3 Determiig te Optimal Stages Number of Module ad te Heat Drop Distributio
3. Aalytical Solutios A importat obective i te desig of a multi-stage axial turbie is to determie te optimal umber of stages i te module ad te distributio of eat drop betwee stages. Typically, a give quatity is te module s eat drop, ad sould vary te umber of stages ad te rotatioal speed (diameter). It sould be uderstood tat te circumferetial velocity reductio, ad ece te diameters of te stages, reduces te disc frictio losses, icrease eigt of te blades (ad terefore reduce te proportio of ed losses), decrease te flow pat leakage. At te same time it leads to a icrease i te optimal umber of stages, wic causes a icrease i losses due to discs frictio ad a additioal amout of te turbie rotor elogatio. Immediately aggravated questios of reliability ad durability (te critical umber of revolutios), materials cosumptio, icrease cost of turbie productio ad power plat costructio. A special place i te problem of te umber of stages optimizatio is te correct assessmet of te flow pat sape ifluece, keepig its meridioal disclosure i assessig losses i stages. As you kow, te issue is most relevat for te powerful steam turbies LPC. It is terefore advisable for te problem of determiig te optimal umber of stages to be able to fix te form of te flow pat for te LPC ad at te same time to determie its optimal sape i te HPC ad IPC. It sould also be oted tat te coice of te degree of reactio at te stages mea radius (te amout of eat drop also associated wit it) must be carried out wit a view to esurig a positive value tereof at te root. Formulated i tis sectio metods ad algoritms: ttp://www.sciecepublisiggroup.com
Optimizatio of te Axial Turbies Flow Pats may serve as a basis for furter improvemet of te matematical model ad complexity of te problem wit te accumulatio of experiece, metods ad computer programs used i te algoritm to optimize te flow of te axial turbie; allow te aalysis of te ifluece of various factors o te optimal caracteristics of te module, wic gives reaso for teir widespread use i teacig purposes, te calculatios for te uderstadig of te processes takig place i stages, to evaluate te impact of te various losses compoets o a stage operatio; allow to perform eat drop distributio betwee stages ad to determie te optimal umber of stages i a module witi te moderizatio of te turbie, i.e. at fixed rotatioal speeds (diameters) ad a give flow pat sape or at te specified law or te axial velocity compoet cage alog te cylider uder cosideratio. A possible variat of te form settig of stages group of te flow pat ca be carried out by takig te kow axial ad circumferetial velocity compoets i all cross-sectios, wic te umberig will be carried out as sow i Fig. 3.. ttp://www.sciecepublisiggroup.com
Capter 3 Determiig te Optimal Stages Number of Module ad te Heat Drop Distributio Stage Stage Stage Stage Figure 3. Te sectios umberig i te turbie flow part sectio, cosistig of stages. Te axial velocity compoets we refer to te axial velocity at te etrace to te stages group: were z specified values. c,, z zc z, (3.) Te sape of te flow pat ceter lie determied by te itroductio of coefficiet 3 4,, z u u, (3.) By satisfyig te coditios (3.), (3.) after optimizatio usig te cotiuity equatio G c zf c zif,, sape of te flow pat boudaries. we ca determie te Assumig tat we kow te iitial parameters of te workig fluid at te turbie module ilet ad te outlet pressure, i.e. teoretical eat drop i te group of stags is kow. Termal process i te group of stages wit te elp of s-diagram is sow i te Fig. 3.. Periperal efficiecy of te stages group determied by te formula ttp://www.sciecepublisiggroup.com 3
Optimizatio of te Axial Turbies Flow Pats u Lu Lu H i i TT, or takig ito accout (3.) ad (3.) i a dimesioless form accordig to te expressio u c z, u, zctg, u, zctg, (3.3) were u C ; C H ; c z c z u. We take ito accout te loss i te blades by applyig velocity coefficiets,,,. Also, assume tat te output of te itermediate stage a portio of te output eergy may be lost. Tis fact will take ito accout by itroducig a factor by wic te output loss is defied as 4 ttp://www.sciecepublisiggroup.com
Capter 3 Determiig te Optimal Stages Number of Module ad te Heat Drop Distributio P P i + r () s () P P P i P 3 P4 i 4 L u () L u () c / L u () P 4 out c P i out ( ) P P P i L u () H P l l l s r out l P P P P i P l l l l s r l out i i TT, S Figure 3. Te termal process i te s-diagram for te group of stages. ttp://www.sciecepublisiggroup.com 5
Optimizatio of te Axial Turbies Flow Pats by itroducig a factor by wic te output loss is defied as c out out, out ;,, (3.4) Calculatig losses i te guide vae ad te rotor by formulas c w s,, r, takig ito accout te factor of eat recovery te group of stages ca be writte as:, te limit for te eat drop i c A H L 3 u s r out Dividig equatio (3.5) by. (3.5) u, takig ito accout (3.), (3.), as well as well-kow kiematic correlatios betwee velocity ad flow agles after obvious trasformatios we obtai a expressio for te limitatio A 3 i te dimesioless form: A c ctg ctg 3 z, u, z, u, z, zc z ctg, zc z ctg c ctg c ctg, u, z z, u out, z z c ctg. (3.6), z oz Te task of defiitio of te agles z u z so tat, give te parameters, c,,,, ad take o te basis of some cosideratios (or defied by oe of te possible metods), te quatities of velocity coefficiets 6 ttp://www.sciecepublisiggroup.com
Capter 3 Determiig te Optimal Stages Number of Module ad te Heat Drop Distributio,,,, reaces its obective fuctio (3.3) maximum ad out satisfies te costrait (3.6). Matematically te formulated problem reduces to fidig: u L max u,, ctg H uder te costrait (3.6). Usig te pealty fuctios metod, you ca reduce te problem of fidig te extremum i te presece of costraits to te problem witout limitatio for te attaced obective fuctio were pealty coefficiet. I u 3 A, (3.7) ttp://www.sciecepublisiggroup.com 7
Optimizatio of te Axial Turbies Flow Pats out c i P i P P () r () () s () s () L u () i i, т P out c r () i i, T i, 3 out c P Figure 3.3 Te termal process i s-diagram for a itermediate -t stage. Give te values of te velocity coefficiets, alog te module, solutio of te problem is simplified due to te possibility of its decisio by idefiite Lagrage multipliers metod. Differetiatig te Lagrage fuctio by variables ctg,, L A, (3.8) u were Lagrage multiplier, we fid te followig ecessary optimality coditios, z c z ctg,, ; (3.9), u 3 8 ttp://www.sciecepublisiggroup.com
Capter 3 Determiig te Optimal Stages Number of Module ad te Heat Drop Distributio c, u Expressig all ctg, z out zctg,, ; (3.), z czctg. (3.), u troug ctg, ad excludig accordig to te tird formula, we get:, u, z ctg ctg,, z c z, u, ; (3.), u, z ctg ctg, out, z c z, u,, Were ;. Substitutig foud i tis maer ctg ad ctg i te equatio (3.6), we get te quadratic equatio i te parameter ctg opt were D E F, (3.3) opt opt ctg ctg D, z coz, u, u u,, u out ; E, z coz, u, u u,, u out ; ttp://www.sciecepublisiggroup.com 9
Optimizatio of te Axial Turbies Flow Pats c z F, z, z out, z, u, u 4 out, u Usig te solutio of tis equatio, te defie all te optimal agles ctg opt, wit (3.), as well as optimal efficiecy as a fuctio of te set parameters wit te elp of (3.3). Cosider te importat special case we,, ; out out;, ;,,, (3.) will ave te form.. I tis case, te formula u z ctg ctg,, ; c z ctg ctg,,, out (3.4) Were. After te substitutio of (3.4) ito (3.6) we ave a quadratic equatio of te form (3.5) wit te followig values of te coefficiets: c z D ; out c z E ; out ttp://www.sciecepublisiggroup.com
Capter 3 Determiig te Optimal Stages Number of Module ad te Heat Drop Distributio 4 out c z F, of wic tere is a optimal value ctg (3.3) te optimal efficiecy, te ctg, ad from u c zctg out out, (3.5) optimal velocity ratios of te stages u c zctg c z ctg c z z ctg ctg c c, (3.6) optimal reactios c z ctg R c z ctg out. (3.7) If adopted above coditios, we see tat all te stages except te last, are te same. Te fial stage is differet from all tat is coected wit te eed to reduce te exit velocity loss tat is completely lost at tis stage out. Usig formulas (3.4) (3.7) for values.96,.9 over a wide rage c z from. to. ad out from to te calculatios were carried out, te results of wic at values c z.4 ad out. are sow i Fig. 3.4. ttp://www.sciecepublisiggroup.com
Optimizatio of te Axial Turbies Flow Pats Calculatios ave sow tat for eac value uc, i.e. eat drop for a give amout H at a fixed circumferetial speed u, a optimum umber of stages exists at wic te maximum efficiecy of te module is reaced. Figure 3.4 Te calculated optimal exit flow agles, velocity ratios of itermediate ad last stages, module efficiecy for differet values of te eat drops ( c z.4,.96,.9, out. ;, ). Te umbers o te curves idicate te umber of stages i te module. Te bold lie sows te evelope of te parameters, correspodig to te maximum efficiecy. Assumig full utilizatio of te output velocity of te itermediate stages te rotor exit agles of te itermediate stages out very differet from 9. ca be ttp://www.sciecepublisiggroup.com
Capter 3 Determiig te Optimal Stages Number of Module ad te Heat Drop Distributio Te last stage flow exit agle i accordace wit te calculatio results must be doe close to 9, wic correspods wit a miimum loss of output velocity. Agles dowstream of te guide vaes lie i te rage...7, te optimum value of te velocity ratio i te rage of.48...58. Wit icreasig of umber of stages i te module te rage of acceptable cages of tese values is arrowed. I te case of output velocity loss i te itermediate stages te picture somewat cages. Icreases te value of te eat drop, i wic it is advisable to go to a larger umber of stages, agles dowstream of te itermediate stages are also close to 9. Tere is a decrease i te velocity ratio values, te exit flow agles of te guide vaes, resultig i a sligt drop i te optimum degree of reactio for te itermediate ad for te last stages. I te case of a sigle stage, assumig = periperal stage efficiecy is give by out L u c u c u i i u u u TT. From (3.3) we obtai i te dimesioless form c ctg ctg. (3.8) u z z u z u z For restrictios A, A ad A 3 from equatios (3.3), (3.6) is writte (see otatio o Fig. 3.): A i i c ; A i i Lu c ;. (3.9) A i i L c. 3 TT u s r ttp://www.sciecepublisiggroup.com 3
Optimizatio of te Axial Turbies Flow Pats Drawig o te kiematic relatios betwee velocities ad flow agles, usig te velocity triagles ave i k ctg A. (3.) z i k ctg A i k ctg z i k ctg u z uz k ctg ctg. (3.) k c ctg oz uz A3 uzc z ctg c z ctg Z c zz ctg c z ctg u. (3.) Here for coveiece itroduced a dimesioless ratio C, (3.3) a were C stage ilet velocity defied by c z ad ; velocity, equivalet to te critical value, for te ideal workig fluid. a k i k I te case of a perfect gas is a reduced velocity at te stage ilet. Te stage optimizatio problem is solved usig cougate gradiet metod by maximizig te attaced obective fuctio coefficiet. I A, were pealty u 3 4 ttp://www.sciecepublisiggroup.com
Capter 3 Determiig te Optimal Stages Number of Module ad te Heat Drop Distributio I te case of fixed velocity ratios ad from (3.) we obtai te opt opt relatiosip betwee ad aalytically ctg opt u z opt ctg z c z u, (3.4) were deoted. To determie te opt ctg is eeded to use te quadratic equatio (3.3), wic coefficiets i te case of sigle stage are give by: c D z z u u ; c E u z z u u ; c F 4 u z z z u z. Te reactio degree R of sigle stage is give by: c z ctg c z R. (3.5) C c z ctg I te case were ad are fuctios of flow parameters, for a sigle stage te solutio of te problem of determiig te optimal parameters ca be simplified by usig te metod of successive approximatios:. Set te iitial approximatio, ad defie te parameters for te stage usig derived formulas. ttp://www.sciecepublisiggroup.com 5
Optimizatio of te Axial Turbies Flow Pats. Te velocity coefficiets are recalculated accordig to te obtaied parameters ad calculatios are reewed from te item. Calculatios ave sow tat tis process coverges wit ig accuracy i a few iteratios. To ivestigate te ifluece of dimesioless parameters o te optimum stage performace computatioal study was coducted uder various assumptios about te loss i te stage. Te velocity coefficiets were take ito accout as a costat or depedet of te flow parameters. I te latter case, teir determiatio was made usig simplified depedecy [8] wit a bit icreased losses o te rotor blades:.5 ; 9. (3.6).4. 9 Te icrease i losses o te rotor blades i te presece of egative degree of reactio produced artificially by te formula, determied usig (3.6), if w w;, if w w. w w (3.7) Te most complete calculatios are made for te importat special case we,. z u Computatioal study as allowed to determie te optimal parametric depedecies of efficiecy, agles ad, reactio R, velocity coefficiets 6 ttp://www.sciecepublisiggroup.com
Capter 3 Determiig te Optimal Stages Number of Module ad te Heat Drop Distributio, ad loss factors s, r, out o c z ad. Te calculatio results are sow i Fig. 3.5 3.7. Figure 3.5 Optimal caracteristics of te turbie stage (.96,.9 ). Te umbers refer to c z values. Circles mark te optimal parameters at optimal eat drop i te stage. ttp://www.sciecepublisiggroup.com 7
Optimizatio of te Axial Turbies Flow Pats Figure 3.6 Optimal caracteristics of te turbie stage wit velocity ratio, calculated by te formula (3.6). Te umbers refer to z c values. Circles mark te optimal parameters at optimal eat drop i te stage. 8 ttp://www.sciecepublisiggroup.com
Capter 3 Determiig te Optimal Stages Number of Module ad te Heat Drop Distributio Figure 3.7 Optimal caracteristics of te turbie stage (, calculated by te formula (3.6)) wit recalculated accordig to (3.7). Te umbers refer to c z values. Circles mark te optimal parameters at optimal eat drop i te stage. 3. Prelimiary Desig of te Multistage Axial Flow Turbie Metod Descriptio I te early stages of te flow pat (FP) desig of te turbie, we determied te diameter, te blade eigts, eat drops ad oter mai ttp://www.sciecepublisiggroup.com 9
Optimizatio of te Axial Turbies Flow Pats caracteristics of te stages, required to study alteratives wit a view to te desig solutio, i te best sese of a quality criterio. Most effectively, tis problem is solved witi te created turbie flow pat CAD systems, because maage: to acieve a ratioal divisio of te desiger, defiig te strategy ad computer, quickly ad accurately perform complex calculatios ad presets te results i uma readable umeric or grapical form; to take ito accout may differet factors ifluecig te efficiecy, reliability, maufacturability, cost ad oter idicators of te quality of te desig beig created; orgaize dialogue or fully automatic determiatio of optimal parameters, etc [9]. Most metods of te multi-stage turbie parameters optimizatio is desiged to select te umber of gas-dyamic ad geometric parameters o te basis of te kow prototype, te caracteristics of wic are take as te iitial approximatio. We usig complex matematical models, a large umber of variables ad costraits, te solutio of suc problems requires cosiderable computer time ad for te purposes of CAD tat require quick respose of te system is ofte uacceptable. It is desirable to ave a metod of desig tat combies simplicity, reliability ad speed of obtaiig results wit a accuracy of te matematical model, a large umber of factors take ito accout ad optimized, te dept of fidig te optimal variat. Tis ievitably certai assumptios, te most importat of wic are: te sytesis parameters of "good", competitive structure witout attractig accurate calculatio models; i-dept aalysis ad refiemet of te parameters are ot take ito accout at te first stage; optimizatio of te basic parameters by repeatedly performig te steps of te sytesis ad aalysis. 3 ttp://www.sciecepublisiggroup.com
Capter 3 Determiig te Optimal Stages Number of Module ad te Heat Drop Distributio Desig of te FP i suc a formulatio will be called prelimiary (PD). PD does ot claim to suc a detailed optimizatio of parameters, as i te above-metioed metods of optimal desig. Its goal to offer a workable, effective eoug desig, te caracteristics of wic, if ecessary, ca be selected as te iitial approximatio for more accurate calculatios. Maor calleges i creatig a PD metod are: a ratioal approac to te problem of te prelimiary desig, te selectio of te quality criteria ad te costraits system; developmet of a metod for te multi-stage flow pat basic parameters selectio; formatio of requiremets for a matematical models complex describig differet aspects of turbies ad teir efficiet umerical implemetatio; selectio of te appropriate algoritm for fidig te optimal solutio; a flexible software creatio for a dialog based solutio of te desig problems i various statemets ad visual represetatio of te results. It is assumed tat FP PD will be coducted immediately after te calculatio of te turbie termal cycle uder kow for eac of te cyliders steam parameters i, P at te ilet, te backpressures for modules P, mass flows mod,, ad rotor frequecy. Te task is selectig te umber of stages i te modules, root diameter D ad stages blades eigts so as to acieve te maximum power of te cylider, wile esurig reliability, maufacturability, or ay oter (material cosumptio, cost, size, etc.) pre-specified requiremets. Te miimum acceptable reliability limits regulated (icludig safety factors) by static stresses i te blades ad diapragms, as well as detuig rotor blades G ttp://www.sciecepublisiggroup.com 3
Optimizatio of te Axial Turbies Flow Pats of costat cross-sectio of te resoace. Tecological costraits are reduced to a certai FP embodimet, task specific surface fiis, as well as te use of stadardized compoets profiles, saks, etc. Commo to powerful steam turbies HPC ad IPC is te requiremet of bladig uificatio, we all stages are formed by trimmig te top of te ozzle ad rotor blades of te last stage of te module. At te same time it maitaied a costat root diameter, agles ad, as well as te root degree of reactio R at te uiform eat drops distributio betwee te stages ad te costat axial velocity compoet i sectios. Cosider ways of formig te cylider FP, cosistig of sectios, satisfyig, i particular, te above requiremets of te uificatio. Te idea of te metod repeatedly expressed earlier. We apply it to te computer-aided FP desig ad make some modificatios ad geeralizatios. 3.. Metods of te FP Sytesis Cosider oe of te formulatios of te PD problems, wic we call te task I, i relatio to te module. Suppose tat te root diameter D, root degree of reactio R ad agle are kow. Te ozzle ad rotor blades are cosidered to be twisted by law cr cost, wic gives: u r ctg cost; r tg cost, (3.8) ad to cage te degree of reactio alog te radius te relatio is applicable R m or te approximate formula [] c u z C D D R D l D l 3 ttp://www.sciecepublisiggroup.com
Capter 3 Determiig te Optimal Stages Number of Module ad te Heat Drop Distributio R m l R D l.8. (3.9) First of all, te estimated process of steam expasio i te module is build. As we kow eiter te umber or te geometrical caracteristics of te stages, it ca be doe oly very approximately, evaluatig te module efficiecy im, suc as by metod [3]. Tis makes it possible to fid te parameters of steam at te ed of te actual process of expasio ad, takig tis process as liear, to evaluate te termodyamic parameters at ay pressure P P P., mod To select te umber of stages i te module let allow approximately uiform breakdow of te eat drops by te stages. Te, by settig te velocity ratio uc or evaluatig its "optimal" (i.e. correspodig to te axial outlet flow from te stage = 9) value, for example, by te formula [] u cos or C R C m cos R u, (3.3) you ca get ci H D c 8 i, (3.3) were H module disposable eat drop; c i velocity at te ilet of te module; stage umber rouded up to te earest iteger. Velocity c i, wic is equal to te axial compoet is determied by takig ito accout (3.3) accordig to te formula ttp://www.sciecepublisiggroup.com 33
Optimizatio of te Axial Turbies Flow Pats c c c si H R si i z cos 4u R si D R tg, (3.3) were is assumed equal to.96...98. Itroducig te otatio ci i i mod S S mod i ; S, o te proposed steam expasio process for eac of te stages te parameters i te gap betwee vaes witout muc error is determied based o te relatiosips: ci c z P Pi i i c z Rm, S S Rm S, im m (3.33) i i P, S S R S, (3.34) Pressures dowward te stages are equal P, i,,,. (3.35) c i P Pi i, S S. Nozzle vaes eigts are determied from te cotiuity equatio were c z take from (3.3). z G l D l c, (3.36) 34 ttp://www.sciecepublisiggroup.com
Capter 3 Determiig te Optimal Stages Number of Module ad te Heat Drop Distributio Solvig (3.36) as a quadratic equatio, we fid Sice te value of R m 4G l D D. (3.37) c z, eterig ito (3.33) (3.35), depeds o te eigt of te blades, te iterative refiemet of all te values determied by formulas (3.9), (3.33) (3.35), (3.37) is eeded. Takig as a iitial approximatio R typically acieve covergece of 4 iteratios. Istead of may be set, for example, te ratio Dm l of te -st stage. We call tis formulatio as te problem II. I tis case, immediately we fid te eigt of te blades of te -st stage l D Dm l ad te -st stage degree of reactio at te mea radius usig (3.9). Agle of te -st stage is determied based o te cotiuity equatio te obvious relatio m G c si D l l, (3.38) D l R m c C Rm (3.39) 4 ad te coditios (3.3), wat after simple calculatios gives tg G D l l R Furtermore, deletig C from (3.39) usig (3.3), we obtai m. (3.4) ttp://www.sciecepublisiggroup.com 35
Optimizatio of te Axial Turbies Flow Pats D l R c ; c c si. (3.4) m i cos Te most advatageous umber of stages i a module is determied by (3.3). Oterwise metod II does ot differ from te metod I. Wit te itroductio of te coefficiet z c c metods I ad II ca be geeralized to te case were te axial velocity compoets liearly vary from stage to stage. For tis purpose, i te equatios z, (3.33), (3.36) ad (3.37) c z sould be replaced by te value c, z c i i z. m It sould be bore i mid tat we z te blade system uificatio coditio ( cost, cost ) is ot satisfied. Tus, as a result of solvig PD problems i te statemet I (II) certai basic caracteristics of te FP are defied: te umber of stages, stages couterpressures P, root level reactio degrees R, root diameter D, eigt of ozzle vaes l, agles (ratio Dm l of te -st stage). 3.. Detailed Termal Calculatio Next, to a more accurate assessmet of te created desig quality criteria ad calculatio of all required parameters is proposed to solve te iverse oedimesioal problem of termal calculatio of te FP for eac of te stages of te cylider. ow at tis poit data is ot eoug for tis calculatio. 36 ttp://www.sciecepublisiggroup.com
Capter 3 Determiig te Optimal Stages Number of Module ad te Heat Drop Distributio Additioally, you eed to specify te eigt of rotor blades, te geometric caracteristics of te cascades, seals, etc. Selectio of missig values must be based o te desig adopted, stregt, tecological ad oter requiremets. For example, te eigt of te rotor blade ca be obtaied o te groud of iformatio about te stadard overlap or troug te strict implemetatio of te coditios cost i te group of stages. Usig stadard profiles data or geeralized depedecies for te profile caracteristics of arbitrary sape allows you to create a cascade, satisfyig te requiremets of efficiecy, reliability ad maufacturability. Selectio of te mai cascade parameters (a cord, stagger agle, pitc, etc.) it is advisable to carry out durig te refiemet of te velocity coefficiets of crows i te oe-dimesioal iverse problem of te FP termal calculatio. It s quite a complicated idepedet problem wic deserves special cosideratio. Te results of tis calculatio are te kiematic parameters of te flow i te gaps, te effective agles, cascade s compoets of te kietic eergy loss ad power parameters of stages. It also calculated te magitude of stresses i te elemets of desig, weigt, size ad oter caracteristics. Tis iformatio is sufficiet to draw a coclusio about te quality of te built structure ad te eed to cotiue te desig process. A proximity i te selectio of te basic FP parameters i te first PD stage compesated wit te detailed accout of te most factors affectig te quality parameters of te turbie i te model of termal calculatio. However, it sould be bore i mid tat durig te sytesis of te FP sould be set a umber of parameters wic are precisely determied oly at te secod calculatio step. Terefore, tere may be some differeces i te parameters uc,, i, mod. ttp://www.sciecepublisiggroup.com 37
Optimizatio of te Axial Turbies Flow Pats Te most sigificat differeces betwee te set ad te refied value, wic ca reac.5... or more because of te stages umber roudig to te earest iteger i te formula (3.3) ad, as a cosequece, te deflectio uc i te formula (3.3) from te optimum. For tis reaso, ad also because of metodologically iappropriate to set as te iitial parameter, wic subsequetly must be determied (agle seems more ratioal. ), te PD problem formulatio II 3..3 Optimizatio Te desire to automate te PD process leads to te developmet of a algoritm for fidig te optimal combiatio of te basic parameters of te flow pat. Wit regard to te formulatio II it is D, R,, D l of te -st stage, ad i case of failure of te uificatio also m z. Te total umber of variables to variable cylider cosistig of m modules tus does ot exceed Tey imposed restrictios 5 mod. D mi D D max ; R mi R R max ; mi max ; D l D l D l ; mi max z mi z z max. (3.4) We selectig cascade s profiles durig te detailed termal calculatio may be preseted restrictios o static stregt of te diapragm ad te rotor blades of te type desig, (3.43) 38 ttp://www.sciecepublisiggroup.com
Capter 3 Determiig te Optimal Stages Number of Module ad te Heat Drop Distributio mi mi max ; ; (3.44) ad oter. To automatically desig te FP, optimal i terms of te selected quality criteria, te desiger must specify rages of variable parameters ad te required umber of poits i te searc space defied by te coditios (3.4). Samplig poits geeratio is coducted usig te LP sequeces. Clarificatio of te optimal solutio is acieved by reducig te rages i te searc. Typically, te amout of te searc poits rages from a few doze to several udreds. Sice te sytesis ad termal desig of oe poit takes a few secods, te maximum time to fid te optimal variat is ot more ta a few miutes o a stadard PC. ttp://www.sciecepublisiggroup.com 39