Labyrinth Seal Design Optimization Based on Quadratic Regression Orthogonal Experiment

Transcription

1 Enrgy and Powr Enginring, 2017, 9, ISSN Onlin: ISSN Print: X Labyrinth Sal Dsign Optimization Basd on Quadratic Rgrssion Orthogonal Exprimnt Lihua Cao, Hyong Si, Pan Li, Yong Li School of Enrgy and Powr Enginring, Northast Elctric Powr Univrsity, Jilin, China How to cit this papr: Cao, L.H., Si, H.Y., Li, P. and Li, Y. (2017) Labyrinth Sal Dsign Optimization Basd on Quadratic Rgrssion Orthogonal Exprimnt. Enrgy and Powr Enginring, 9, Rcivd: Fbruary 23, 2017 Accptd: March 30, 2017 Publishd: April 6, 2017 Abstract Th influnc of labyrinth sal structur paramtrs and thir intraction on th charactristics of lakag amount ar numrically invstigatd by conducting a quadratic rgrssion orthogonal xprimnt. To dtrmin th optimal structur paramtrs of th stam sal for minimizing th lakag amount, a rliabl rgrssion quation that dos not lack of fit is stablishd. Th flow charactristics of th fluid in th labyrinth sal ar analyzd in dtail. Rsults show that th lakag amount is gratly influncd by sal cavity dpth, convx platform hight, sal tooth thicknss, and tooth tip claranc, with th tip claranc having th most significant ffct. Th intraction among th four itms xrts a crtain impact on th lakag amount. Th proposd rgrssion quation xhibits a good significanc and dos not lack of fit. Aftr optimization, th labyrinth sal dmonstrats incrasd ntropy and nrgy dissipation at th tip of th sal tooth, as wll as dcrasd spd and inrtia ffct in th cavity, suggsting that th rsistanc lakag prformanc of th optimizd labyrinth sal is improvd. Kywords Labyrinth Sal, Structur Paramtrs, Rgrssion Orthogonal Tst, Lakag Amount, Optimization 1. Introduction Th diaphragm sal of stam turbin plays an important rol in prvnting stam lakag and rducing stam turbin loss. Th diaphragm sal is widly usd bcaus of its simpl structur, convnint installation, and high fficincy [1] [2] [3]. Labyrinth sals hav various kinds, and diffrnt factors affct thir fficincy. Studis on th lakag charactristics of th labyrinth, rub-groov labyrinth, radial annular sal, and honycomb labyrinth sals hav dmonstratd that th lakag amount positivly changs with th prssur ratio [4] [5] [6]. DOI: /p B025 April 6, 2017

2 Whn all othr conditions ar constant, th rotor spd, in contrast to th sal structur, has a ngligibl ffct on th lakag amount [7] [8] [9]. Th lss is th numbr of labyrinth sals, th gratr is th lakag amount bcaus of th numbr of stam sal tth; morovr, th lakag amount incrass with an incras in sal tip claranc [10] [11]. On this basis, a rsarchr pointd out that th lakag cofficint is linar to th saling tip claranc [12]. A qualitativ analysis using carbon dioxid as working fluid was conductd on th rlationships among labyrinth sal lakag, tooth tip claranc, cavity width, cavity dpth, and so on; th findings rvald that th lakag amount incrasd with an incras in tooth tip claranc, dcrasd with an incras in cavity width, and initially dcrasd and subsquntly incrasd with an incras in cavity dpth [13]. Svral domstic scholars hav positd that diffrnt typs of sal hav diffrnt flow fild charactristics and lakag amounts; ths studis found a trnd that th lakag amount incrasd with an incras of cavity dpth, and dcrasd with an incras in th thicknss of th sal tth [14]. Howvr, a rsarch on th high-prssur labyrinth sal of th F gas turbin found that th lakag cofficint dcrasd with a dcras in th thicknss of th sal tth [15]. Analyzing th changs in saling tooth top prssur and turbulnt kintic nrgy is on of th ways to study th sald ffct and th influnc of diffrnt tooth structurs on th lakag amount [16]. Many studis hav invstigatd th influnc of labyrinth sal structur paramtrs on th lakag amount, and rliabl conclusions hav bn obtaind. Howvr, most of ths studis wr conductd by controlling a singl variabl factor. Choosing an appropriat rsarch mthod is on of th important stps in analyzing saling lakag charactristics. Orthogonal xprimnts hav bn widly usd in biology, nginring, and othr filds to study th ffcts of multipl factors on th tst indx [17] [18] [19]. Th rgrssion orthogonal xprimnt is basd on th orthogonal xprimnt to stablish a rgrssion quation and implmnt a sris of vrification and analysis. This xprimnt has a high supriority in th study of quantitativ rlationship btwn multi-factors and th tst indx [20] [21]. Thrfor, a rgrssion quation is stablishd in this papr by using quatrnion quadratic rgrssion orthogonal xprimnt to analyz th ffcts of multipl structur paramtrs on th lakag amount in considration of th notion that basic gomtrical paramtrs of th labyrinth sal hav diffrnt influncs on lakag charactristics. In addition, th solution of th rgrssion quation is calculatd by programming to confirm th optimal structur paramtrs that corrspond to maximum saling fficincy. 2. Rgrssion Orthogonal Exprimnt 2.1. Numrical Calculation Mthod Th cavity dpth, convx platform hight, tooth thicknss, and tooth tip claranc paramtrs of a 300 MW condnsing stam turbin ar quotd according to th actual labyrinth sal structur paramtrs to idntify th variation rang of 205

3 factors in rgrssion orthogonal xprimnt that nsurs th availability and rliability of th xprimnt in nginring. Th concrt structur paramtrs ar shown in Figur 1. Calculation tim is rducd by using an arc sction with a priodic surfac rotatd 3.6 dgrs around th cntr rathr than th whol cycl sction. Undr th sam grid numbr, th modl has a highr grid quality. In th calculation, watr vapor is st in accordanc with th ral condition, which prssur boundary conditions ar adoptd in th inlt and outlt of th calculation domain. Th prssur and tmpratur in th inlt is 10.7 MPa and K, whras that at th outlt is MPa and K. Standard k-ε quation is combind with SIMPLE algorithm in th 3D flow fild of th labyrinth sal to solv th N-S quation. Gambit softwar is usd to modl th arc sction, and th grid has a ngligibl ffct on th calculation rsults of th stam lakag whn th grid spacing rachs and th grid numbr xcds 2 million, as shown in Figur 2. Th y plus is 35 to nsur that th turbulnc modl has highr accuracy. Thrfor, th numbr of msh usd in this papr is x3 x4 x1 x2 Figur 1. Schmatic diagram of labyrinth sal structur. Figur 2. Th lakag quantity varis with th numbr of grid clls. 206

4 on th rang of million to improv th calculation spd Dsign of th Rgrssion Orthogonal Exprimnt Rgrssion orthogonal xprimnt is chosn to arrang th simulatd calculation statistics in a rasonabl rang of factors, and a rgrssion quation with a crtain dgr of confidnc is stablishd by using finit xprimnt points. Consquntly, th lakag amount can b stimatd by using th rgrssion quation whn th structur paramtrs ar dtrmind, and th thortical optimum valu of tst factors that corrsponds to th minimum lakag amount of stam can b obtaind in a crtain rang by solving th rgrssion quation. Such an approach avoids a grat dal of computation, rducs th tsting workload, and improvs th fficincy of th labyrinth sal dsign optimization. Thrfor, this papr uss th quatrnion quadratic rgrssion orthogonal 1/2 implmntation mthod and dtrmins th variation rang of four control paramtrs on th basis of th gnral principls of labyrinth sal dsign: th cavity dpth x 1 is 5-13 mm; th convx platform hight x 2 is mm; th tooth thicknss x 3 is mm; th tooth tip claranc x 4 is mm. If x ( = 1, 2, 3, 4) is th lvl of ach factor in th rgrssion orthogonal xprimnt, and x 1 and x 2 ar uppr and lowr limits of th factors, thn th zro lvl of th factor can b rprsntd by x 0 = (x 1 + x 2 )/2. Basd on th variation rang of th factors, th lvl cod is as shown in Tabl 1. Th numbr of numrical calculations of th xprimnt program is dtrmind by Equation (1), which, compard with a comprhnsiv xprimnt with fiv lvls and four factors (i.., 5 4 = 625 tims), only nds 17 tims. Th calculation priod is rducd to 2.72% of th original on. n = m0 + mc + mr (1) whr m 0, th cntr tsts numbr, is 1; m c, th tims of two lvl tst, is 2p-1; m r, th astrisk tst, is 2p; p, th numbr of factors, is 4; r, th astrisk arm 1/2 lngth, is r = ( nmc mc) / 2 (i.., r = 1.353). Th chang stp of th tst lvl can b dtrmind by th uppr and lowr limits of th factors and astrisk arm lngth. x1 x2 = (2) 2r Tabl 1. Encod form of two lvls and four factors. Lvls Factors cavity dpth x 1 /mm convx platform hight x 2 /mm tooth thicknss x 3 /mm tooth tip claranc x 4 /mm (+r) (+1) (0) ( 1) ( r) Δ

5 Th factors should b kpt in a consistnt dimnsion and should not substantially diffr in th orthogonal xprimnt. Thus, th natural variabls of ach factor should b tratd in th cntralization. Suppos z is cod valu obtaind aftr th cntralization of ach factor lvl valu, thn z ( x x0 ) = +. Th tst rsults and analysis ar prsntd in Tabl 2. A unifid form of rgrssion quation can b obtaind by th data procssing of th orthogonal tabl: 1 S = y ( y ) n n 2 2 = 1 n = 1 (3) Hr, y dnots th stam lakag amount, which is th inspction indx of th tst obct. Givn that th lakag amount of th partial arc sction of th labyrinth sal modl is small, th stam lakag of that arc sction is convrtd to a lakag of th whol cycl sction; i.., th lakag is magnifid on hundrd tims. b is a constant trm. b is th first rgrssion cofficint. b i is th partial rgrssion cofficint of th intraction trm. b is th quadratic rgrssion cofficint. Tabl 2. Analysis of rsults of rgrssion orthogonal xprimnts No. z 0 z 1 z 2 z 3 z 4 z 1 z 2 z 1 z 3 z 1 z 4 z 2 z 3 z 2 z 4 z 3 z 4 z1 ( z 1) z2 ( z 2) z3 ( z 3) 4 ( 4) z z Y r r r r r r r r D B b S F significanc * * ** *** * ** ** * ** * 208

6 According to th abov cofficints and th data in th tabl, th spcific calculation quation can b xprssd as follows: n ( ) 2 n B S / f D = zi, B = zi y, bi =, S = Bb, F = D S / f = 1 = 1 whr f = 1. S is th rror sum of squars, which is obtaind by m0 2 S = (y 0 y0). = 1 To conduct th significant tst and th lack of fit tst of th rgrssion quation, thr cntr xprimnts ar conductd in this papr. Th rror sum of squars S is 0.002, and its dgr of frdom f is 2, whn y 0 ar , , and Th cofficints of th rgrssion quation ar dtrmind by th tst of th rgrssion cofficint of th orthogonal xprimnt. And th factors that corrspond to th cofficint significantly affct y if F (, ) > F0.25 f f. In Tabl 2, * indicats that th significanc of th lvl satisfis F (, ) 0.25 f f ; i.., th confidnc lvl rachd Similarly, ** indicats that th significanc of th lvl satisfis F (, ) 0.05 f f, and *** indicats that th significanc of th lvl satisfis F 0.01 ( f, f ). In this study, all rgrssion cofficints ar significant xcpt for b 13, b 24, b 11, and b 22, of which th lvl of satisfis F (, ) < F0.25 f f. Taking insignificant rgrssion trms in th rmaindr of th rgrssion trms and obtaining th rsidual sum of squars is SR = S Sh = , and th lack of fit squar sum is Slf = Sh S = , n n whr S is total sum of squars and its formula is S = y ( y ). = 1 n = Tst of Significanc and Not Lack of Fit of th Equation Th influnc of factors on th inspction indx can b arrangd in th valus of b, b i, and b by tsting th rgrssion cofficint of th quation. Th array of linar trms is z4 > z3 > z2 > z1. Th array of intraction trms is zz 1 4 = zz 2 3 > zz 3 4. b 4 is th largst, suggsting that z 4 has a vry larg ffct on th lakag of stam, and th main factor that influncs lakag in th stam turbin sparator is th tooth tip claranc. Morovr, b 4 is positiv, indicating that th stam lakag is smallr whn th tooth tip claranc is smallr. This rlationship is consistnt with th thory that th flow ara and th lakag is proportional to that drivd from th mpirical quation of stam lakag. In addition, cavity dpth, convx platform hight, tooth thicknss, and thir intraction xrt crtain ffcts on th lakag of stam. Positiv rgrssion cofficints indicat th positiv corrlation btwn th inspction indx and th factors, whras ngativ cofficints indicat th convrs. Th ffcts of th tst trms on th inspction indx ar shown in Figur 3. According to th significanc tst quation of th rgrssion orthogonal xprimnt: Sh / fh = > F0.01( fh, fr) = (4) S / f R R Th quation accords with th significanc tst of quatrnion quadratic r- 209

7 Figur 3. Th variation of th rgrssion cofficint of tst itms. grssion orthogonal xprimnt indicating that th confidnc lvl of th rgrssion quation is (1 α) 100% = 99% and that th significanc lvl is α = Similarly, according to th quation of th lack of fit: Slf / flf = < F0.25( flf, f ) = (5) S / f Th rgrssion quation stablishd in this papr can b dtrmind to hav good not lack of fit, indicating that th lakag, th linar trm, th quadratic trm, and th intraction of factors hav a vry good fit, and it has no significant rlationship with highr-ordr trms Transformation and Vrification of th Rgrssion Equation Th rgrssion quation can b obtaind by th abovmntiond mthod of calculation and data. y= z z z z zz 1 2 (6) zz zz zz 0.096z z In viw of th vrification tst and th optimization dsign, th cod of rgrssion quations should b transformd into natural variabls, and th tst cod is transformd invrsly into th function of x by th cntral procssing quation: y = x x x x xx xx xx 2 3 (7) xx 0.384x x To vrify th accuracy of th rgrssion quation, th calculation of thortical quation is implmntd through a group of paramtrs in th rasonabl rang of ach factor. Whn x 1 = 9.5 mm, x 2 = 3.5 mm, x 3 = 0.7 mm, and x 4 = 0.5 mm, 210

8 L. H. Cao t al. th calculation rsult of th rgrssion quation is kg/s, and th rror is 6.14% than th rsult calculatd by Flunt. Ths findings suggst that th quation has high accuracy and can mt th nds of nginring application to a crtain xtnt. 3. Rsults and Analysis of Optimal Dsign Th rgrssion Equation (7) is calculatd by C# undr th constraints of a givn rang of factors and is locally optimizd to dtrmin th stam lakag. In this way, th bst valu that corrsponds to th minimum valu can b obtaind for ach factor to raliz dsign optimization. Th calculatd stp siz is st to 0.1, and ach factor x is calculatd to dtrmin th minimum valu of ymin. Thn, th program is improvd to calculat ymin invrsly, so that th corrsponding natural factors in which th lakag is th minimum valu can b dtrmind. Whn x1 = 13 mm, x2 = 4.5 mm, x3 = 0.2 mm, and x4 = 0.2 mm, th minimum lakag of ymin is kg/s aftr calculation, which is th optimal structur paramtr in th rang of ach factor. At this point, th bst structur ratio is x1 : x2 : x3 : x4 = 65 : 22.5 :1:1. Figur 4 shows th variation of prssur contours of tooth tip claranc in th Figur 4. Prssur contours of flow fild in th sald cavitis. (a) Bfor optimization; (b) Aftr optimization. 211

9 mridional plan bfor and aftr optimization. As shown in th diagram, th chang in th prssur gradint mainly occurs at th top of th sal, showing that th kintic nrgy of working fluid at th tooth tip claranc is gratr than that in th cavity. Comparison of (a) and (b) in Figur 4 rvals that th cavity prssur distribution of th optimizd labyrinth sal is mor uniform in (b) than in (a). Th incras in cavity dpth and th dcras in tooth thicknss caus an incras in th filling dgr of th fluid in th volum of th cavity. Th improvmnt of prssur man valu in ach cavity aftr optimization indicats that th kintic prssur is ffctivly convrtd into static prssur, rducing th vlocity of stam flow and making th non-stagnant stam flow out dirctly from nxt tooth lss, ultimatly wakning th inrtia ffct of th stam. Thus, th saling ffct of th sal on th working fluid is improvd, and th lakag amount is rducd. At th sam tim, th nlargd pictur of th tip claranc at th inlt and th outlt shows that th prssur drops vnly whn th working fluid flows through th sal tooth aftr optimization. Howvr, th prssur of tip claranc at th inlt and th outlt initially dcrasd and subsquntly incrasd bfor optimization. This shows that part of th kintic nrgy of th working fluid in th acclrating procss is not dissipatd but rconvrtd into prssur nrgy. Consquntly, th throttling ffct of th saling tth is not obvious, and th lakag incrasd bfor optimization. Figur 5 shows a graph of ntropy of trac drivd from th fixd monitoring points at th inlt bfor and aftr th optimization of th labyrinth sal. Comparison of ntropy curvs bfor and aftr optimization rvals that th ntropy of stam shows a surging trnd. Howvr, th pak of ntropy aftr optimization is gratr than that bfor optimization, indicating that th working fluid flowing through th optimizd sal tth xprincs gratr nrgy loss. Th two curvs dcrass aftr th pak of ntropy, but th ntropy of th optimizd Figur 5. Entropy graph of trac bfor and aftr th optimization. 212

10 modl dclins fastr. Entropy changs smoothly bfor arriving at th nxt sal tooth. Thn, th nrgy loss of th stam is rducd. Th main rason of this is that th stam is stagnant, th vlocity of th stam is smallr, and most of th kintic nrgy is convrtd into prssur nrgy, lading to th rduction of nrgy dissipation. Howvr, th rducd nrgy dissipation dos not caus incrasd lakag, bcaus th inrtia ffct almost disappard du to th stagnant stam, which gratly rducs th lakag amount. Figur 6 shows th lakag of labyrinth sal bfor and aftr optimization undr diffrnt loads. Th lakag amount of th labyrinth sal incrass with an incras in th load rgardlss of optimization, but th lakag tndncy of th optimizd labyrinth sal incras with an incras in load is small. Th lakag amount of optimizd labyrinth sal is rducd by 0.466, 0.567, 0.812, and kg/s for loads of 40%, 50%, 75%, and 100%, accounting for 78.85%, 78.75%, 78.61%, and 78.32% whn compard with that bfor optimization. In conclusion, th lakag amount of th optimizd labyrinth sal has th sam small lakag amount undr diffrnt loads. 4. Conclusions Th rgrssion quation of th structur paramtrs and lakag amount of labyrinth sal is stablishd by prforming a quatrnion quadratic rgrssion orthogonal xprimnt. Th quation satisfis th tst of significanc and not lack of fit. In this papr, th thortical calculation rsults ar in good accordanc with th simulation rsults, which can mt th nds of th actual proct. Th cavity dpth, convx platform hight, tooth thicknss, and tooth tip claranc of th labyrinth sal xrts a crtain influnc on th saling prformanc. Th tip claranc is th main factor that influncs th lakag amount, and lakag amount was positivly corrlatd with th chang of tip claranc. Undr th influnc of multipl factors, th lakag amount of th labyrinth sal has th minimum valu. Figur 6. Th lakag of labyrinth sal bfor and aftr optimization undr diffrnt loads. 213

11 Aftr th optimization of th structur paramtrs of th labyrinth sal, th lakag amount of optimizd labyrinth sal is rducd by approximatly 78% whn th optimal structur ratio is x1: x2 : x3 : x 4 = 65 : 22.5 :1:1. Th ntropy incrass and th pak of ntropy is highr than that bfor optimization whn stam flows through th tip claranc, nhancing th nrgy dissipation ffct and improving th throttling ffct. Morovr, th largr volum of th cavity improvs th stam stagnation, and th convrsion of dynamic prssur into static prssur is promotd. Th inrtia ffct is gratly rducd, such that th optimizd labyrinth sal has a dcrasd lakag amount undr diffrnt loads. Acknowldgmnts This work was supportd by National Natural Scinc Foundation of China ( ). Rfrncs [1] Jun, L.I., Yan, X., Song, L.M., Dng, Q.H., Fng, Z.P.,& Jiang, S. K., t al. (2008)A rviw of saling tchnologis in Turbomachinry. Thrmal Turbin, 37, [2] Dnck, J., Färbr, J., Dullnkopf, K., & Baur, H. J. (2008) Intrdpndnc of Discharg Bhavior, Swirl Dvlopmnt and Total Tmpratur Incras in Rotating Labyrinth Sals. ASME Turbo Expo 2008: Powr for Land, Sa, and Air, Brlin, Grmany, 9 13Jun, 2008, [3] Ivanov, A. V., & Moskvicv, A. V. (2015) Influnc of Gomtry on Vortx Configuration and Dimnsion in Lr Turbopump Labyrinth Sal. Procdia Enginring, 106, [4] Subramanian, S., Skhar, A. S., & Prasad, B. V. S. S. S. (2015) Influnc of Combind Radial Location and Growth on Th Lakag Prformanc of a Rotating Labyrinth Gas Turbin Sal. Journal of Mchanical Scinc and Tchnology, 29, [5] Cao L., Liu Y., Zhang Y., Li Y. (2014)Numrical Invstigation of Lakag Effcts Of Rub-Groovs on Labyrith Sal. Lubrication Enginring, [6] Zhang, W. F., Yang, J. G., Tian, Y. W., Cao, H., & Gu, J. G. (2013) Rsarch on th Lakag and Dynamic Charactristics of a Nw Kind of Radial Annular Sal and Comparisons With Labyrinth Sals. Procdings of th Institution of Mchanical Enginrs Part A Journal of Powr & Enrgy, 227, [7] Zhigang, L.I., Lang, J., Jun, L.I., & Fng, Z. (2011) Exprimnt on Lakag Flow Charactristics of Labyrinth Sal. Journal of Xian Jiaotong Univrsity, 107, [8] Z. Li, J. Li, & Z. Fng. (2010) Effcts of Gap Prssur Ratio and Rotational Spd on Discharg Bhavior of Labyrinth Sal. Journal of Xian Jiaotong Univrsity, 44, [9] Bozzi, L., D Anglo, E., Facchini, B., Micio, M., & Sogh, R. D. (2011) Exprimntal Invstigation on Lakag Losss and Hat Transfr in a Non-Convntional Labyrinth Sal. ASME 2011 Turbo Expo: Turbin Tchnical Confrnc and Exposition, British Columbia, Canada, 6 10 Jun, 2011, [10] Jun, L.I., Qiang, L., Fng Z.P. (2006). Rsarch on th Lakag Flow Charactristics in th Stppd Labyrinth Sal. Journal of Mchanical Enginring, 42,

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13 Submit or rcommnd nxt manuscript to SCIRP and w will provid bst srvic for you: Accpting pr-submission inquiris through , Facbook, LinkdIn, Twittr, tc. A wid slction of ournals (inclusiv of 9 subcts, mor than 200 ournals) Providing 24-hour high-quality srvic Usr-frindly onlin submission systm Fair and swift pr-rviw systm Efficint typstting and proofrading procdur Display of th rsult of downloads and visits, as wll as th numbr of citd articls Maximum dissmination of your rsarch work Submit your manuscript at: Or contact p@scirp.org