Investigation on the Flow in a Rotor-Stator Cavity with Centripetal Through-Flow

Transcription

1 Internional Journal Turbomachery Propulsion Power Article Investigion on Flow a Ror-Star Cavity Centripetal Through-Flow Bo Hu *, Dieter Brillert, Hans Josef Dohmen Friedrich-Karl Benra ID Department Mechanical Engeerg, University Duburg-Essen, Duburg, Germany; dieter.brillert@uni-due.de (D.B.); hans-josef.dohmen@uni-due.de (H.J.D.); friedrich.benra@uni-due.de (F.-K.B.) * Correspondence: bo.hu.1987@stud.uni-due.de; Tel.: Th paper an extended version our paper Proceedgs European Turbomachery Conference ETC12, 2017, Paper No. 97. Academic Edir: Claus Sieverdg Received: 31 July 2017; Accepted: 6 Ocber 2017; Publhed: 19 Ocber 2017 Abstract: Daily Nece dtguhed four flow s an enclosed ror-star cavity, which dependent on circumferential Reynolds number dimensionless axial gap width. A diagram different flow s cludg respective mean priles both tangential radial was developed. coefficients different flow s have also been correled. In centrifugal pumps turbes, centripetal through-flow quite common outer radius impeller impeller eye, which has a strong fluence on radial pressure dtribution, axial thrust frictional rque. fluence centripetal through-flow on cavity flow different circumferential Reynolds numbers dimensionless axial gap width not sufficiently vestiged. It also quite important convert 2D Daily Nece diagram 3D by troducg through-flow coefficient. In order vestige impact centripetal through-flow, a test rig designed built up University Duburg-Essen. design test rig described. impact above mentioned parameters on prile, pressure dtribution, axial thrust frictional rque presented analyzed th paper. 3D Daily Nece diagram troducg through-flow coefficient also organized th paper. Keywords: ror-star cavity; centripetal through-flow; axial thrust; frictional rque 1. Introduction In radial pumps turbes, leakage flow (centripetal through-flow) quite common outer radius impeller impeller eye, which has a major impact on pressure dtribution, axial thrust (F a ) frictional rque. Von Kármán [1] Cochran [2] gave a solution ordary differential equion steady, axymmetric, compressible flow. Daily Nece [3] examed flow an enclosed rotg dk both analytically experimentally. y dtguhed four flow s, 1, by correlg different empirical equions moment coefficients. Kurokawa et al. [4 6] studied cavity flow both centrifugal centripetal through-flow. Schlichtg Gersten [7] organized an implicit relion based on Goldste [8] moment coefficient under turbulent flow conditions. Poncet et al. [9] studied centripetal through-flow a ror-star cavity obtaed an equion core swirl rio K based on local flow re coefficient (C qr ) Bchelor type flow [10]. For Bchelor type flow, centrifugal dk layer centripetal layer separed by a central core. Debuchy et al. [11] derived an explicit equion K which valid over a wide range C qr. Launder et al. [12] provided a review current understg stability ptern th creed ror-star cavities leadg transition eventually turbulence. Recent experimental Int. J. Turbomach. Propuls. Power 2017, 2, 18; doi: /ijtpp

2 Int. J. Turbomach. Propuls. Power 2017, 2, th Int. J. Turbomach. creed Propuls. ror-star Power 2017, cavities 2, 18 leadg transition eventually turbulence. Recent 2 17 experimental vestigions up circumferential Reynolds numbers Re out through-flow have been conducted by Coren et al. [13], Long et al. [14] Barabas et al. [15]. scope vestigions present upstudy circumferential Reynolds 1. numbers Re out through-flow have been conducted by Coren et al. [13], Long et al. [14] Barabas et al. [15]. scope present study Flow s accordg Daily Nece [3]. 1. Flow s accordg Daily Nece [3]. ma ma dimensions ror-star cavity cavity illustred 2. Th 2. Th study study focuses focuses on on fluence non-pre-swirl centripetal through-flow on on cavity cavity flow. flow. Based Based on on previous studies, C D (through-flow coefficient) may have have a large a large fluence fluence on on moment moment coefficient, coefficient, noted asnoted C M. Uncerties as still ext effect C. Uncerties still ext effect D onc M on different different Re G Re G (dimensionless axial axial gap). gap). Th Th study study aimed aimed provide more more crease daset better underst fluence above parameters on on C p (pressure coefficient), C F (axial (axial thrust thrust coefficient) C M.. defitions defitions significant significant dimensionless parameters parameters th study th study given given Equions Equions (1a) (1k). (1a k). 2. Ma dimensions test rig. a: Hub radius; b: outer radius dk; r: radial 2. Ma dimensions test rig. a: Hub radius; b: outer radius dk; r: radial coorde; coorde; s: axial gap front chamber; s s: axial gap front chamber; : axial gap b : axial gap back chamber; t: thickness dk; back chamber; t: thickness dk; z: axial z: axial coorde. coorde. Re Ω b2 v Re ϕ Ω r2 v G s b (1a) (1a) (1b) (1b) (1c)

3 Int. J. Turbomach. Propuls. Power 2017, 2, (1c) Int. J. Turbomach. Propuls. Power 2017, 2, C m D 2 ( ) d µ b b 2 π (p C F b p) rdr ρ ω 2 b 4. a 2. (1d) (1d) (1e) (1e) (1f) C qr Q Re ϕ π Ω r 3 (1g)(1f) ζ z s x r b 2 C M 2 M ρ Ω 2 b 5 p p ρ Ω 2 b 2 C p p (x 1) p (1) ()(x) (1g) (1h) (1h) (1i) (1i) (1j) (1j) (1k) (1k) 2. oretical Analys 2. oretical Analys In th study, Q (volumetric through-flow re), C D C qr negive In th study, Q (volumetric through-flow re), centripetal through-flow. Usg a two-component Laser Doppler Anemometer negive (LDA) system, centripetal through-flow. Usg a two-component Laser Doppler Anemometer (LDA) system, Poncet Poncet et al. [9] correled Equion (2a) evalue core swirl rio K ( rio angular et al. [9] correled Equion (2a) evalue core swirl rio K ( rio angular fluid th dk ζ 0.5) centripetal through-flow when C qr 0.2. fluid th dk 0.5) centripetal through-flow when Debuchy et al. [11] determed Equion (2b) calcule K a 0.2. Debuchy et wider range C qr 0.5 al. [11] determed Equion (2b) calcule K a wider range a two-component LDA system. Equion (2b) smaller 0.5 a than those two-component LDA system. Equion (2a) large Cqr, Equion (2b) smaller than those Equion compd 3. (2a) large, compd 3. K 2 ( 5.9 C qr ) (2a) (2a) [ ] Cqr K (2b) (2b) e ( 1.45C qr) Comparon Comparon Equion Equions (2a,b). (2a) K: core (2b). swirl K: core rio swirl rio 0.5; ζ : 0.5; local C flow qr : local re flow coefficient; re coefficient; Eq: equion. Eq: equion. A number studies, such as those by Kurokawa et al. [6], Poncet et al. [9], Coren et al. [13], Barabas et al. [15], show th pressure dtribution along radius dk can be

4 Int. J. Turbomach. Propuls. Power 2017, 2, estimed core swirl rio K both out through-flow. Will et al. [16 18] determed Equion (3) evalue pressure dtribution along radius dk compressible, steady flow. It obtaed directly radial momentum equion when turbulent shear stress neglected. In a ror-star cavity, cross sectional a changes radial direction. Consequently, pressure must also change sce mean changes radial direction accordg contuity equion. ( ) p r vϕ 2 ρ v r v r ρ K 2 Ω 2 ρ Q r + 2 r r 4 π 2 s 2 r 3 (3) difference ce on both sides dk ma source axial thrust, noted as F a, calculed Equion (4a). F a f (ce on front dk; calculed Equion (4b)) C F f (C F on front ) respectively represent ce thrust coefficient on front dk ( front chamber, 2), while F ab (ce on back dk; calculed Equion (4c)) C Fb those on back dk ( back chamber). a p b represent radius hub (see 2) pressure x 1, respectively. back chamber (G 0.072), 2, supposed be an enclosed cavity. C Fb (C F on back ) obtaed when C D 0 axial gaps both cavities have same size different Re (under th condition C F f C Fb ). After obtag those, C F f different C D can be calculed Equion (5). F a F ab F a f F a f π p b b 2 C F f ρ Ω 2 b 4 ( F ab π p b b 2 a 2) ( C Fb ρ Ω 2 b 4 a 4) (4a) (4b) (4c) 3. Test Rig Design Experimental Set-Up C F f F a + C Fb ρ Ω 2( b 4 a 4) + πp b a 2 ρ Ω 2 b 4 (5) design test rig 4. cross section test rig depicted 4a. centripetal through-flow (volumetric through-flow re Q), by black arrows 4a, supplied wer by a pump system. view along A direction sketched 4b. shaft sealg back cavity depicted 4c (r seal 10 mm). A picture test rig 4d. shaft driven by an electric mor. A frequency converter used adjust speed rotion (0~2500/m) absolute uncerty 7.5/m. In th study, only axial gap front chamber changed by stallg six sleeves different length. re 24 channels guide vane (stead entirely open periphery) get more unim centripetal through-flow, 4b. a each channel m 2. In th study, channels guide vane radial directed (see 4b). Or parameters experiments th study given Table 1. transducers test rig clude two pressure transducers (36 pressure tubes), a rque transducer three tension compression transducers. A thrust ple fixed by a ball bearg a nut both sides convey axial thrust tension compression transducers. A lear bearg used mimize frictional restance durg axial thrust measurements. Durg measurements axial thrust, calibrions axial thrust transducers permed when changg axial gap width front chamber. When measurg rque, shaft out dk rotg different speeds rotion subtracted. measured R z dk 1 µm. R z on all or s test rig below 1.6 µm.

5 Int. J. Turbomach. Propuls. Power 2017, 2, Int. J. Turbomach. Propuls. Power 2017, 2, Int. J. Turbomach. Propuls. Power 2017, 2, Test rig design. : axial thrust; radius shaft seal. 4. Test rig design. Fa : axial thrust; rseal : :radius shaft seal. Table 1. Parameters experiments. Table 1. Parameters experiments. b (mm) b (mm) (/m) Q (m3/s) s (mm) 4.n Test rig design. : axial thrust; 110 0~ ~ ~0 n (/m) Q (m /s) s (mm) a (mm) t (mm) :(mm) radius shaft seal sb (mm) a (mm) n: speed rotion; Q: volumetric through-flow re. Table 1. Parameters experiments ~0 t (mm) 110 0~2500 2~ /s) measurements axial Q thrust clude two steps. cavities each step depicted b (mm) n (/m) (m s (mm) (mm) a (mm) t (mm) n: speed rotion; Q: volumetric through-flow re. 5. first measure axial ce 110 step0~ ~0 2~8 imposed 8 by 23drive end 10 mor when shaft out dk rotg different speed rotion air. For second step, all n: speed rotion; Q: volumetric through-flow re. measurements axial thrust clude two steps. cavities each step depicted modified by subtractg above obtaed first step accordg speed measure axial thrust clude two steps. cavities end eachstep depicted 5.rotion. measurements first step axial ce imposed by drive mor shaft head 5b considered as a part front ( a when ). 5. first step measure axial ce imposed by drive end mor when shaftout dk rotg different speed rotion air. For second step, all geometry nut ignored. n, thrust coefficient on a sgle can be shaft out dk rotg different speed rotion air. For second step, all Equion (5). calculed modified by subtractg above obtaed first step accordg speed modified by subtractg above obtaed first step accordg speed rotion. shaft head 5b considered partfront front ( a ).Ω b2 ). rotion. shaft head 5b considered as as aapart ( a geometry nutnut ignored. thrust thrust coefficient a sgle can be geometry ignored.n, n, coefficient on on a sgle can be calculed Equion (5).(5). calculed Equion 5. Cavities measure axial thrust. 5.5.Cavities Cavities thrust. measure measure axial axial thrust.

6 Int. J. Turbomach. Propuls. Power 2017, 2, relive error, noted as e T, pressure transducers 1% (full scale (FS)). value e T rque transducer 0.1% (FS). value e T axial thrust transducers 0.5% (FS). All experimental ensemble average 1000 samples. uncerties measured, noted as N, differences between real measured. y estimed root sum squd method. uncerties calculed Equion (6). N T uncerty due transducers. N D uncerty due da acquition system. measurg range (M r ) rque meter 0~10 Nm. measured range pressure transducer 0~2.5 bar (absolute pressure). measured range thrust transducers 100~100 N. put voltage signals followg ranges: 0~10 V pressure transducers rque transducer, 10 V~10 V axial thrust transducers. absolute accuracy da acquition system ( NI USB-6008) 4.28 mv th study. rom noe zero order uncerty neglected because y very small. dtributions considered as normal dtributions normal dtribution coefficient selected as 1.96 (95% confidence level). n T represents number transducers used obta one result ger. To evalue F a M on front, tal, on back (obtaed when C D 0 s s b ) when shaft rotg out dk measured transducers. Hence, measurg times obta one result, noted as n M, 3 axial thrust, frictional rque (M) Re (by measurg n). uncerties measurements summarized Table 2. N N 2 T + N 2 D ; N T n T n M (e T M r ) ; N D n T n M (e D M r ) (6) Table 2. Uncerty analys measurements. p (Bar) F a (N) M (Nm) Re C D N n T n M N: uncerty measured ; n T : number transducers; n M : measurg times obta one result; p: pressure; M: frictional rque; Re: global circumferential Reynolds number; C D : through-flow coefficient. 4. Numerical Simulion To predict cavity flow, numerical simulions carried out usg ANSYS CFX 14.0 code [15]. re 24 channels guide vane. Considerg axial symmetry problem, a segment (15 degree) whole doma modeled a rotional periodic condition applied. Structured meshes genered ICEM doma numerical simulion when G depicted yellow color 6. simulion type set as steady ste. Barabas et al. [15] found th simulion shear stress transport (SST) k ω turbulence model combion scalable functions good agreement measured pressure a ror-star cavity air. deviions pressure measurement less than 1%. Hence, th study, same turbulence model functions used. conditions let outlet pressure let mass flow outlet, respectively. pressure let set accordg pressure sensor pump outlet. convergence criteria all numerical simulions set as 10 5 maximum type. turbulent numeric set as second order upwd. maximum value y + all simulion models 13.4.

7 Int. J. Turbomach. Propuls. Power 2017, 2, Int. J. Turbomach. Propuls. Power 2017, 2, Doma Doma numerical numerical simulion simulion when when G Results Dcussion Simulion Simulion Results Results Velocity Velocity Dtributions Dtributions priles priles sensitive sensitive condition condition let. let. priles priles front front chamber chamber dcussed dcussed th th part part because because re re a small small jet jet flow flow through through each each channel channel guide guide vane. vane. All All velocities velocities made made dimensionless dimensionless by dividg by dividg m by m Ω by b.. V z positive when positive y when have y a direction have a direction dk. dk. priles three priles radial positions three radial Re positions Re 1.9 G (wide G gap) (wide gap) 7. dimensionless 7. dimensionless radial velocities radial velocities not exactly not zeroexactly zero central cores, central cores, 7a c. From7a c. dtribution From dtribution tangential, tangential re, central re cores central all cores vestiged all radial vestiged positions radial where positions where V ϕ almost constant, almost constant, 7d f. At x d f. At x , x 0.79, tangential tangential smaller ζ smaller 0.5 when 0.5 CD crease when crease , 3787 depicted 5050, depicted 7d,e. priles 7d,e. priles special, because special, accordg because accordg mer mer (such as (such Poncet as et al. [9] Poncet Debuchy et al. [9] et al. Debuchy [11]), et crease al. [11]), CD crease will result will an crease result an crease core swirl rio core K swirl centripetal rio K through-flow. centripetal through-flow. n, tangential n, tangential should crease should stead crease decrease. stead Probably decrease. th Probably can be tributed th can be tributed jet flow through jet flow channel through channel let, which let, stronger which large stronger CD. At large x 0.955,. At x 0.955, V z become smaller become larger smaller CD general, larger general, 7g. direction 7g. Vz direction dk wards dk wards C D 0 C D x 0.955, while 1262 it x 0.955, while it wards dk wards CD 3787 dk CD re axial circulions re axial fluid circulions front chamber. fluid directions front chamber. axial circulions, directions however, axial strongly circulions, fluenced however, by strongly fluenced CD, depicted by 7g i., depicted 7g i. priles priles three three radial radial coordes coordes Re Re G (small (small gap) gap) dimensionless dimensionless radial radial velocities velocities V r vary vary along along ζ,, 8a c. 8a c. V r decrease decrease crease crease CD general. general. tangential tangential Vϕ decreases decreases constantly constantly crease crease ζ,, which which charactertic charactertic,, 8d f. 8d f. At At x x x x 0.79, 0.79, tangential tangential much much smaller smaller large large CD,, 8d. 8d. reason reason th th impact impact jet flow jet flow let let becomes becomes greer greer smaller smaller G. G. priles priles V quite z quite different different x x g, compd 8g, compd those those 7g. 7g. V z less less than than those those 7h,i. 7h,i. V z dice dice th th axial axial circulions circulions fluid fluid sensitive sensitive G. G.

8 Int. J. Turbomach. Propuls. Power 2017, 2, Int. J. Turbomach. Propuls. Power 2017, 2, Velocity priles Re G (wide gap). G: dimensionless axial gap. 7. Velocity priles Re G (wide gap). G: dimensionless axial gap.

9 Int. J. Turbomach. Propuls. Power 2017, 2, Int. J. Turbomach. Propuls. Power 2017, 2, Velocity priles Re (small gap). 8. Velocity priles Re G (small gap) Pressure Dtributions 5.2. Pressure Dtributions Due construction geometry, re no pressure tube x 1. closest tube Due construction geometry, re no pressure tube x 1. closest tube x pressure x 1 (reference pressure) taken numerical simulion. Sce x pressure x 1 (reference pressure) taken numerical simulion. p numerical simulion x close those experiments, small Sce p numerical simulion x close those experiments, errors neglected. small errors neglected. pressure coefficient positive because pressure drops wards shaft. In pressure coefficient C 9, plotted p positive because pressure drops wards shaft. versus dimensionless radial coordes three In 9, C p plotted versus dimensionless radial coordes three Re G. experimental show th crease creasg. Re G. experimental show th C p crease creasg CD. decrease crease Re G general. When Re Re 2.79 C p decrease crease Re G general. When Re Re , 10 6, uncerties respectively , which very small uncerties C p respectively , which very small compd compd measured. Hence, y neglected 9d i. measured. Hence, y neglected 9d i.

10 Int. J. Turbomach. Propuls. Power 2017, 2, Int. J. Turbomach. Propuls. Power 2017, 2, Influence C D on C p (pressure coefficient) dependence Re G Axial Thrust Coefficient Based on axial thrust measurements (direction axial thrust see 4a), a correlion C F f determed, given Equion (7). Equion (7) constent experimental, plotted 10. With crease, CD, CF f crease, which can be tributed drop p. C F f smaller higher G Re ln() [ ] ln() 0.67 (7) C F f ln(re) e ( C D ) [0.122 ln(g) 0.67] (7)

11 Int. J. Turbomach. Propuls. Power 2017, 2, 18 Int.J. J.Turbomach. Turbomach. Propuls. Power 2017, Int. Propuls. Power 2017, 2, 2, Mean curves dependence Re G. : axial thrust coefficient; : on Mean CF f -C-D 0 curves dependence ReRe G. G. CF : axial thrust coefficient; CF f : C F on Mean curves dependence : axial thrust coefficient; : on front. front front D Daily Nece Diagram D 3DDaily Daily Nece NeceDiagram Diagram typical tangential priles (merged dk layer tangential priles (merged dk typical typical tangential priles layer (merged dk layer layer given layer) IV (separed dk layer) layer) IV (separed dk layer layer) given layer) IV (separed dk layer layer) given Typical tangential priles : (a) (b) IV. 11. Typical tangential priles : (a) (b) IV. 11. Typical tangential priles : (a) (b) IV. Based Basedon on simulion simulion tangential tangential,,part part Daily Daily Nece Necediagram diagram Based on simulion tangential, part Daily Nece diagram (see 1) extended 3D by dtguhg tangential priles (see 11) (see 1) extended 3D by dtguhg tangential priles (see 11) 0 (see 1) extended 3D by dtguhg tangential priles (see 11) xx0.955, D 5050, 5050, 0.955,xx xx scope scope th th study study followg followgparameter parameterranges: ranges: C 6 7 namely 5050, x 0.955, Re scope study followg parameter ranges: Gth y cegorized two s, x Re 3.17 x G y cegorized two s, namely 6 Re G y cegorized two s, namely (below dtguhg les) IV (above dtguhg les). Currently, (below dtguhg les) IV (above dtguhg les). Currently, five 0 (above dtguhg les). Currently, five dtguhg (below dtguhg les) five les found different 12b., CDIV 12b. dtguhg dtguhgle le dtguhg les found different 0, 12b. dtguhg le les found different Cdtguhg 0 almost equal th Daily Nece [3]. dtguhg les become steeper D 0 almost equal th Daily Nece [3]. dtguhg les become steeper 0 0 almost equal th Daily Nece [3]. dtguhg les become steeper higher C. approxime dtguhg drawn through les, higher D. approxime dtguhg drawn through les,. approxime dtguhg drawn IV, through les,near higher 12a.Below Below above 12a. above IV,respectively. respectively. Near 12a. Below above IV, respectively. Near

12 Int. J. Turbomach. Propuls. Power 2017, 2, 18 Int. Int. J.J. Turbomach. Turbomach. Propuls. Propuls. Power Power 2017, 2017, 2, 2, dtguhg, re mixg zone where IV coext front dtguhg,, re re aaamixg mixgzone zonewhere where IV IVcoext coext front front dtguhg chamber. In th study, it not plotted 12. chamber. In th study, it not plotted 12. chamber. In th study, it not plotted Part 3D 3D Daily Nece diagram: (a) Dtguhg (b) Dtguhg les. Nece diagram: (a) (b) 12. Part Part 3D Daily Daily Nece diagram: (a) Dtguhg Dtguhg (b) Dtguhg Dtguhg les. les Moment Coefficient Moment Moment Coefficient Coefficient Accordg experimental Han et al. [19], moment coefficient on cylder Accordg Han moment Accordg (C experimental Han et et al. [19], dks. moment coefficient coefficient on on dk ) can be estimed Equion (8)al. [19], smooth Mcylexperimental cylder )) can cylder dk dk (( can be be estimed estimed Equion Equion (8) (8) smooth smooth dks. dks. 222 Mcyl C Mcyl ρ Ω2 b π t b (( lg Ωυ b )).. (8) (8) (8) When,, G ( IV) WhenC 0 0,00 G G ( ( ) G( 0.072IV) ( IV) When C ( ) ) G 0.072G compd D M compd 13a 13b, differences between experimental compd 13a 13b, respectively. respectively. differences between experimental 13a,b, respectively. differences between experimental those those correlions by Daily Nece [3] both IV, those correlions by Daily Nece [3] both IV, correlions by Daily Nece [3] both IV, colossal. k s equivalent colossal. equivalent roughness, defed Equion (11). Accordg Schlichtg colossal. equivalent roughness, defed Equion (11). Accordg Schlichtg roughness, defed Equion (11). Accordg Schlichtg et al. [7], k sl ( limition ( hydraulic be (12). et ( limition limition estimed hydraulic smooth) smooth) can can be estimed estimed Equion Equion dk dk et al. al. [7], [7], smooth) hydraulic can be Equion (12). dk ree can be(12). considered ree can be considered hydraulic smooth when 38.2 μm. To expla wide gap, ree can be considered hydraulic smooth when 38.2 μm. To expla wide gap, hydraulic smooth when Rz 38.2 µm. To expla wide gap, a rougher dk aa rougher dk (( 22 μm) also plotted comparon. much closer rougher dk 22 μm) also plotted comparon. much closer (Rz 22 µm) also plotted comparon. much closer those equion those those equion equion by by Daily Daily Nece Nece [3]. [3]. Hence, Hence, differences differences can can be be tributed tributed by Daily Nece [3]. Hence, differences can be tributed difference roughness. difference roughness. Equions (9) (10) ree determed sfy difference roughness. Equions (9) (10) ree determed sfy Equions (9) (10) ree determed sfy experimental (Rz 1 µm experimental experimental (( 11 μm μm dk). dk). dk). 13. Comparon (moment coefficient) Comparon Comparon C M (moment (momentcoefficient) coefficient) GG G GG G CD

13 Int. J. Turbomach. Propuls. Power 2017, 2, Int. J. Turbomach. Propuls. Power 2017, 2, (. ) (9) C M G 6 1 Re 1 4 [e ( C D ) ] (9) G Re (. ) C M G 10 1 Re 5 1 [e ( C (10) D ) ] (10) k s π ε, 8, ε R (11) z (11) ν k sl (12) (12) (1 (1 ) K) r Ω To To troduce troduce fluence fluence C D on on moment moment coefficient, coefficient, C M both both experiments experiments equions equions plotted plotted versus versus Re Re With With crease crease Re, Re, flow flow may may change change IV (see IV (see dtguhg dtguhg les les 12). For 12). G For G G 0.036, G most 0.036, most flow s flow s close those close Equion those (9) Equion general, (9) general, 14a,b. flow s change IV crease Re C D 0 C 14a,b. flow s change IV crease Re D G G b. For G b. For G 0.072, most G 0.072, flow most s flow s IV IV close those close Equion those (10) Equion general, depicted (10) general, depicted 14c,d. 14c,d. C M equions good agreement those experiments. C M crease crease C equions good agreement those experiments. crease D, while crease decrease, while crease decrease Re. At crease large Re. At Re, large impact C Re, D on C impact M becomes lesser. At same on becomes lesser. At CD same, tersection, pots tersection curves pots Equion curves (9) those Equion (9) Equion those (10) close Equion those (10) close 12b. those difference 12b. can be tributed difference can be extence tributed mixg extence zone. mixg zone. 14. Curves C 14. Curves M dependence C dependence D different Re G. different Re G. Equions (9) (10) dtguhg les should be equal. C M3 /C / M4 ( (C M /C M IV) a non-dimensioned gap width G dtguhg les (see 12b) presented differences, tributed

14 Int. J. J. Turbomach. Propuls. Power 2017, 2, 2, Int. J. Turbomach. Propuls. Power 2017, 2, extence mixg zone, less than 4%. dice th dtguhg les ( extence mixg zone, less than 4%. dice th dtguhg les ( extence 12), Equions mixg (9) zone, (10) less reasonable. than 4%. dice th dtguhg les 12), Equions (9) (10) reasonable. ( 12), Equions (9) (10) reasonable. 15. Results / ( / IV) dtguhg les Results C M3 /C / M4 (C ( M /C / M IV) dtguhg les. An example provided on applicions th paper. Shi et al. [20] studied An example provided on applicions th paper. Shi et al. [20] studied axial An thrust example a sgle provided stage well on pump applicions based both numerical simulion th paper. Shi experiments. et al. [20] studied axial thrust a sgle stage well pump based on both numerical simulion experiments. pressure axialactg thrust a sgle impeller stage well pump based 16a. onbased both numerical on Equion simulion (7), thrust coefficient experiments. pressure actg impeller 16a. Based on Equion (7), thrust coefficient pressure 1 actg 2 can impeller be calculed when leakage 16a. Based flow onestimed. Equion (7), volumetric thrust coefficient leakage 1 2 can be calculed when leakage flow estimed. volumetric leakage through-flow 1 re considered 2 can be calculed as 5% when flow re leakage flow pump. estimed. ce volumetric impeller leakage eye ( through-flow re considered as 5% flow re pump. ce impeller eye ( through-flow 3) re numerical considered simulion. as 5% n, flow axial re thrust pump. impeller ce can be calculed impeller when eye 3) numerical simulion. n, axial thrust impeller can be calculed when ( axial ce 3) numerical shaft simulion. estimed. y n, predicted axial thrust ce on impeller all s can be calculed impeller when axial ce shaft estimed. y predicted ce on all s impeller axial shaft ce calcule shaft axial estimed. thrust. y axial predicted ce ce shaft on all obtaed s different flow impeller re shaft calcule axial thrust. axial ce shaft obtaed different flow re shaft simulion calcule by axial Shi thrust. et al. [20]. axial maximum ce difference shaft obtaed between ir different simulion flow simulion by Shi et al. [20]. maximum difference between ir simulion re measurements simulion by axial Shithrust et al. [20]. 5.9%. maximum difference between ir simulion measurements axial thrust 5.9%. plotted versus plotted versus measurements axial thrust 5.9%. F 16b. experimental ab F a f plotted versus C D 16b. experimental obtaed by subtractg ces on obtaed by subtractg ces on rest 16b. s experimental ( numerical simulion). F ab F a f obtaed byequion subtractg (7) ces better onagreement rest rest s ( numerical simulion). Equion (7) better agreement s ( numerical simulion). Equion (7) better agreement experimental than those equion by Kurokawa et al. [4] when experimental than those equion by Kurokawa et al. [4] when ranges ranges experimental than those equion by Kurokawa et al. [4] when C D ranges Axial Axialthrust a a centrifugal sgle sgle stage stage well well pump pump [20]: [20]: (a) Pressure (a) Pressure dtribution dtribution (b) 16. Axial thrust a centrifugal sgle stage well pump [20]: (a) Pressure dtribution (b) Comparon (b) Comparon F ab F a [20]. f [20]. F ab : ce : ce on on back back dk; F a f : : ceon on front Comparon [20]. : ce on back dk; : ce on front dk. dk. re re still still some some limitions th th work. work. correlions limited non-preswirswirl centripetal centripetal through-flow. through-flow. centripetal centripetal centripetal leakage leakage flow radial pumps turbes, leakage flow flow radial pumps radial pumps turbes, however, turbes, however, contas a certa amount angular momentum, which deserves furr vestigion. All however, acontas certa a amount certa amount angular angular momentum, momentum, which which deserves deserves furr furr vestigion. vestigion. All All experimental experimental obtaed smooth dk ( experimental obtaed obtaed smooth smooth dk (R z dk 1 µm). ( 1 μm). applicions 1 μm). applicions applicions equions equions still limited all fluenced by roughness equions still limited still because limited all because all fluenced by fluenced roughness by dks. roughness Some more dks. Some more will be presented rough dks equions will be modified by dks. Some more will be presented rough dks equions will be modified by

15 Int. J. Turbomach. Propuls. Power 2017, 2, will be presented rough dks equions will be modified by troducg impact roughness dks next step. Currently, dtguhg les IV obtaed by evalug tangential flow component based on numerical simulion. Th will be put on an experimental level by measurg components both tangential radial direction a Laser Doppler Velocimetry (LDV) system future. 6. Conclusions fluence centripetal through-flow on, radial pressure dtribution, axial thrust frictional rque a ror-star cavity different axial gaps illustred be strong. A correlion determed, which enables predict fluence G, Re C D on thrust coefficient C F a smooth dk (R z 1 µm). For first time, part 3D Daily Nece diagram obtaed by dtguhg tangential priles. Currently, flow s cagorized two s, namely IV. Five dtguhg les approxime dtguhg presented. Two correlions determed predict fluence C D on C M two s good accuracy smooth dk (R z 1 µm). At dtguhg les, two equions very close. Usg equions axial thrust coefficient moment coefficient, fluence centripetal through-flow can be better considered when designg radial pumps turbes smooth impellers. Some more tention will be drawn future impact dk roughness. 3D Daily Nece diagram will be modified based on measurements a LDV system. Acknowledgments: Th study funded by Cha Scholarship Council chair turbomachery University Duburg-Essen. Author Contributions: Bo Hu, Dieter Brillert, Hans Josef Dohmen Friedrich-Karl Benra deved designed experiments; Hans Josef Dohmen guided design construction test rig; Dieter Brillert was responsible lab safety rk assessment; Friedrich-Karl Benra contributed funds experiments was supervor durg research; Bo Hu did all measurements wrote paper while Friedrich-Karl Benra reviewed paper. Conflicts Interest: authors decl no conflict terest. Nomenclure L Symbols a Hub radius b Outer radius dk C D Through-flow coefficient C F Axial thrust coefficient C F f C F on front C Fb C F on back C M Moment coefficient C Mcyl Moment coefficient on cylder dk C M3 C M C M4 C M IV C p Pressure coefficient C qr Local flow re coefficient e T Relive error transducer e D Relive error due da acquition device F a Axial thrust F a f Force on front dk F ab Force on back dk

16 Int. J. Turbomach. Propuls. Power 2017, 2, G Dimensionless axial gap K Core swirl rio ζ 0.5 k s Equivalent roughness k sl Limition k s hydraulic smooth M Frictional rque M cyl Frictional restance on cylder dk M r Measured range. m Mass flow re N D Uncerty da acquition system N T Uncerty transducer N Uncerty measured n Speed rotion n T Number transducers n M Measurg times obta one result p Pressure p b Pressure r b p Dimensionless pressure Q Volumetric through-flow re Re Global circumferential Reynolds number Re ϕ Local circumferential Reynolds number r Radial coorde r seal Radius shaft seal s Axial gap front chamber s b Axial gap back chamber t Thickness dk V r Dimensionless radial V z Dimensionless axial V ϕ Dimensionless tangential x Dimensionless radial coorde z Axial coorde Greek Symbols ε Diameter spheres ζ Dimensionless axial coorde µ Dynamic vcosity wer v Kemic vcosity wer ρ Density wer Ω Angular dk Abbreviions FS Full scale LDA Laser Doppler Anemometer LDV Laser Doppler Velocimetry SST Shear Stress Transport References 1. Kármán, T.V. Über lam und turbulente Reibung. Z. Angew. Mh. Mech. 1921, 1, [CrossRef] 2. Cochran, W.G. flow due a rotg dk. Proc. Camb. Philos. Soc. 1934, 30, [CrossRef] 3. Daily, J.W.; Nece, R.E. Chamber dimension effects on duced flow frictional restance enclosed rotg dks. J. Basic Eng. 1960, 82, [CrossRef] 4. Kurokawa, J.; Toyokura, T. Study on axial thrust radial flow turbomachery. In Proceedgs 2nd Internional JSME Symposium Fluid Machery Fluid Mechanics, Tokyo, Japan, September 1972; pp

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