearances beween roors: Equidisan profile mehod Equidisan helical surface mehod Equidisan cuer blade geomer mehod Reducing cener disance beween he cuer and roor mehod Increasing cener disance mehod. In pracice running onl he driving face of he male roor conac wih he female roor and ransfer orque. The orque was produced b conac force in he conac poin. The posiion of he wo maing roors can be confirmed afer confirming he posiion of he conac poin. The auhor confirms he posiion of he conac poin and conac sae b he iner-lobe clearance calculaion mehod. Calculaing he iner-lobe clearance disribuion along he conac line b he mehod presened as before if he minimum clearance of he iner-lobe clearance in he driving face of he male roor is d he coordinae of he poin wih he minimum clearance d is Q in he coordinae ssem fied o male roor. The disance beween he r + θ d / r λ Q poin and he shaf of male roor is hen roae he male roor wih an angle of Q according o formula 4 and formula 5 calculae he minimum clearance again ill he minimum clearance is less han ε where λ is amending coefficien ε is desined minimum clearance given before. If he minimum clearance is less han ε we consider he wo roors are in conac sae. 6. EAPLE
C Page 6 Figure 3: The GHH screw roors Figure 4: The male roor and he conac line a b Figure 5: The clearance disribuion along he conac line b equidisan lobe profile mehod a b Figure6: The clearance disribuion along he conac line b cener disance increasing mehod
C Page 7 This paper calculaes an eample wih GHH roors. Fig 3 shows a 5/6 roor pair. The cener disance is 45.5 mm and ou diameers are 65 mm and 5.94 mm for he male and female roor respecivel. A uniform clearance beween he wo roors in he end plane is.3 mm. Figure.4 shows he male roor and he conac line. Figure.5 shows wo graphs of he clearance disribuion along he conac line obained b equidisan lobe profile mehod. Figure.6 shows wo graphs of he clearance disribuion along he conac line obained b cener disance increasing mehod. Figure.5 a and Figure.6 a show graphs of he clearance disribuion along he conac line in he OZ plane. Figure.5 b and Figure.6 b show he relaionship beween iner-lobe clearance and he lengh of conac line. As i can be seen a Figs 56 clearance ump eiss along he conac line he iner-lobe clearance disribuion along he conac curve is quie differen according o various clearance generaing mehod and his algorihm obain full iner-lobe clearance wih higher accurac. 7. CONCLUSIONS A mahemaical apparaus is presened in his paper in which he iner-lobe clearance is calculaed as he minimum disance beween he wo roors along he conac line. Tha is he disribuion of iner-lobe clearance is ransformed o he calculaion of clearance disribuion along he conac bel. And he acual posiion of he roors in running sae was considered in which he roors conaced wih each oher in he drive surface. The resuls show ha he iner-lobe clearance disribuion in real conac sae has obvious difference wih he inerlobe clearance disribuion designed and clearance ump eiss along he conac line and he iner-lobe clearance disribuion along he conac curve is quie differen according o various clearances generaing mehod. And he opimiaion mehod presened in his paper can calculae he iner-lobe clearance beween he wo screw roor surfaces wih higher accurac. i n Q r NOENCLATURE gear raio coordinae ransformaion mari normal vecor of roor surface poins in he male roor and female roor screw surface epressed in he coordinae ssem fied o he screw roor R R Z screw surface of roor epressed in he coordinae ssem fied o he compressor housing roaing coordinae ransformaion mari parameer of he helical surface coordinaes of he helical surface in he coordinae ssem faed o screw roor coordinaes of he helical surface in he coordinae ssem faed o he compressor housing. θ roae angle abou -ais Subscrips ϕ roae angle of male roor male roor female roor REFERENCES ing Z. W. Screw compressor heor design and applicaion echanical indusr press Beiing China: p. 5-63. Tang. Fleming J. S. e al. 994 Clearances beween he roors of helical screw compressors: heir deerminaion opimiaion and hermodnamic consequences Proceedings of he Insiuion of echanical Engineers Par E: Journal of Process echanical Engineering 8E p.55-7.
C Page 8 Sosic N Smih lan K. Roor inerference in screw compressors. Proceedings of he 3rd Inernaional Compressor Technique Conference i an Jiaoong universi i an China p.34-4 Fong Z. H. Huang F. C. Fang H. S. Evaluaing he iner-lobe clearance of win-screw compressor b he iso-clearance conour diagramiccd echanism and achine Theor 36:p. 75-74