REPORT No. 842 SOME EFFECTS OF COMPRESSIBILITY ON THE FLOW THROUGH FANS AND TURBINES

Transcription

1 REPORT No. 842 SOME EFFECTS OF COMPRESSIBILITY ON THE FLOW THROUGH FANS AND TURBINES By W. PEEL and H. T. EPSTEIN SUMMARY The law of conervation of ma, momentum, and energy are applied to the compreible flow through a two-dimenional cacade of airfoil. A fundamental relation between the ultimate uptream and downtream flow angle, the inlet Mach number, and the preure ratio acro the cacade i derived. Comparion with the correponding relation for incompreible flow how large difference. The fundamental relation reveal two range of flow angle and inlet Mach number, for which no ideal preure ratio exit. One of thee nonideal operating range i analogou to a imilar type in incompreible flow. The other i characteritic only of compreible flow. The effect of variable axial-flow area i treated. Some implication of the baic conervation law in the cae of nonideal flow through cacade are dicued. INTRODUCTION In the endeavor to obtain high preure rie from axialflow compreor, the operating peed have been increaed in recent year to compreibility peed. The precie extent of validity at thee peed of the incompreible-flow method of deign of axial-flow compreor blading appear not to have been conidered heretofore. In thi paper, which decribe reult of a theoretical invetigation made at the NACA Cleveland laboratory during 944 and 945, the extent of validity i determined in the ientropic cae by applying the compreible-flow law to the ultimate uptream and downtream flow condition in a two-dimenional cacade of airfoil. The reulting equation relating the preure ratio acro the cacade, the flow angle, and the inlet Mach number are compared with the correponding incompreible-flow relation. The reult apply to both fan-type and turbine-type cacade. Although compreible-flow relation are generally ued in the deign of turbine cacade and, le generally, in the deign of fan cacade, it i thought that the compact form of the reult preented herein will render them more eaily applicable than the other method now in ue. P S S t u tatic preure ga contant entropy cacade pacing temperature velocity in»-direction velocity in y-direction reultant velocity; alo mean velocity»-direction (tangential) force per blade per unit pan y-direction (normal) force per blade per unit pan ratio of pecific heat angle of flow meaured from axial direction denity V w X Y 7 P Subcript: uptream of cacade 2 downtream of cacade tagnation VELOCITY-DIAGRAM RELATIONS FOR COMPRESSIBLE CASCADE FLOW GOVERNING EQUATIONS Conider the ultimate uptream and downtream compreible-flow condition, or the velocity diagram, for the cacade of airfoil repreenting a fan or a turbine-blade arrangement (fig. ). The conervation law applied between the up- SYMBOLS The following ymbol are ued in thi report: A flow area perpendicular to axial direction c f pecific heat at contant preure M Mach number of flow FIOURE. Two-dlmanlonl cacade

2 24 REPORT NO. 842 NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS tream ection and the downtream ection where the flow i aumed to be uniform and to pa through equal area are: Continuity Energy- Momentum PlVt = Pi V2 () Normal, y-direction: =pi Pi+PiV* p%vt 2 Tangential, ar-direction: =pi»i(«i u?) In addition, there i the ga law and the ientropic relation Pi p= P Rt Pi (The aumption of the ientropic exponent y prevent the direct application of the reult to fan or turbine blading. It appear, however, that ue of a uitable polytropic exponent n?^7 allow direct application in many cae.) Subtitution of equation (), (5), and (6) in equation (2) yield the fundamental relation between the flow angle Xi and t, the preure ratio ptlpi, and the inlet Mach number Mi: w co X. co X 2 (7-DM* ^ - (7) J.O (2) (3) (4) (5) (6) For the cae of incompreible flow, the expreion correponding to equation (7) may be derived from the equation correponding to () and (2); namely Continuity (incompreible): Vi=v t Bernoulli (incompreible): #i + o PiWi l =Pr"n PW*' The reult i in which the ubtitution = l r ( Pi l) co X a "V TMJ 2 VJ»! M, 2 pi (8) (9) (0) (ID ha been made for comparion with the compreible-flow relation (equation (7)). Equation (0) may alo be derived a the limiting form of equation (7) when the pecific-heat ratio 7 approache infinity. (The incompreible fluid may be regarded a the limiting form of a_compreible perfect fluid in which the velocity of ound -v T approache infinity. Inamuch a p and p remain finite, y mut become infinite.) GRAPHICAL REPRESENTATION OF EQUATIONS For a given inlet Mach number M L and preure ratio Ptlpi, equation (7) how that the flow angle Xi and X2 are not pecified but only their coine ratio. A plot of the coine ratio co X t co X» againt the flow deflection X 2 Xj, with Xi or X 2 a a parameter (fig. 2), illutrate the poible individual flow angle and deflection for a given coine ratio A d- 5 '- - X 2^ '*v A" ( - ' %. 2.0 %. * 0 * V 0* V* V c * I i *7 ^, ' i,y r, VI T, ix. ivy * X^ - V "-X- "' IJ2 ' V«M,-' ' "^ 0.-' -- >*< ^ ^. --._ >. ^.'-'" vy ^*.'I* xs" "--_,0 A >* ' -. *.8 * - y ' ~y y' **J* ^ ~^r "» U'o V ^ ** "SI i, t^ 'A : ^ '*'? - y *-*» ':o > >? ' % y' V". 4 i Ä *J ^ * * VN n ' S t X -SO -ÖD -rfc. -20 O 2D 40 60' - 80 foo t20 I Flow deflection, X^-Xt FIGUEK 2. Relation of 52L-j to turning angle of flow.

3 ie l.4.2 <n 0 I.O 0 0).8 a SOME EFFECTS OF COMPRESSIBILITY ON THE FLOW THROUGH FANS AND TURBINES 25 % c o,q y on " i ^. I» 0^ ^^ ä 4 f' C^- j*^is_, M -LB 4? V N X "fi? ^ ^. L 2 ß, 5p5f-~ K N N y A A '+ V y tr,'a <A f M i 5 t * W J- -tin 0 i w. 7 h>, v 'Er AO S, Preure ratio. p<p> FIGUKE 3. Relation of compreible Sow angle to ptpi for variou Inlet Mach number. The flow angle Xi and Xj may be viualized a the tagger angle of the tangent to the camber line of the cacade blading at the leading edge and the trailing edge, repectively. For low-peed flow, at leat, and for hock-free entry, thi correpondence i approximately valid for cacade oliditie of the order of unity (olidity=blade chordblade pacing) and increae in validity with the olidity. The contour repreenting equation (7) are hown in figure 3 and 4. Several of the contour of figure 3 and 4 are compared with the correponding contour in the incompreible cae (equation (0)) in figure 4 and 5. COMPARISON OF COMPRESSIBLE AND INCOMPRESSIBLE EQUATIONS The comparion between the compreible and incompreible equation (fig. 4 and 5) reveal coniderable difference except in the preure-rie range (p2pi)>l at low inlet Mach number (ilf t <0.4). Outide thi range conider, for example, the determination of the blade camber of a high-olidity preure-rie cacade deigned to operate at an inlet-flow angle of 45, an inlet Mach number of 0.80, and a preure ratio of.2. (Camber i defined a the difference in angle between leading-edge and trailing-edge camber-line tangent. Camber therefore approximate X t Xj for high-olidity cacade.) By the incompreible contour ili"i=0.8 in figure 5 and by ue of figure 2, the camber hould be According to the compreible curve, on the other hand, the camber hould be 9.4, or 65 percent maller. If the inlet Mach number i increaed to unity, the reult are: Incompreible camber, degree L 8 Compreible camber, degree L 5 A an example of the difference between compreible and incompreible flow in the cae of a preure drop through the cacade, aume Xi=0, Mi=0.80, and Pi(pi=0.7. Then from figure 2 and 5 Incompreible-flow deflection, degree Compreible-flow deflection, degree 3. Thu, the effect of compreibility in the normal operating range of moderate preure ratio i to lower ubtantially the required flow deflection for a given preure ratio. On the other hand, for given flow angle, that i, given co Xico Xa, figure 4 how that for normal compreor operation, (coxicoxaxland (pi?i)>l, the compreible preure ratio increae with the inlet Mach number Mi at a greater rate than the incompreible preure ratio. For example, if co Xico X2=0.8 in normal compreor operation and ili=0.8, the incompreible preure ratio i.6 and the compreible preure ratio i.27. FURTHER DISCUSSION OF THE COMPRESSIBLE CASE The urface repreenting the fundamental equation (7) i the locu of all poible ientropic -(lo-free) operating condition of the cacade and no other. Thecontourprojection of thi urface how that certain range of operating^ condition are unattainable in ientropic flow; thu, in figure 5, there i no inlet Mach number M r to correpond to coine ratio co Xjco Xj and preure ratio ppi in the plane region AOB and COD. The reaon for uch nonideal region can be een by analogy with the incompreible cae. The incompreible Afi contour entirely exclude the firt and the third quadrant, relative to point O a origin. Conider the cae

4 *NV 26 REPORT NO. 842 NATIONAL ADVISORY COMEOTTEE FOR AERONAUTICS.a JidJE «Ä^,#*' V tys^ o^ -** -^^l L»,- t ^ ^7 _^f" ^i ( ***.^i '..[ ' o W J «4 C-JC ),- ^ *2^»- * fc.-' Cv i ' --,' ''3 f»*-5? ^ «3 * % 3 ^ ^,'' y C3 ' t j A II <A ^ " yy - 2 A ^, ^ ' > V V» ffl J ' V X - I VI f v m I f i ' <r ^ ' _ COS yi,, C 'co A, Compre*'" c i"^--. cp A i ' Ificoffpe&ioc ; u lv(..4.ff.8.<?.2.4.ff Preure ratio ptp, FIGURE i Belatlon of compreible and Incompreible Inlet Mach number to pipi for variou flow angle. 8rr. **^ fc B ^..6 "***.^ h Co npre Ino ompt eit le e ible.4.2 J- % -8 " * ***v #? T' ~*^»»^ *' -- '" «., * ** -. v r ^»^^ f. ^ ' u *"*»* ^ Mr = 00 - o «*> SV-. # y ^ ^ *^~ :.. ' V V V A 4 V 4- V k^, <5 M J 'i o ' V u ', V ^ I 4 [ I I Vo c 2.4.S Preure ratio, p,p, FIGURE 5. Comparion of compreible and Incompreible Sow angle a function of pjpi for variou inlet Mach number.

5 SOME EFFECTS OF COMPRESSIBILITY ON THE FLOW THROUGH FANS AND TURBINES 27 (co Xico M> Inamuch a the exitangle Xa i greater than the entrance angle Xi, the exit tangential-velocity component i greater than the entrance tangential-velocity component. Becaue the axial velocity i contant, a preure drop (PiPi)^! i required in accordance with Bernoulli' theorem, and the operation point for (co Xico XO^l cannot lie in the firt quadrant but mut lie in the econd quadrant. Similarly, operating point for (co Xico Xa)>l mut He in the fourth quadrant. The partial overlapping of the compreible-flow curve into the firt and third quadrant i due to the influence of compreibility on the axial component of velocity. Another type of nonideal region i hown in figure 4. For fixed (co Xico Xa)>l, an ientropically impoible range of inlet Mach number exit, which increae with co Xico X. At co Xico X2=l.l, for example, the excluded Mi range i 0.695<Mi<.37. Thi type of nonideal range i not preent in the incompreible cae hown a tne dahed curve of figure 4. The lower limiting inlet Mach number of thi range correpond to maximum flow through the cacade, for, becaue co Xico Xa i contant along each curve in figure 4, the maxima of Mi alo correpond to the maxima of M x (co Xico Xa). For a given high-olidity cacade, Xj varie only lightly a the inlet condition are varied; therefore, Jli co X? which i proportional to the flow through the cacade, i likewie a maximum at thee point. Thi particular limiting Mach number i independent of the blade-hape detail and, hence, of the ratio of minimum flow area to inlet area. If a cacade i deigned to operate in an ientropically impoible region, preumably much greater loe, caued by flow eparation and hock wave, will occur than would be encountered at an ideally permiible operating point. Such nonideal deign appear to be not at all unuual on the bai of incompreible-flow theory. Of coniderable interet i the fact that, at the maxima and the minima of all co Xico Xj contour, the exit Mach number M 2 i unity, which may be hown a follow: The maxima and the minima of the co Xico X a contour in figure 4 correpond to the maxima of the Mi contour in figure 3. The loci of thee maxima are indicated in both figure. The condition for maximum co Xico X» along an Mi contour i obtained by differentiation of equation (7) a»+ f coxigylt" co X 2 XPi) The downtream Mach number M 2 M,= Pi IT Pi (2) (3) can now be written, with the aid of equation (), (2), (5), and (6), a co Xi Mi *r COS X3 M*= 3*- a Pi2y (4) Pi Subtitution of equation (2) in equation (4) yield the deired reult, namely, M 2 =l. In the limiting cae Mi=0, equation (2) and (7) how that co XA = ^ COS 2 Mi-0 Thi limiting preure ratio i the ame a the critical preure ratio for local Mach number of unity in onedimenional compreible flow. Point to the right of the maximum on an M t contour (fig. 3) correpond to ubonic exit condition M 2 <; wherea point to the left of the maximum correpond to uperonic exit condition M 2 >. In figure 4 the ubonic exit region i to the right of the locu of maxima and minima and the uperonic exit region to the left. The proof i a follow: From general principle, the ideal continuou flow through a cacade i reverible; that i, equation (7) remain valid if the ubcript and 2 are interchanged. Thi validity may alo be hown directly by ubtitution of equation (4) in equation (7) to eliminate Mi. Conequently, the exit Mach number M 2 can be determined from the curve that repreent equation (7); thu, in figure 3, the exit Mach number M 2 for the operating condition Mi, p 2 pi, and co Xico Xj i obtained by interpolation from the M contour at the point for which the coordinate are the reciprocal quantitie pi!p 2 and co Xaco Xi. Thi reciprocal relation, applied to the loci of the maximum point on the M l contour (fig. 3), yield the M I = contour (where now M i =M ), by the proof previouly given; therefore, the reciprocal relation applied to a point to the right of a maximum yield a point in the ubonic region relative to the M=l contour and vice vera. VARIABLE AXIAL-FLOW AREA In many multitage compreor and turbine, the axialflow area i continuouly varied in order to maintain the axial-flow velocity contant. The effect of a change in axial-flow area through a cacade can be very imply taken into account in the preceding analyi. The equation of continuity (equation ()) i now or PiViAi=paVüA2 PiV a=p 2 v 2 (5) (6) where A t and A 2 are the axial-flow area uptream and downtream, repectively, of the cacade, and 4l "A 2 (7) The only change required in the fundamental equation (7) i that the flow-angle parameter co Xi co X be replaced by a (co X^co X 2 ).

6 28 REPORT NO. *42 NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS ENERGY RELATIONS FOR CASCADES If the ientropic relation (equation (6)) i dropped, equation () to (5) relate ultimate condition on each ide of a cacade within which loe may occur. The equation may alo be regarded a relating condition acro two ide of a line of dicontinuity eparating two region of uniform flow, with the force denitie X, Yf acting on the fluid along the dicontinuity. The oblique hock wave in uperonic flow repreent the pecial cae in which X=Y=0. Without the ientropic relation there i, of coure, no longer a unique dependence of the flow-angle ratio co Xtco X 2 on the preure ratio pipi and the inlet Mach number Mi. (See equation (7).) Equation () to (5), however, yield information concerning the relation between the preure ratio that might actually occur, the force on the blade, and the loe in the cacade. Suppoe, for example, that a high-olidity compreor cacade were deigned for a certain ideal operating point. ' The ideal force on the blade could be calculated from equation (3) and (4). In the actual flow the preure ratio would be le than the ideal value. The flow angle, however, becaue of the high olidity, could be expected to remain nearly unaltered. On the aumption of an ideal flow-angle ratio and a maller than ideal preure ratio, the actual blade force could be computed by equation (3) and (4). The vector difference between the actual reultant force and the ideal reultant force i a difference force caued by friction and hock-wave loe. The correponding tagnation-preure lo or, equivalently, the entropy increae acro the cacade, can alo be calculated. An example of the type of computation jut outlined follow: Among other tiling, it Will be hown that the difference force i neither at right angle to the ideal reultant force nor in the direction of the mean relative flow. Although, a will be hown, the incompreible equation predict that the difference force i alway in the axial direction, the compreible equation how that thi i far from being the cae. The loe in cacade flow between an ultimate uptream and an ultimate downtream uniform-flow condition may be expreed in term of the actual tatic-preure ratio ptpi and the denity ratio p a pi a follow: In the downtream condition, the ratio of tagnation preure #,, 2 to tatic preure jpt i ientropically related to the correponding temperature ratio a -l P2 U Similarly, in the uptream condition, P,,l y _!«( Pi) k If the preceding two equation are^divided, remembering that,,i=f,, becaue no energy i upplied from the outide, and if preure and denity are ubtituted for temperature by the ga law, tbere reult 2*?_ T (8) In ientropic flow, equation (8) reduce, of coure, to p,.i=p,,. In an actual flow p, fi i le than p,, correponding to an entropy increae St Si, given ;by the wellknown formula S i -S =Rlog. p f (9) Pi* Subtituting the nonnal velocity ratio v t jvi into equation (8) yield, by the continuity equation (), P;i_ 2>..i or : 7- (20) The normal velocity ratio p»t>i can bo expreed in term of the tatic-preure ratio p%p y the inlet Mach number Mi, and the coine ratio co X t co X2 by equation (), (2), and (5). The reult i fl 2 - '- P2 Lv '+2( 7 -)M 2^4T L= w COS 2 X a (7-) il, alo Ui V2 Vi tli tan X2 Vi tan Xi Vi (22) The expreion (3) and (4) for the blade force may be written in nondimenional form a X^-g-^WX^-l) (23) X p A example of the ue of the preceding equation, loe add blade force for a preure-rie and a preure-drop cacade have been computed and the reult are lited in table I. For both cae an inlet Mach number of unity wa aumed with inflow and outflow angle of 55 and 45, repectively.

7 SOME EFFECTS OF COMPRESSIBILITY ON THE FLOW THROUGH FANS AND TURBINES 29 Thee flow angle, though, poible for preure rie, are impoible for preure drop by incompreible-flow theory (under the condition of contant axial-flow area aumed here). The ideal quantitie were firt calculated and then for the lo cae, new preure ratio were aumed, which yielded the tagnation-preure loe hown. It will be noted that the tatic-preure ratio of 0.5, which yield a tagnation-preure ratio of in the preure-drop cae, i greater than the ideal tatic-preure ratio. In incompreible flow it would have to be lower. contant axial-flow area aumed here), ü2ci= and, in higholidity cacade, U^TM. remain unchanged from the ideal cae by equation (22), it i evident from equation (3) and (4) that only the tatic preure can change in going from the ideal to the lo cae. The incompreible difference force mut therefore be in the axial direction. Thi concluion for the incompreible cae wa reached in reference by another method. The loe in high-peed cacade flow mut evidently be coniderably influenced by compreibility effect. CONCLUSIONS. For inlet Mach number le than 0.4 in the normal operating range of an axial-flow preure-rie tage, the relation for compreible and incompreible flow yield very nearly the ame reult. For larger inlet Mach number the dicrepancie become coniderable. 2. The compreible equation how the exitence of nonideal operating range, which are not indicated by the incompreible equation. Converely, certain operating range, which are excluded by the incompreible equation, are poible according to the compreible equation. 3. In certain range of the flow angle, two poible preure ratio are predicted for given inlet Mach number and flow angle. <-~y ALRCKAFT ENGINE RESEARCH LABORATORY, NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS, CLEVELAND, OHIO, Augut, 945. REFERENCE. Mutterperl, "William: High-Altitude Cooling. VI Axial-Flow Fan and Cooling Power. NACA ARR No. L4Ille, 944. TABLE I-LOSSES AND BLADE FORCES FOR CASCADES IN COMPRESSIBLE FLOW IX l =65 ; X. =45 0 ;M = ] Preure rie Preure drop (a) Preure rie. (b) Preure drop. FIGUEE 6. Typical cacade force and velocltlre. The cacade-flow angle and force are illutrated in figure 6. The reultant force I, the difference force d, and the mean velocity w bear no imple directional relation to each other. Becaue, in incompreible flow (under the condition of Stagnation preure ratio, p.,tlp,.i- Tangential forcr, Xi pi Normal force, YJtpi Batlo of reultant force to difference force Aumed valne. Ideal L With loe».4 a a 307 JL7 Ideal 0.«-0.«0.9» m With loe»

8 TITLE: Some Effect of Compreibility on the Flow Through Fan and Turbine AUTHOR(S); Perl. W.j Eptein, H.T. ORIGINATING AGENCY: Aircraft Engine Reearch Lab., Cleveland, O. PUBLISHED BY: National Adviory Committee for Aeronautic, Wahington, D. C. Mit 946 BSTR CT. DOC OJUS. CCWCTOY AM6UA6I Uncla. U.S. Englih 'If" lumtbailoni diagr, graph ATI- 4H55 ItVltlCN (None) OIIO. AOCNOr NO. R-842 HI IISNMO AOCNCT NC (Same) The law of conervation of ma, momentum, and energy are applied to the compreible flow through a two-dimenional cacade of airfoil. A fundamental relation between the ultimate uptream ftnd downtream flow angle, the inlet Mach number, and the preure ratio acro the cacade i derived. Comparion with the correponding relation for incompreible flow how large difference. The fundamental relation reveal two range of flow angle and inlet Mach number, for which no ideal preure ratio exit. One of thee nonideai operating range i analogou to a imilar type in compreible flow. The other I characteritic onw to compreible flow. The effect of variable axial-flow area i treated. Some implication of the baic conervation law in the cae of nonideal flow through cacade are dicued. DISTRIBUTION: Requet copie of thi report only from Publihing Agency DIVISION: Aerodynamic (2) SUBJECT HEADINGS: Flow, Compreible (40458); SECTION: Fluid Mechanic and Aerodynamic Airfoil - Cacade (08000); Airfoil - Compreibility Theory (9) effect (08032); Compreibility - Aerodynamic effect (24095) ATI SHEET NO.: R Air Documontt Dtvitlon, Intolligonco Dopartment Air Materiel Command AIB TECHNICAL INDEX Wrieht. PoHoron Air For«BOM Dayton, Ohio

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