The Design and Analysis of Helium Turbine Expander Impeller with a Given All-Over-Controlled Vortex Distribution

Transcription

1 Plasma Scence and Technology, Vol.16, No.3, Mar. 214 The Desgn and Analyss of Helum Turbne Expander Impeller wth a Gven All-Over-Controlled Vortex Dstrbuton LIU Xaodong ( ), FU Bao ( ), ZHUANG Mng ( ) Insttute of Plasma Physcs, Chnese Academy of Scences, Hefe 2331, Chna Abstract To make the large-scale helum cryogenc system of fuson devce EAST (expermental advanced super-conductng tokamak) run stably, as the core part, the helum turbne expander must meet the requrement of refrgeraton capacty. However, prevous desgns were based on one enson flow to determne the average flud parameters and geometrc parameters of mpeller cross-sectons, so that t could not descrbe real physcal processes n the nternal flow of the turbne expander. Therefore, based on the nverse proposton of streamlne curvature method n the context of quas-three-ensonal flows, the all-over-controlled vortex concept was adopted to desgn the mpeller under specfed condton. The wrap angle of the mpeller blade and the whole flow dstrbuton on the merdan plane were obtaned; meanwhle the performance of the desgned mpeller was analyzed. Thus a new desgn method s proposed here for the nverse proposton of the helum turbne expander mpeller. Keywords: helum turbne expander, streamlne curvature method, merdan plane, allover-controlled vortex, nverse proposton PACS: 7.2.Mc DOI: 1.188/19-63/16/3/21 (Some fgures may appear n colour only n the onlne journal) 1 Introducton Two refrgeraton cycles have been adopted by the helum refrgeraton system of EAST [1 3]. The refrgeraton power and the lquefacton at 3.5 K and 4.5 K are produced by a Claude cycle. Lqud ntrogen s employed to pre-cool helum to 8 K. In ths cycle, the frst two turbnes (T1,T2) are n seres dstrbuton wth a mass flow of 1 g/s, and then are n parallel dstrbuton wth another J-T turbne (T3) wth 11 g/s. The refrgeraton at 8 K s completed by a Reverse-Brayton cycle. Turbne T4 s equpped n such refrgeraton cycle. Ths cycle s used to cool the 8 K thermal sheld. Therefore, there are 5 coolng levels n the Claude cycle, such as pre-coolng level, the frst level of expanson coolng (T1), the second level of expanson coolng (T2), J-T turbne expanson coolng (T3), and the level of throttlng coolng. Snce the performance of the helum turbne expander plays a qute mportant role n the stable runnng of the large scale cryogenc system of EAST, t s qute necessary to make the turbne expander run stably and effcently. Then we have to desgn the turbne expander based on real physcal processes. As to the desgn method of nverse proposton about turbomachnery, some effectve methods were proposed, such as one-ensonal flow [4], quas-threeensonal flow [5,6] and three-ensonal flow [7]. Prevous desgn was based on one enson flow to determne the average flud parameters and geometrc parameters of mpeller cross-sectons, whch was stll very mportant to the aerodynamc desgn of turbomachnery, but such desgn could not completely conform wth the real physcal process of nternal flow through the mpeller. The three-ensonal desgn could adopt physcal models whch are closest to the real physcal process, however such method needs a lot of computatonal resources. Compared to the three-ensonal desgn, the streamlne curvature method, as one quasthree-ensonal desgn, s economcal and accurate enough to conduct relevant desgns. 2 The all-over-controlled vortex method based on the nverse proposton of the streamlne curvature method Based on the streamlne curvature method, WANG Shangjn proposed a convenent method for the desgn of an mpeller wth a new concept, all-overcontrolled vortex. Ths method made use of the physcal model closer to the real flow process, rather than one-ensonal flow s approxmate assumpton. So t s qute promsng to complete the desgn of a turbne mpeller n the future. Heren, some new deas have been combned wth 288

2 LIU Xaodong et al.: The Desgn and Analyss of Helum Turbne Expander Impeller the all-over-controlled vortex method, such as the dentfcaton of the flud helum state, whch s used to specfy whether the flud helum could be vewed as an deal gas or not. Ths s the frst tme such method has been adopted to conduct the desgn of helum turbne expander mpellers, whch s benefcal to the research and development of helum turbne expanders. Durng the desgn va nverse proposton, t s qute crucal to provde the ways to control aerodynamc performance. In 1974, WANG Shangjn proposed the essental dea to handle the twsted blades of radal, mxed flow mpellers, and then gradually the all-overcontrolled vortex method was formed whch could control all the aerodynamc performances. Consequently, ths method has been proved qute effectve and advanced. So here, ths method wll be appled to the blade desgn of a helum turbne mpeller. The desgn crteron s that the vortex on the merdan plane was gven frst, so as to control all the flow on the merdan surface. After solvng the velocty gradent equaton, the wrap angle of the blade and the velocty dstrbuton on mean S 2 stream surface [6], namely mean stream surface, could be obtaned convenently. 2.1 Control equaton (1) Velocty gradent equaton [8 1] where, B = d(rc θ) dw m dq = AW m + B + C W m, (1) A = dq d(rc θ) dq cos(α ψ) r c, rc θ r 2 ω r 2 W m C = dh dq ω d(rc θ) T ds dq dq W m = W cos(β). + dw m sn(α ψ), Here, as shown n Fg. 5, α s the angle between the merdan streamlne and the Z-axs, n radans; ψ s the angle between the quas-orthogonal and the radal drectons, n radans; ω s the rotatonal speed, n 1/s; r c s the radus of curvature of the merdan streamlne, n m; m s the dstance along the merdan streamlne, and at the nlet the dstance s wrtten as m n ; q s the dstance along the arbtrary quas-orthogonal on the merdan plane, n m; θ s the relatve angular coordnate of the merdan stream surface, n radans; rc θ s the vortex on the merdan stream surface, and s (rc θ ) or λ at the nlet, n m 2 /s; β s the angle between relatve velocty vector and merdan plane, n radans; S s the entropy on the merdan stream surface, n J/(kg K); h s the stagnaton enthalpy at the nlet, n J/kg; (2) Mass flow equaton Ths equaton could be used to check the reasonableness of velocty feld, calculated from the velocty gradent equaton, so that the velocty could be revsed n turn. W T = N q W n = W m cos(α ψ). ρw n r θdq, (2) Here, W T s the weght flow, n kg/s; N s the number of blades; ρ s the densty of the flud helum, n kg/m 3 ; θ s the angle between blade surfaces at a gven pont, n radans. (3) Angular coordnate equaton θ = = rc θ r 2 ω r 2, (3) W m = tanβ, (4) r m m n + θ n. Here, θ n s the angular coordnate of the stream surface at the nlet, n radans. 2.2 The specfcaton of refrgerant helum physcal parameters through the expander mpeller Snce physcal parameters are nvolved n the mass flow equaton, t s necessary to specfy related thermal physcal parameters, such as the densty ρ. About the helum turbne expander mpeller, the gven nlet pressure s P =.292 MPa, and nlet temperature, T =17.45 K (here, subscrpt represents the nlet parameters of mpeller); and expected outlet pressure s P o =.118 MPa, and outlet temperature T o =13.8 K (here, subscrpt o represents the outlet parameters of mpeller). Accordng to the desgn parameters, the compresson factors of flud helum at the nlet and outlet are Z =.9929 and Z o =.9894, respectvely. Obvously, the state of the flud helum at the nlet and outlet s qute close to the deal gas. Then, we could demonstrate that the flud helum through such mpeller could be convncngly vewed as an deal gas The specfcaton of the state of deal gas Gven that the loss of cold energy s qute small through the helum turbne expander, the turbne wall generally could be consdered adabatc. So t s qute reasonable that the expanson process through the mpeller could be regarded as an sentropc process, as shown n Fg. 1, f the nternal flow loss s not taken nto consderaton. 289

3 Fg.1 Smplfed chart of the sentropc process The followng wll provde an approach to demonstrate that the flud could be vewed as an deal gas: a. Through the sentropc process, the parameter of flud helum at any grd pont could be obtaned. Inlet condton: P =.292 MPa, S =1451 J/(kg K); outlet condton: P o =.118 MPa, S o =1451 J/(kg K); b. Through the turbne mpeller, the pressure s contnuously decreased. Therefore, we could splt the pressure range nto N equal parts, namely N+1 state pont, and the pressure at every pont could be specfed, wth the entropy at every pont equal to the gven S n =1451 J/(kg K); c. The compresson factor at any pont s a functon of the gven pressure and entropy, Z n =Z(P n, S n ). So by callng Hepak, the compresson factor at the specfed pont could be obtaned. The whole process has been programmed and computed on a computer and the compresson factor at every pont s qute close to 1.. Accordngly, the flud through the helum turbne expander mpeller could be reasonably vewed as an deal gas. Then, the densty could be obtaned from formula ρ = P RT. The followng has gven the specfcaton of temperature T at any pont The specfcaton of temperature at any pont through the mpeller h = h ωλ + ω2 r 2 W 2, 2 T = T (ωλ ω2 r 2 W 2 )/C p, (5) 2 where: h s the enthalpy at any pont; T s the temperature at any pont; W s relatve velocty at any pont; T s stagnaton temperature at the nlet; C p s specfc heat capacty at constant pressure of helum. Velocty gradent equaton s derved accordng to sentropc process, then the rreversble loss could be presented n the form of pressure loss [5]. Plasma Scence and Technology, Vol.16, No.3, Mar Energy loss from pressure loss Here, only energy loss from pressure loss s ncluded at present, and the loss from the axal vortex wll be taken nto consderaton later. At frst, we assumed that the pressure loss along the streamlne ncrease lnearly, namely dp = K; where K s Lnear coeffcent; m s the length of the streamlne. Then the energy loss E loss through every flow tube could be calculated as follows, E loss = o dp ρ = o K ρ o = K ρ. (7) Then, the related numercal ntegraton program could be compled, so that the energy loss could be obtaned exactly. 2.4 Numercal procedure Here, accordng to the work condton of the second level of expanson coolng (T2), frst, the vortex dstrbuton on the merdan stream surface was specfed; then the all-over-controlled vortex method was adopted to desgn the target mpeller. The frst step n the analyss s numercal evaluaton of parameters α, β, ψ, r c, dw m /, from Eq. (1). In order to evaluate parameters α, ψ and β, related streamlne geometry should be establshed. About dw m /, n ntal calculaton, the ntal velocty feld on the whole merdan plane s consdered to be equal to the W mh calculated accordng to the weght flow of the mpeller. At the begnnng, the boundary condton, the velocty along the hub W mh could be obtaned from the followng equaton. W mh = W T ρ r l θ, (8) where: ρ s stagnaton temperature at the nlet, n kg/m 3 ; r s the mpeller radus at the nlet, n m; l s the blade heght at the nlet, n m; θ s the angle between blade surfaces at the nlet, n radans. Frstly, fxed straght lnes, namely quas-orthogonal lnes, are drawn from hub to shroud. For an ntally assumed streamlnes, each quas-orthogonal can be dvded nto some equal spaces, as shown n Fg. 3. ρ = ( T T ) 1/(γ 1) ρ [( T T )( T T )]1/(γ 1) p RT ( T ), (6) T where: ρ s stagnaton densty at the nlet, n kg/m 3 ; p s total pressure loss, n Pa; T s the temperature at the pont where the relatve velocty s m/s, n K; γ s the rato of specfc heat. Fg.2 The profle of merdan plane and quas-orthogonal lnes around the mpeller 29

4 LIU Xaodong et al.: The Desgn and Analyss of Helum Turbne Expander Impeller Accordng to prevous desgn experences, the pressure loss s p =.15 MPa, the rotatonal speed s n=147 r/mn. Other nput parameters needed are as follows: the profle coordnates θn at hub and shroud, as shown n Fg. 2, the vortex dstrbuton along hub and shroud, as shown n Fg. 3, and the angular coordnate dstrbuton along the nlet of stream surface. The vortex rcθ dstrbuton on the merdan stream surface could be acqured by lnearly nterpolatng along the quas-orthogonal lnes between the vortex on the hub and the one on the shroud [11 13], as shown n Fg. 4. Fg.3 The vortex dstrbuton on hub and shroud Secondly, we should specfy all vortex rcθ dstrbuton on the merdan stream surface. Once the vord(rcθ ) tex s specfed, then such parameters as rcθ,, d(rcθ ) could be evaluated. The detaled prncple and dq whch s used to specfy the vortex wll be gven n secton 3.1. Here, some crucal detals should be ponted out. θ) a. The dervatve of rcθ, d(rc, along the merdan drecton, at the leadng and tralng edges, must be set to zero so as to satsfy the Kutta-Joukowsky condton. b. From Eq. (4), t s possble to see that when the radus r and merdan velocty Wm are small (as on the hub streamlne), may be unacceptably hgh so that the blade surface s twsted excessvely. Therefore, to obtan reasonable wrap angle θ, the vortex rcθ on the hub should be close enough to the r2 ω at the same grd pont. Therefore, after all the coeffcents are gven, accordng to the boundary condton Wmh along the hub, the velocty gradent equaton could be solved on the bass of the Runge-Kutta method so that the whole velocty feld of the turbne mpeller could be obtaned. As to the convergence crtera, the velocty feld must meet the contnuty equaton by requrng that the calculated weght flow across any quas-orthogonal from hub to shroud s equal to the known weght flow through the helum turbne expander. For ths requrement, the densty has been specfed by Eq. (6). Once the velocty output could not meet the contnuty equaton, the boundary condton for velocty Wmh needs be modfed slghtly. 3 Contours of specfed rcθ dstrbuton on merdan 3.2 Results and analyss Wth the above parameters gven, the whole process has been programmed and computed usng a computer. After convergence, the flow on the merdan stream surface of the mpeller s shown n Fg. 5, whch shows that the real streamlnes are obvously dfferent from the ntally assumed ones. Fg.5 Flow on merdan plane of the mpeller after convergence The mpeller desgn of helum turbne expander 3.1 Fg.4 plane The wrap angle of the mpeller blade Through numercal calculaton, the wrap angle of the mpeller blade could be calculated. The wrap angle at 3 partcular streamlnes was shown n Fg. 6, where the length of streamlne s non-ensonalzed by mhub, the length of the hub streamlne. Obvously, along quas-orthogonal lnes, the wrap angle s reduced at frst, and then ncreased, leadng to a twsted blade. However, we should notce that the blade could not twst excessvely. Related parameters Desgn parameters: Mass flow, W T =.1 kg/s; Stagnaton temperature at the nlet, T =21.42 K; Stagnaton pressure at the nlet, P =.4867 MPa. 291

5 Plasma Scence and Technology, Vol.16, No.3, Mar. 214 conforms to the real physcal process. Snce the pressure surface s the work surface and the sucton surface s not, so the pressure on the sucton s smaller than that on the pressure surface. Moreover, the relatve velocty dstrbutons at the shroud and on the mean surface of revoluton are smlar to the one at the hub. However, negatve velocty has been found n Fg. 7, whch ndcates an eddy, resultng n turbulence and mxng losses. Further studes should focus on how to reduce the eddy loss. The relatve veloctes at hub and shroud, along the mean streamlne, and on the mean stream surface are shown n Fg. 8(a). From Fg. 8(b), the relatve velocty at the shroud s the largest, and the one at the hub s the smallest, whch also conforms to the real physcal process. The contours of relatve veloctes on the mean stream surface non-ensonalzed by C, the speed of sound at the nlet, are shown n Fg. 8(b), whch s also consstent wth Fg. 8(a) Fg.6 The wrap angle of mpeller of helum turbne expander The flow around the desgned mpeller At the hub, mean surface of revoluton and shroud, the relatve velocty of sucton, the merdan stream surface, and the pressure are respectvely shown n Fg. 7. Fg.8 (a) Relatve velocty dstrbuton along 3 specfc streamlnes of mean S2 stream surface, (b) Contours of relatve veloctes on mean S2 stream surface Analyss of pressure, temperature dstrbutons of the mpeller Through the numercal calculaton, we analyze the pressure dstrbuton of the mpeller. The pressure dstrbuton s shown n Fg. 9(a). From Fg. 9(a), the pressure level s always decreasng along the streamlne, complyng wth the work prncple of the turbne expander. Then pressure contours non-ensonalzed by P, the stagnaton pressure at the nlet, are shown n Fg. 9(b). Fg.7 (a) Relatve velocty dstrbuton at hub, (b) Relatve velocty dstrbuton on mean surface of revoluton, (c) Relatve velocty dstrbuton at shroud From the above dagram, the relatve velocty on the sucton surface s the largest at the hub, and the relatve velocty on the pressure surface s the smallest, whch 292

6 LIU Xaodong et al.: The Desgn and Analyss of Helum Turbne Expander Impeller the dstrbutons of velocty, temperature, and pressure. In addton, n ths paper, the boundary layer effect s not taken nto consderaton at the moment. 4 Summary and concluson Here, based on the nverse proposton, the all-overcontrolled vortex method was adopted to desgn the turbne expander mpeller, and a new method was utlzed to specfy the state of the flud helum. To optmze the blade shape, the compressble turbulent boundary layer effect wll be consdered n further studes. References 1 Fg.9 (a) The pressure dstrbuton on mean S2 stream surface, (b) The contours of pressure on mean S2 stream surface 2 Accordngly, the temperature dstrbuton s shown n Fg. 1. From the above fgures, the pressure and temperature at the nlet are.292 MPa and K, and then reduced to the values at outlet,.126 MPa and K, through the mpeller Fg.1 surface 8 The temperature dstrbuton on mean S2 stream Analyss of pressure loss Z o Z o K dp = = Based on Eq. (7), E = p p Z o Z o K = K, the energy loss of every p p(m) stream tube E could be obtaned va numercal cal culaton, then the total loss of the mpeller s E =ΣE. So the effcency of the mpeller s η = E E =1. h ho Cp (T To ) Based on the output, the effcency can be obtaned as η=99.5%. Therefore, the pressure loss s acceptable for the desgn of the mpeller. Based on the above analyss, the streamlne curvature method could be modfed to desgn the mpeller of the helum expander, and the numercal results are qute consstent wth the real physcal process, such as Ba Hongyu. 22, Thermodynamc Analyss and desgn of the helum refrgeraton system for HT-7U superconductng tokamak [Ph.D]. Insttute of plasma physcs, Chnese Academy of Scences, Hefe (n Chnese) Ba Hongyu, B Yanfang, Zhu Png, et al. 26, Fuson Engneerng and Desgn, 81: 2597 Ba Hongyu, B Yanfang, Wang Jnrong, et al. 22, Fuson Scence and Technology, 42: 162 J Guanghua. 1989, Turbo-expander. Chna machne Press, Bejng (n Chnese) Wang Shangjn. 1991, Three-ensonal flow theory and applcaton to Centrfugal compressor. X an jaotong Unv. Press, X an (n Chnese) Wu Chung-Hua. 1952, A General Theory of ThreeDmensonal Flow n Subsonc and Supersonc Turbomachnes of Axal-, Radal-, and Mxed-Flow Types. NACA TN 264 Borges J E. 199, A three-ensonal nverse method for turbomachnery: Part 1-Theory. ASME J Turbomachnery, 11: 346 Katsans, Theodore. 1964, Use of Arbtrary QuasOrthogonals for Calculatng Flow Dstrbuton In the Merdonal Plane Of A Turbomachne. NASA TN D2546 Mchael R Vanco. 1972, FORTRAN Program for Calculatng Veloctes n the Merdonal Plane of a Turbomachne Ⅰ- Centrfugal Compressor, NASA TN D761 John D Stantz and Vasly D Pran. 1951, A Rapd Approxmate Method for Determnng Velocty Dstrbuton on Impeller Blades of Centrfugal Compressors. NACA TN2421 Zangeneh M, Goto A, and Takemura T. 1996, Suppresson of secondary flows n a mxed-flow pump mpeller by applcaton of three-ensonal nverse desgn method. 1: Desgn and numercal valdaton. Transactons of ASME, 118: Zangeneh M. 1991, Internatonal Journal of Numercal Methods n Fluds, 13: 599 L Chao, Zhang Rucheng. 23, Journal of Power Engneer, 23: 2845 (n Chnese) (Manuscrpt receved 19 July 212) (Manuscrpt accepted 25 March 213) E-mal address of LIU Xaodong: luspace@163.com 293