Avalable onlne at www.scencedect.co Poceda Engneeng 39 (0 ) 75 85 XIIIth Intenatonal Scentc and Engneeng Coneence HERVICON-0 Desgn and Flow Paaetes Calculaton o the Tuboachne Channels Mykola Kalnkevych a, Oleksand Gusak b, Andy Skoyk c, Oleg Shchebakov d, a* a,c,d Suy State Unvesty, Depatent o techncal theal physcs, Rsky-Kosakov St., Suy 40007, Ukane b Suy State Unvesty, Depatent o appled gydo- and aeoechancs, Rsky-Kosakov St., Suy 40007, Ukane Abstact The low calculaton ethod and desgn pocedue o axsyetc channels o tuboachnes ae pesented n the pape. On the bass o the pesented pocedue, a specal copute poga has been developed. To con the adequacy o the technque, the esults o the calculaton wee copaed wth the esults o the nuecal expeent. 0 Publshed by Elseve Ltd. Selecton and/o pee-evew unde esponsblty o Suy State Unvesty Open access unde CC BY-NC-ND lcense. Keywods: Tuboachne; axsyetc channel; low paaetes; channel desgn; CFD.. Intoducton The geneal eleents o tuboachnes ae the channels o deent type. Cuvlnea axsyetc channels have to be attended, n whch the low chaacte stongly aect the copesso ecency. Cuvlnea axsyetc channels ae the pat o nput and output devces, etun bend and etun channel o tuboachne. The channel cuvatue causes the stong low nonunoty n coss-secton, whch causes the ntaton o sepaaton ones and loss nceasng. That s why t s potant to evaluate the low chaacte to ceate the shape o cuvlnea axsyetc channel whch povdes the low level o loss. * Coespondng autho. Tel.: 38-098-439-5054. E-al addess: vkola58@uk.net. 877-7058 0 Publshed by Elseve Ltd. do:0.06/j.poeng.0.07.034 Open access unde CC BY-NC-ND lcense.
76 Mykola Kalnkevych et al. / Poceda Engneeng 39 ( 0 ) 75 85 In the conceptual stage t s sensblee to use pogas, n whch the sple atheatcal low odels based on the physcal oundatonn o the low chaacte ae used. Lght pogas, as copaed wth had pogas lke ANSYS CFX and FlowVson, allow to desgn channels aste and cheape. It s convenent o channel desgn to apply the analytcal o to dene the channel shape. The ecency o such appoach cons n the wokss o Puyewsk and Flasynsk [] and Stant [].. Radal-axal axsyetc channel low calculaton ethod Developed calculaton ethod s based on ollowng assuptons: gas low s copessble, statonay and adabatc. Inlet low paaetes dstbuton s uno. Accodng to the accepted physcal low odel, the low coe s nonvscous and a the vscosty was taken nto account o the bounday laye calculaton. The cuvlnea coodnate syste, whch s shown n Fgue, conssts o edonal pojectons o stealnes s and othogonals to the n; c s the edonal pojecton o velocty; R s the adus o cuvatue o stealne nn the edonal plane; c /R and c u / ae the centugal oces appled to the unt ass o gas. Fg.. Schee o the gas low n axsyetc channel The deental equatons o otonn o the patcle o nonvscous gas ae: p c ρ s p cu ρ n c s u c c cosθ R sn θ 0, 0. () () Fo the equatons () and () wth allowance o the Benoull equaton o noncopessble gas: c c c n 0 and 0. (3) n R c R
Mykola Kalnkevych et al. / Poceda Engneeng 39 ( 0 ) 75 85 77 Ate the ntegaton o the equaton (3): dn R c e c0 A c 0 e, (4) whee c 0 s the velocty at the nne suace o the channel, whch s dened o the contnuty equaton. I the values o the ntegal Adn/R ae dened, the elaton (4) allows to dene the velocty dstbuton along the noal. I the dstbuton o the stealne cuvatue n coss-secton along the noal s lnea, the ntegal becoes A j n R j n n j b R n R out, (5) whee R n and R out ae the aduses o cuvatue o the geneatx o the nne and oute suaces, espectvely; b s the dstance o the nne suace to the oute one along the noal; n j - the dstance o the nne suace to the pont whee velocty c 0 s beng estated. The ass low s estated o the aveage low velocty at the nlet o the channel. Each othogonal s dvded nto a cetan nube o equal pats. Fo each pat the aveage velocty s estated such that the total ass low wll be equal to the gven ass low o the channel. It should be noted that the total pessue and the total tepeatue ae assued as a constant along the channel and equal to the value at the channel nlet. 3. Radal-axal channel edonal contou desgn technque The channel edonal contou desgn ncludes next steps: - the cente lne constucton; - the cente othogonal coodnates denton accodng to the gven law o coss secton aea vaaton along the channel; - the nne and oute suace geneatx constucton; - constucton o the othogonals to the cente stealne and denton o the aduses o cuvatue at the coss ponts o nne and oute suaces and othogonals. The souce data o channel desgn s geoetcal chaactestcs shown n Fgue (a).
78 Mykola Kalnkevych et al. / Poceda Engneeng 39 ( 0 ) 75 85 (a) Medonal contou schee (b) Cente lne constucton schee Fg.. Desgn o the edonal contou o channel The channel length can be speced n ange o 0,5 L/D 0 0,35. It s ecoended to tun the low n the edonal plane wth axal adus o cuvatue. Ths condton s satsed by ccula ac [3]. The cente lne s the cobnaton o ccula ac 04 and lnea segent 4 (Fgue (b)). The equaton o staght lne whch s cossed the pont 0 at an angle o to the axs : 0 0 ) tg ( γ. (6) The equaton o staght lne whch s cossed the pont at an angle o cl to the axs : ) tg cl ( γ. (7) The coss pont o those staght lnes (pont 3) s dened o the soluton o the set o equatons (6) and (7): 3 0 0tgγ tgγ tgγ tgγ cl cl, 3 ( 0 ) tgγ 0tgγ cl tgγ. tgγ tgγ cl (8) Coodnates o pont 4 ae estated o the equalty o staght lnes 03 and 34. Coodnates o the cente o the ccle (pont 5) ae estated by the coss pont o staght lnes whch ae pependcula to the staght lnes 03 and 34. The adus o cuvatue o the ccula ac wth a pont 5 as a cente 5 4 ) ( 5 4 ) R (. (9) The coodnates o the cente othogonal ae estated accodng to the gven law o coss secton aea vaaton along the channel. Theeate coodnates o the coss ponts o nne and oute suaces and cente othogonal ae used to estate unknown coecents o those geneatxes.
79 Mykola Kalnkevych et al. / Poceda Engneeng 39 ( 0 ) 75 85 The law o coss secton aea vaaton along the adal axal channel s gven by the quadatc dependence 0 l l F, (0) whee l s the cente lne length cuent value; 0,, ae channel aea coecents. Coecents 0, ae estated o the bounday condtons. The value s gven by the ntal data. The nne and oute suace geneatxes ae gven by the second-ode cuve: 0 E D C B A. () To dene unknown coecents A,B,C,D,E the set o ve equatons was copled. The sets o equatons o nne and oute suaces ae esolved by Kae s ethod [4]. The equaton o the othogonal to the cente lne dened plctly ( ) ( ) 0 cl cl cl cl cl cl. () The quadatc equaton wth espect to s estated by the substtuton o devatve cl / cl and cl / cl nto equaton (): 0 s q p, (3) whee constants p,q,s ae unctons o A,B,C,D,E. Coss ponts o othogonals and nne and oute suace geneatxes ae obtaned n knd. The adus o cuvatue at the abtay pont o nne and oute suace geneatx: 3 R s. (4) 4. Bounday laye nluence accountng Bounday laye thckness s calculated n the st appoxaton by the lat plate oula [5] 5 0,37 ν δ l u l, (5) whee: - bounday laye thckness at length l, whch s counted o the channel nlet along the nne and oute suaces; - kneatc vscosty coecent; u - low velocty at the bounday laye lt along the geneatx.
80 Mykola Kalnkevych et al. / Poceda Engneeng 39 ( 0 ) 75 85 The powe law dstbuton o the bounday laye velocty s assued. Fo Re 0 5 the exponent s n /7. I one takes nto account that the densty change s neglgble, dsplaceent thckness can be obtaned: δ δ * u y δ dy δ 0 u 0 y dy δ δ 8 7 7 7 7 δ 8 δ. (6) 8 To dene the eal veloctes value, the calculaton s pleented o the estcted channel by the coespondng dsplaceent thcknesses. The nne and oute suace geneatxes ae obtaned usng dsplaceent thcknesses o the cente and outlet coss-sectons. 5. R-A Channel sotwae o adal-axal channel desgn Accodng to the pesented ethod the R-A Channel sotwae was ceated. It allows to desgn and calculate the low o the adal-axal channels autoatcally. The nlet and outlet geoetcs, channel length, theophyscal paaetes and velocty at the channel nlet ae the ntal data. Also the law o coss secton aea vaaton along the channel s gven (the value o coecent s gven). The sotwae povdes the channel edonal contou desgn and constucton o othogonals to the cente lne autoatcally. The esults o velocty and pessue calculaton ae pesented gaphcally by the supecal edonal velocty dstbuton c /a c (a c s ctcal velocty) and supecal pessue dstbuton ()p/p * ( p * s stagnaton pessue) along the channel (Fgue 3). The supecal edonal velocty dstbuton gaphcs along the nne and oute suace geneatxes ae pesented n Fgue 3 by ed and bown colou espectvely. The supecal edonal velocty dstbuton and supecal pessue dstbuton along the nne and oute suace geneatxes ae used o the calculatng o o-paaete value, whch allows to dene the bounday laye sepaaton wll be o not.
Mykola Kalnkevych et al. / Poceda Engneeng 39 ( 0 ) 75 85 8 Fg. 3. Soe steps o the calculaton by the R-A Channel sotwae As s known, t s ecoended to desgn tuboachne channels wth the nal axal densons. The developed sotwae allows to desgn the channel elatvely ast wth the nal axal densons o the gas low wthout sepaaton. It s obvous that loss wll be nal n that case, whch avou the nceasng o tuboachne ecency. 6. Radal-axal channel low analyss by eans o ANSYS CFX sotwae Fo the calculaton o the low, the sold odel o the adal-axal channel has been desgned. Because the low s axsyetc, not the whole aea o the odel, but a patcula secto wth the angle between the planes o the lat sdes equal 5 0 was calculated to be.fo the lat sdes an nteace bounday condton was gven. Geoetcs o the odel (accodng to the Fgue ) ae D0300, b00, D00, d00, 0n 90,, n 78, 0out 90,, out 78, L98, 0, geneatxes o the edonal contou o the channel wee gven as the second-ode cuves. Inlet low paaetes wee speced accodng to the ntal data gven o the calculaton by eans o R-A Channel. These paaetes ae c 0 60 /s, p 0 0,0 MPa, T 0 73. The SST tubulence odel and Total Enegy heat tanse odel wee used o the calculaton. A, deal gas was speced as wokng lud. Statc pessue, statc tepeatue and low velocty wee speced at the nlet bounday and the ass low ate was speced as the outlet condton. Fo dsplay puposes the esults o the calculaton ae pesented as velocty and pessue elds (Fgue 4). Velocty dstbuton and pessue dstbuton n a channel coespond to theoetcal concepton o low chaacte n axsyetc cuvlnea channels. As you can see, n the low coe velocty nceases
8 Mykola Kalnkevych et al. / Poceda Engneeng 39 ( 0 ) 75 85 o the nne suace to the oute one. Also thee s a velocty peak at the ntal pat o the oute suace. It s caused by sgncant cuvatue n ths aea. Fg. 4. Velocty and pessue elds estated by ANSYS CFX The edonal velocty dstbutons along the nne and oute suaces ae shown n Fgue 5. Ths dagas show the dstbuton o the low velocty at the bounday laye lt. Copason analyss o the veloctes calculated by R-A Channel and by ANSYS CFX v. shows good ageeent, wth the excepton o the shot secton at the nlet, whee dscepancy s about 0%. It occus because uno velocty dstbuton at the nlet was speced o the calculaton by R-A Channel. Whle usng ANSYS CFX an nluence o the stealne cuvatue on the nlet low paaetes was taken nto account (.e. the low was calculated as o nonuno velocty and pessue dstbuton at the nlet). Fo the est o the channel (L el 0, ) dscepancy s nsgncant. Pessue dstbutons along the nne and oute suaces ae shown n Fgue 6. Dscepancy does not exceed %. Fg. 5. Medonal velocty dstbuton along the nne and oute suaces
Mykola Kalnkevych et al. / Poceda Engneeng 39 ( 0 ) 75 85 83 Fg. 6. Pessue dstbuton along the nne and oute suaces 7. Radal-axal channel low analyss by eans o FlowVson sotwae The calculaton was povded usng the SST tubulence odel and totally copessble lqud odel o lud. The bounday condtons ae total pessue and statc tepeatue at the nlet and aveage velocty and aveage pessue at the outlet o the channel. Fo the cleaness the esults o the calculaton ae shown as a velocty eld and pessue eld (Fgue 7). Fg.7. Velocty and pessue elds estated by FlowVson The pessue values n Fgue 7 ae excessve values o pessue elatvely to the eeence value 0000 Pa. It should be noted that the esults o calculaton by FlowVson sotwae consdeably depend on bounday condtons. The conducted nuecal expeent showed the best coespondence o the calculaton esults and theoetcal concepton o the low chaacte the bounday condtons wee aveage velocty and aveage pessue at the outlet o adal-axal channel. The edonal velocty dstbutons along the nne and oute suaces, whch ae estated by R-A Channel and FlowVson v..05, ae shown n Fgue 8, the pessue dstbutons ae shown n Fgue 9.
84 Mykola Kalnkevych et al. / Poceda Engneeng 39 ( 0 ) 75 85 Fg. 8. Medonal velocty dstbuton at the nne and oute suaces along the longtudnal axs Z Fg. 9. Pessue dstbuton at the nne and oute suaces along the longtudnal axs Z As shown n Fgue 8, thee s a good copason between velocty dstbutons calculated by FlowVson and R-A Channel sotwae. The quanttatve copason shows acceptable accuacy. Maxal dscepancy s not exceedng 4%. The pessue dstbutons have a hgh ageeent (Fgue 9). Maxal dscepancy s not exceedng %. The adal-axal channel calculaton by ANSYS CFX and FlowVson sotwae shows a good qualtatve and quanttatve ageeent wth the developed R-A Channel calculaton. 8. Concluson The ethod o low calculaton and desgn ethod o adal-axal channels o tuboachnes ae pesented n the pape. The ethod was used n the copute poga R-A Channel. The copason between the esults o R-A Channel and ANSYS CFX and FlowVson calculaton was conducted. The copason shows a good ageeent, so the developed R-A Channel sotwae can be used o the adal-axal channel desgn. Reeences [] Puyewsk R. and Flasynsk P. Hypebolcally shaped centugal copesso, J. o Theal Scence, Vol., N3, (003). [] Stant J.D. Desgn o two-densonal channels wth pescbed velocty dstbutons along the channel walls. NACA Repot 5, (953), pp. 53-9.
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