Preliminary Design Efficiency Optimization of 1D Multistage Axial Flow. Compressor for a Fixed Distribution of Axial Velocities

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1 Prelmary Deg Effcecy Optmzato of 1D Multstage Axal Flow Compressor for a Fxed Dstbuto of Axal Veloctes Lge Che, Ju Luo, ad Fegru Su Postgraduate School, Naval Uverty of Egeeg, Wuha , P. R. Cha To whom all correspodece should be addressed. E-mal address: lgeche@hotmal.com ad lgchea@yahoo.com (Lge Che) Fax: , Tel: Abstract. A prelmary deg effcecy optmzato of a multstage axal flow compressor s studed ths paper ug oe-dmeoal flow theory. A model for the optmum deg of a multstage axal flow compressor, assumg a fxed dstbuto of axal veloctes s preseted. The absolute let ad ext agles of the rotor of every stage are tae as the deg vaables. Aalytcal relatos betwee the setropc effcecy of the multstage compressor ad the flow coeffcet, the wor coeffcet, the flow agles ad the degree of reacto of every stage are obtaed. The results have some geeralzatos ad ca provde some gudace for the optmum deg of multstage compressors. Key words: multstage axal flow compressor; effcecy; optmzato Nomeclature c absolute velocty m/s F through-flow area m G Ar mass flow rate g/s h ethalpy se J/g specfc ethalpy J/g velocty coeffcet umber of stage P pressure Pa R deal gas costat s specfc etropy J/(gK) u wheel velocty m/s w relatve velocty m/s Gree symbols flow agle relatve to a stator, 0 z reheat coeffcet flow agle relatve to a rotor, 0 flow coeffcet adabatc effcecy rato of specfc heats dety degree of reacto 3 g / m loss coeffcet wor coeffcet Subpts a axal drecto c total performace of a multstage compressor th stage j jth stato opt optmum r rotor s stator; setropc process u tagetal Superpts stagato codto 1

2 1. Itroducto The deg of the axal flow compressor s ot a precse ece. The lac of accurate predcto flueces the deg process seously. Utl today, there are o methods curretly avalable that guaratee to predct the absolute values of these quattes to a suffcet accuracy for a ew deg. Some progresses have bee obtaed va the applcato of umecal optmzato techques to gle- ad mult-stage axal flow compressor deg, see Refs. [1-]. Especally wth the developmet of computatoal flud dyamcs (CFD), may more accuracy methods of calculatg have bee obtaed may refereces whch the techques of computatoal flud dyamcs have bee appled to tow-dmeoal ad threedmeoal optmum deg of axal flow compressor, see Refs. [17-0]. However, t s stll of worthwhle gfcace to calculate ug oe-dmeoal flow-theory for the optmum deg of compressors. Boo [3] preseted detaled mathematcal model for optmum deg of gle- ad mult-stage axal flow turbes for assumg a fxed dstbuto of axal veloctes, ad obtaed the correspodg aalytcal results. Ug the mlar dea, Che et al. [] preseted a mathematcal model for optmum deg of gle-stage axal flow compressors. I ths paper, the results of Ref. [] are exteded to the optmum deg of multstage compressors. The model for the optmum deg of multstage axal flow compressor for the fxed dstbuto of axal veloctes s preseted. Aalytcal relatos betwee the setropc effcecy of the multstage compressor ad the flow coeffcet, the wor coeffcet, the flow agles ad the degree of reacto of every stage are obtaed ug oe-dmeoal flow-theory. Numecal examples are provded to llustrate the effects of vaous parameters o the optmum performace of the multstage compressor.. Fudametal equatos for elemetal stage compressor Coder a -stage axal flow compressor. Fg.1 shows the flow path of the -stage axal flow compressor. Fg. shows the specfc ethalpy-specfc etropy dagram of the -stage compressor. For a -stage axal flow compressor, there are ( 1) secto statos. The stage velocty tagle of a termedate stage (th stage) s show Fg.3. The correspodg specfc ethalpy specfc etropy dagram s show Fg.4. The calculato of mult-stage performace s performed by ug oe-dmeoal flow theory. The aalys begs from the eergy equato ad cotuty equato, ad the axal flow veloctes of the worg flud ad wheel veloctes at the dfferet statos the compressor are ot codered as costat, that s, deotes jth stato. u u ad c c ( j) j j, where deotes th stage, ad j Fg.1 Flow path of a -stage axal flow compressor

3 Fg. Ethalpy-etropy dagram of a -stage compressor 3

4 Fg. 3. Velocty tagle of termedate stage Fg. 4 Ethalpy-etropy dagram of termedate stage The major assumptos made the method are as follows: 1) The worg flud flows stably relatve to vaes, stators ad rotors, whch rotate at a fxed speed. ) The worg flud s compresble, o-vous ad adabatc. 3) The mass flow rate of the worg flud s costat. 4) The compreso process s homogeeous the worg flud. 5) The absolute outlet agle of the worg flud th stage s equal to the absolute let agle of the worg flud (+1)th stage. 6) The effects of tae ad outlet ppg are eglected. Therefore, the eergy equato for oe-dmeoal flow of th stage s: h u c ctg u c ctg (1) a, 1 a,1 1 The cotuty equato s: G 1ca,1F 1 ca,f () The wor requred by the rotor s: r, 1 1 h ( w w ) / ( u u ) / (3) The degree of reacto of the th stage compressor s defed as h / h, hece, oe has r, 4

5 [ a, 1 ( (1 ctg ) u, a, ctg a,1 u,1 ( ctg a,1 1 ctg 1)] 1 ) (4) 3. Mathematcal model for mult-stage compressor The compreso wor requred by every stage s ( 1 ). The total compreso wor requred by the mult-stage compressor s h h c h. The stagato setropc ethalpy se of every stage s 1 The sum of the stagato setropc ethalpy se of every stage s ethalpy se of the mult-stage compressor s h s 1 h. Oe has h 1 h 1 h s,.,, whle the stagato setropc s, ( z ) coeffcet of the mult-stage compressor. The stagato setropc effcecy of the mult-stage axal flow compressor s:, where z s the reheat h / h h / h (5) c 1 Because the flow veloctes of the worg flud are ot costat alog the radal ad axal drectos, at dfferet statos the compressor, the velocty coeffcets, ( j 1 ), are troduced. They are defed as: a j a, j ca, j / ca,1 1F1 j j u, j, a, j / F, u, j u j / u1 (6) It should be explaed that the shape of flow path, vz. the chage of heght of blade, s determed by,, ad the shape of medoal plae, vz. the chage of dameter, s determed by u j. The flow, coeffcet s defed as c a, 1 / u1. The setropc wor coeffcet of the mult-stage s defed as h / u 1. The setropc wor coeffcet of every stage s defed as / u1 h. Wth the help of the stage velocty tagle, t s coveet to rewte Equato (5) terms of velocty coeffcets, flow coeffcet, setropc wor coeffcet ad flow agles. /[ ( ctg ctg )] (7) 1 u, a, u,1 a, Optmzato problem ad ts soluto The optmzato problem s to fd the flow agles ( j 1 ) to mae the stagato setropc j effcecy be maxmum for the fxed u a,,,, ad uder the followg costrat: 5

6 A h h h 0 (8) 1 c r s 1, s where hr ad hs are the total losses of the rotors ad the stators, ad the other symbols are show Fgs. 1 ad. The total profle losses of th stage rotor ad the stator are calculated as followg: where 1 h w /, h c / ( 0, 1) (9) s the total profle loss coeffcet of th stage rotor blade ad s that of th stage stator blade. The total losses of the rotors ad the stators: h w /, h s c / (10) r Accordg to the ethalpy-etropy dagram of the multstage compressor, the costrat A ca be rewtte as A h (1 ) h h h 0 (11) c z r s The, the costrat A ca be rewtte terms of velocty coeffcets, flow coeffcet, setropc wor coeffcets, flow agles ad profle loss coeffcets of the blades. A ( u, [( u,1 a, a, ctg a,1 (1 ctg ) u,1 ctg The blade profle loss coeffcets 1 a,1 ) / ctg 0 1 a,1 ) (1 ) ] / z (1) ad are fuctos of parameters of the worg flud ad blade geometry. They ca be calculated ug vaous methods. If they are codered as costats, the optmzato problem ca be solved ug a Lagraga fucto. The fucto L A ca be costructed, where s Lagraga multpler. The partal devatve equato groups ca be obtaed ug Euler- Lagraga equatos as follows: L / ( ctg 1) /[ ( u, a,ctg u, 1 a,1ctg 1)] [ r 1(1 ctg 1) ] 0 1 L / ( ctg 1) u, 1 a,1 /[ ( u, a,ctg u, 1 a,1ctg 1)] ( ) 1 u,1 a,1 a,1( u, 1 a,1ctg 1) 0 L / ( ctg ) /[ ( u, a, u, a,ctg u, 1 a,1ctg 1)] 1 a, u, a, ctg 0 ( 1 ) (13) 6

7 Solvg Equato (13) gves: ctg ctg u, (1 ctg1) /( a, )( 1 ) 1 [ u, 1 u, 1 (1 ctg1)] /( a, 1 )( 1 ) (14) Substtutg Equatos (14) to (1) gves the quadratc equato for ctg 1 as follows: ctg [ (1 ) ( 1) 0 u, 1 1 / [ z u, u, / / u,1 / ) ( u,1 u,1 u / ] ctg [ u )/ / ( )/ s1 s1 1 ( a,1 s1 a, 1 1 / u, u, u,1 u,1 / / ] / ) /( )] (15) The soluto for ctg 1 from Equato (15) s the optmum value ( opt ctg ) of ctg 1 1. The the opt optmum values of ( j 1 ) ca be obtaed ug Equatos (14). The maxmum of the objectve fucto j, the stagato setropc effcecy of the mult-stage axal flow compressor, ad the optmum effceces, wor coeffcets, ad degrees of reacto of th stage ca also be obtaed. The stagato setropc effcecy of th stage compressor s s, h / h, the, s, ( h h 1[ /[ ( a,1 u, h )/ h ( a, u,1 ctg a,1 u,1 a,1 ctg ctg 1 1 )] ) a, (1 ctg ) ] (16) The setropc wor coeffcet of th stage compressor s h u / 1, the [ ( / u, u,1 u,1 a,ctg ctg a,1 u,1 1 ) ctg 1 (1 ctg a,1 a, a,1 ) ] (17) Accordg to the above aalys, for the fxed values of the optmzato problem ad to obta the obvous aalytcal equatos. Whe ad, t s easy to get the soluto of ad are fuctos of the parameters of the worg flud ad blade geometry, the loss coeffcets ca be calculated ug the method of Ref. [4], whch was used ad debed detal Ref. [1]. The optmzato problem ca be solved ug the terato method: Frst, select the ogal values of ad, ad the calculate the parameters of the 7

8 mult-stage compressor from the equatos deduced above. Secodly, calculate the values of the calculated valves ad the ogal oes are small eough. ad, ad repeat the frst step utl the dffereces betwee 5. Numecal examples To fd the fluece of vaous dmeoless parameters o the optmum performace of the mult-stage stage axal flow compressor, umecal examples are provded ug the uversal mathematcal model establshed earler. The relato of the optmzato objectve wth dmeoless parameters has bee studed for the fxed values of,, a,, ad u,. It should be poted out that there wll be some fluece o the relato of the optmzato objectve wth these dmeoless parameters f, a,, ad, u, are fuctos of worg flud parameters ad geometry parameters of the flow path cofgurato. However, the relato obtaed wll ot chage qualtatvely. I the calculatos, 0. 1, 1 ( 1 ), 0. 1 ad 3 are u set. The optmum stagat setropc effcecy vs. the wor coeffcet ad the flow coeffcet of the 3-stage compressor s show Fg. 5. It ca be see that the charactestc s mlar to that of mult-stage turbe obtaed by Boo [3], that s, the optmum stagat setropc effcecy vs. the wor coeffcet s a parabolc-le oe, there exsts a optmum wor coeffcet correspodg to a maxmum optmum (double-maxmum) stagato setropc effcecy, ad the optmum stagat setropc effcecy vs. the flow coeffcet s a mootoc decreag fucto, see Fgs. 6 ad 7. a z Fg.5 Optmum stagat setropc effcecy vs. wor coeffcet ad flow coeffcet of a 3-stage compressor 8

9 Fg. 6 Optmum stagat setropc effcecy vs. wor coeffcet of a 3-stage compressor Fg. 7. Optmum stagat setropc effcecy vs. flow coeffcet of a 3-stage compressor ( 1opt The flueces of the wor coeffcet ( ) o the optmum absolute let agle of the frst rotor ) versus flow coeffcet ( ) are show Fg. 8. Fgure 8 shows that the optmum absolute let agle s a creag fucto of the flow coeffcet whe the wor coeffcet s less tha some value. Otherwse t s a decreag fucto of flow coeffcet, ad the optmum absolute let agle s larger tha 90. It s obvously that the optmum absolute let agle approaches to 90,. e. axal flow, whe the flow coeffcet creases. 9

10 Fg.8. Effect of wor coeffcet o the optmum let arflow agle of the frst rotor versus flow coeffcet of a 3-stage compressor The optmum absolute let agle of the frst rotor ( 1opt ) versus the wor coeffcet of the 3-stage compressor wth 0. 8 s show Fg. 9. It ca be see that the optmum absolute let agle of the frst rotor ( 1opt ) s a mootoc creag fucto of the wor coeffcet. Ug Eq. (14), oe ca obta the other optmum let agles at all statos. For ths umecal example, because the loss coeffcets ad the velocty coeffcets at all statos are the same, the optmum absolute outlet agles of all rotors are the same, ad the optmum absolute let agles after the secod rotor are the same. Fgures 10 ad 11 show the optmum absolute let agles after the secod rotor versus the wor coeffcet wth 0. 8 ad the optmum absolute let agles after the secod rotor versus the flow coeffcet wth. It ca be see that the optmum absolute let agles after the secod rotor are creag fuctos of the wor coeffcet ad the flow coeffcet. Fgures 1 ad 13 show the optmum absolute outlet agles of all rotors versus the wor coeffcet wth 0. 8 ad the optmum absolute outlet agles of all rotors versus the flow coeffcet wth. It ca be see that the optmum absolute outlet agles of all rotors are decreag fuctos of the wor coeffcet ad creag fuctos of the flow coeffcet. 10

11 Fg.9. Optmum let arflow agle of the frst rotor versus wor coeffcet of a 3-stage compressor Fg.10. Optmum absolute let agles after the secod rotor versus the wor coeffcet of a 3-stage compressor Fg.11. optmum absolute let agles after the secod rotor versus the flow coeffcet of a 3-stage compressor 11

12 Fg. 1 Optmum absolute outlet agles of all rotors versus the wor coeffcet of a 3-stage compressor Fg. 13 Optmum absolute outlet agles of all rotors versus the flow coeffcet of a 3-stage compressor 6. Cocluo I ths paper, the prelmary deg effcecy optmzato of a mult-stage axal flow compressor has bee studed ug oe-dmeoal flow theory. The uversal charactestc relato for a mult-stage axal flow compressor s obtaed. The flueces of vaous parameters o the relatos are aalyzed. Numecal examples are preseted. The results provde some gudace for the performace aalys ad optmzato of compressors. Ths s a prelmary study. It wll be ecessary to use mult-objectve umecal optmzato techques [11-13, 0, 1, 5-9] ad artfcal eural etwor algothms [10, 19, 30, 31] for practce compressor optmzato. 1

13 Acowledgemets Ths paper s supported by Program for New Cetury Excellet Talets Uverty of P. R. Cha (Project No. NCET ) ad The Foudato for the Author of Natoal Excellet Doctoral Dssertato of P. R. Cha (Project No ) Refereces 1. Wall R A. Axal flow compressor performace predcto. AGARD-LS-83, 1976 (Jue): Gu C ad Mao Y. Blade deg of axal flow compressors by the method of optmal cotrol theory. Tras. ASME J. Turbomachary, 1987, 109(1): Hearsey R M. Numecal optmzato of axal compressor deg. ASME paper No.89-GT Tucclle R. A proposal for optmzed deg of multstage compressors. ASME paper No.89-GT Lm J S ad Chug M K. Deg pot optmzato of a axal-flow compressor stage. It. J. Heat ad Flud Flow, 1989, 10(1): Massardo A ad Statta A. Axal flow compressor deg optmzato: Part I-ptchle aalys ad multvaable objectve fucto fluece. Tra. ASME, J. Turbomachery., 1990, 11(): Massardo A, Statta A ad Ma M. Axal flow compressor deg optmzato: Part II-throughflowaalys. Tra. ASME, J. Turbomachery., 1990, 11(): Egorov I N ad Fom V N. Numecal method of optmzato of a multstage axal compressor. Expemetal ad Computatoal Aerothermodyamcs of Iteral Flows. World Publshg Corporato, 1990: Tucclle R. Optmum deg of axal flow compressor. ASME IGTI, 1990, 5: Geoge H ad Stuart B. Prelmary deg of axal compressors ug artfcal tellgece ad umecal optmzato techques. ASME paper No.91-GT Che L. A bef troducto of multobjectve optmzato for axal flow compressor stage. Gas Turbe Tech., 199, 5(1): 11-13( Chese). 1. Egorov I N ad Kre G V. Multcteo stochastc optmzato of axal compressor. ASME IGTI, 199, 7: Egorov I N. Optmzato of multstage axal compressor a gas turbe ege system. ASME paper, 9-GT Che L. Some ew developmets o the optmum deg of trubomachary dug the past decade. J. Egg. Thermal Eergy Pow., 199, 7(4): 14-1( Chese). 15. Egorov I N. Determstc ad stochastc optmzato of a vaable axal compressor, ASME paper No.93-GT Su J ad Elder R L. Numecal optmzato of a stator vae settg multstage axal-flow compressors. Proc. Ist. Mech. Egrs., 1998, 1(A4): Calvert W J ad Gder R B. Trasoc fa ad compressor deg. Proc. Ist. Mech. Egrs., 1999, 13(C5): Gallmore S J. Axal flow compressor deg. Proc. Ist. Mech. Egrs., 1999, 13(C5): L J, Satofua N. Optmzato deg of a compressor caade arfol ug a Naver-Stoes solver ad geetc algothms. Proc. Ist. Mech. Egrs., 00, 16(A):

14 0. Be E. Three dmeoal mult-objectve deg optmzato of a trasoc compressor rotor. AIAA J. Propulo Power, 004, (3): Che L, Su F ad Wu C. Optmum deg of subsoc axal flow compressor stage. Appl. Eergy, 005, 80(): Che L, Luo J, Su F ad Wu C. Optmzed effcecy axal-flow compressor. Appl. Eergy, 005, 81(4): Boo A B. Optmum Deg for Flow-Path of Axal Turbes. Harov: Hgher Educato Press, 198( Rusa). 4. Casey M V. A mea le predcto method for estmato the performace charactestcs of a axal compressor stage. Proc. ImechE 1987, Turbomachary Effcecy Predcto ad Improvemet, 1987: Che L, Wu C, Bla D ad Su F. Prelmary deg optmzato of a mae dual tadem gear. It. J. Pow. Eergy Sys., 1997, 17(3): Che L, Wu C, N N, Cao Y ad Su F. Optmum deg of cetfugal compressor stages. It. J. Pow. Eergy Sys., 1998, 18(1): Che L, Wu C, Bla D ad Su F. The multobjectve optmum deg method for a radal-axal flow turbe wth the ctea of optmum twst at the outlet of blade. It. J. Pow. Eergy Sys., 1998, 18(1): Che L, Zhag J, Wu C, Bla D ad Su F. Aalys of multobjetve deco-mag for mae steam turbe. It. J. Pow. Eergy Sys., 1998, 18(): Che L, Zhou S, Wu C, Su F. Prelmary deg optmzato of a steam geerator. Eergy Covero ad Maagemet, 00, 43(13): L B J, Hug C I, Tag E J. A optmal deg of axal-flow fa blades by the machg method ad a artfcal eural etwor. Proc. Ist. Mech. Egrs., 00, 16(C3): Q X, Che L, Su F, Wu C. Effcecy optmzato for a axal flow steam turbe stage ug geetc algothm. Appl. Thermal Egeeg, 003, 3(18):