Calhoun: The NPS Institutional Archive Theses and Dissertations Thesis Collection 1963 Investigation of the influence of supersonic rotor velocities on turbine performance and design. Byington, Melville R. University of Michigan http://hdl.handle.net/10945/12435 NPS ARCHIVE 1963 BYINGTON, M. library U. S. Naval Po*»^tick»ate Monfcerey, Co#t'>' S< INVESTIGATION OF THE INFLUENCE OF SUPERSONIC ROTOR VELOCITIES ON TURBINE PERFORMANCE AND DESIGN Lt. Melville R. Byington, Jr. U.S. Navy Conducted in partial fulfillment of requirements for the Degree of AERONAUTICAL AND ASTRONAUTICAL ENGINEER at the University of Michigan August, 1963 ihy Library U.S. NavaJ Po^aauate icftooi Monterey, Catttomia ACKNOWLEDGMENTS The author is indebted to Professor R. University of Michigan and to Professor M. H. B. Keller of the Vavra of the United States Naval Postgraduate School for their patient and constructive guidance. Partial cost of this research was borne by the United States Naval Postgraduate School. 11 . TABLE OF CONTENTS Page ACKNOWLEDGMENTS ii LIST OF TABLES v LIST OF FIGURES vi vii ABSTRACT I . II. INTRODUCTION 1 ANALYSIS OF LOSSES AND FLOW PATTERNS IN SUPERSONIC TURBINE STAGES 2 1. 2. 3- k. 5. 6. III. A RATIONAL TECHNIQUE FOR DESIGN POINT SELECTION FOR SINGLE-STAGE IMPULSE TURBINES 1 2. 3. h. 5. IV. General Discussion Development of a Method of Loss Prediction in Supersonic Turbine Blading Comparison of Loss Prediction Theory with Available Test Results Comparison of Typical Rotor Velocity Coefficient Estimates Obtained by Different Methods Detailed Analysis of the Effect of Rotor Profile Geometry on Losses Analysis of the Effects of Rotor Incidence On the Supersonic Flow Through Turbine Blading General Discussion Static Efficiency as a Function of Velocity Coefficients and Rotor Deflection Limiting Stress Requirements Combination of Performance and Stress Requirements Procedure for Application ILLUSTRATIVE DESIGN PROBLEM 1. 2. 3- k. 5. 6. Specifications and General Design Philosophy... Design Point Selection Detailed Design Point Calculations Nozzle Flow Analysis and Profile Layout Rotor Flow Analysis and Profile Layout Discussion of Resulting Design 111 2 5 10 11 12 l8 25 26 27 29 30 33 35 ~$h k2 ^5 k-9 51 TABLE OF CONTENTS CONT'D Page V. VI . CONCLUSIONS AND DISCUSSION 53 REFERENCES % APPENDICES A. NOMENCLATURE B. DEVELOPMENT OF EQUATION 57 58 II-3-1 IV 62 LIST OF TABLES Table I-a Page Pressure Coefficient as Function of Y, M. I-b I-c I-d II 7 = 1.1+0 63 Y, M. Velocity Coefficient as Function of 7 = 1.40 6k Y, M. Pressure Coefficient as Function of 7 = 1.25 65 Y, M. Velocity Coefficient as Function of 7 = 1.25 66 Comparison Between Estimated and Experimentally Determined Loss Coefficients of Transonic and Supersonic Turbine Blading Rows 67 LIST OF FIGURES Figure Page Comparison of Theoretical Efficiency Esimates for (Subsonic) Single Stage Impulse Turbines With Results of Supersonic Turbine Tests 68 Profile Loss Coefficients for Conventional Section Blades at Zero Incidence 69 3 Secondary Loss Acceleration Parameter 70 k Pressure Coefficient as Function of M, Y. y - 1.U0. 71 5 Velocity Coefficient as Function of M, Y. 7 = 1.40. 72 6 Estimate of Rotor Velocity Coefficient Impulse Blade in Supersonic Flow - 1 2 Typical 73 7 Cascade Loss Results of Reference 13 7^ 8 Comparison of Rotor Incidence from Experiment and Two Theories - For Turbine of Reference 2 75 Velocity Triangle Nomenclature 76 9 and F" Functions for Turbine Design Problem 10 F' 11 Constant Stress Efficiency Curves for Turbine Design Problem Mean Radius Conditions... 77 78 12 Design Velocity Triangle 13 Sharp-Corner Nozzle Geometry Considerations 80 l4a Design Nozzle Profile 8l lUb Sharp-Corner Nozzle Cascade 8l 15 Design Rotor Profile 82 - VI 79 - ABSTRACT An analysis has been conducted of the capabilities of high single-stage impulse turbines. pressure ratio,, Some recent experi- mental work has shown considerable promise for these machines. This work provides a tentative basis for better understanding the potentialities and peculiarities of such high performance turbines. The present analysis indicates that well designed supersonic rotor impulse turbines can be expected to produce static efficiencies closely approximating those predicted by well-known (subsonic) curves 4..p.p. -p of static efficiency J versus • 4. blade -speed E T lsentropic jet speed -: . . . . . . , Apparently, there presently exists no generally applicable method of blade row loss estimation in the case of supersonic flow. An extension to the Ainley-Mathieson loss prediction procedures to the supersonic case is proposed. Experimental and estimated supersonic -rotor loss coefficients of the present report differ little in magnitude from well-established results for subsonic blades of similar deflections. The effect of rotor incidence, in supersonic flow, has a profound influence on pressure and velocity distributions throughout the stage. theory. The effect is inadequately explained by a recently proposed The present analysis proposes an alternative method for esti- mating the effect. A logical and straightforward procedure is developed for selection of the "optimum" design point for a single-stage impulse VII turbine, given a required RPM, power output, and rotor blade root stress. The preliminary design of a large rocket-turbine is con- ducted as an exercise in application of the foregoing developments. This study was conducted as partial fulfillment of the requirements for the Degree of Aeronautical and Astronautical Engineer at the University of Michigan. vm ) I. INTRODUCTION In recent years., considerable interest has centered around the development of high pressure ratio turbine stages for rocket- Desirable character- pump drive and other specialized applications. istics are reliability and light weight, with high efficiency as a desirable objective of lesser importance. One type of machine attractive for these applications is the single-stage impulse turbine. This turbine combines low axial thrust on the rotor with potential for high pressure ratios and specific work output. The high pressure ratios normally result in supersonic flow through the rotor, and very few experimental results concerning these stages are available. The present analysis is undertaken to explore the potential- ities of such stages with the benefit of some recent test results. study is directed along three parallel, but distinct topics. The These are: (1 analysis of flow patterns and loss distribution within the stage, (2) development of a scheme for selecting the design point to best achieve specified design goals, and (3) a typical design problem intended to illustrate the results of topics -1- (l ) and (2). II. 1. ANALYSIS OF LOSSES AND FLOW PATTERNS IN SUPERSONIC TURBINE STAGES General Discussion While a wide body of theoretical and experimental data are available (e.g. Reference l) for predicting flow patterns and blade losses at low Mach numbers, little information is available at supersonic velocities. As a consequence, turbines designed to operate in these regimes frequently fail to achieve the expected performance and pressure distributions. Two examples are the turbines of Reference 2 and J wherein the flow patterns, pressure distributions, and efficiencies differed considerably from estimated design values. In both cases, the results would indicate efficiency was lower than necessary, had the losses been accurately predicted in advance and compensated for in the design geometry. The method of Reference k was utilized by the N.A.S.A. to successfully correlate rotor blade momentum thickness and "specific blade loss" with "blade surface diffusion parameter" for one family of several transonic-rotor turbines. This method does not include the effect of total rotor deflection angle, and has not been satis- factorily generalized. The method predicts a grossly optimistic rotor velocity coefficient for the turbine of Reference tion angle is 1^0 2. The rotor deflec- in the latter rotor as compared to only 8j in the N.A.S.A. family, explaining the deviation. While the loss prediction methods of Reference 1 are comprehensive, they are only recommended by the authors in cases wherein -2- 3- the flow is everywhere suhsonic. Hence, an extension in generality to include supersonic rotor flow is essential to provide for accurate flow channel layout and a more precise estimate of performance. Figure 1 presents the performance of the two supersonic rotor turbines of Eeference 2 and transonic family of References 5 J, together with that of the N.A.S.A. through 10. For comparison, there are the theoretical curves from References 11 and 12 and an unpublished The theoretical curves presumably curve from the Aerojet-General Corp. are intended to apply to subsonic stages only as no mention is made of Mach number levels. Slight deviations between the curves are explained by minor differences in the geometry and loss assumptions used in deriving them. They may be assumed to fairly approximate the range of efficiencies to be expected for single stage impulse turbines having negligible Mach number effects. Comparison of these curves with the experimental results of the high performance turbines shows agreement which is enlightening, if not surprising. It seems that no appreciable deterioration in performance may be attributed to Mach effects. The mediocre effic- iencies of the turbines of References 2 and J are evidently due to very high leaving losses, while the N.A.S.A. family has low leaving losses. The turbine of Reference 2 shows performance that is quite encouraging when compared to theoretical values . It is no surprise that this is the most recent and probably the most carefully designed of the three. This turbine is designed for high (l40°) rotor deflec- tion and maximum static efficiency. On the other hand, the N.A.S.A. h- family of turbojet research turbines are designed for high total efficiencies, which average about 85$The relatively inferior performance of the turbine of Reference 3 appears to result from causes other than the high Mach number level, although insufficient data are reported to provide for a precise analysis of losses. The performance evidently is degraded by extremely severe flow separation from the rotor suction surface. This explanation is substantiated by the almost negligible change; in flow characteristics which resulted from a major modification of the rotor profile. The modification consisted of reducing the blade thickness about by removing metal from the suction surface. hO^o The negligible resulting flow change indicates the flow was separated in both cases, and demonstrates the ineffectiveness of profile modifications downstream of the separation point. Since the profile violates the criteria of Reference 13, as discussed in able separation is to be expected. consider- In addition to the separation problem, the design apparently suffers large tip clearance and "lap". II-5., somewhat from relatively "Lap" refers to a sudden annulus enlargement immediately downstream of the nozzle exit. Returning to considerations of the turbine of Reference Figure 1 also illustrates the margin of improvement percentage points in static efficiency - - 2, about two obtained by increasing rotor flare enough to compensate for actual rotor losses. This flare increase allowed design impulse conditions to be closely approached, and illustrates the effect of accurate loss estimates upon geometry and performance. Comparison of theoretical performance -5- with the best performance of this turbine strongly indicates that any losses associated with supersonic effects must indeed be This deduction diametrically conflicts with standard negligible. works (e.g. Section I+.5 of Reference l) which indicate a rapid increase in losses as the rotor flow begins to exceed local sonic velocity. Apparently a phenomenon exists which is similar to the "sound barrier" drag increase of aircraft wings. Evidently, when the flow just exceeds local sonic velocity, strong and near-normal shocks exist which cause separation. Once the flow is fully super- sonic, the flow field appears to stabilize to the extent that separa- tion losses need be no more severe than in subsonic rotors of similar geometry. The design criteria for minimizing rotor flow separation are considered in II-5- For the present, it seems reasonable to assume that the combined profile, secondary, and tip losses of the supersonic rotor far outweigh any losses credited to supersonic effects. 2. Development of a Method for Loss Prediction in Supersonic Turbine Blading In view of the foregoing examination of Mach number effects, the loss prediction methods of Reference 1 bear re -examination. The basic Ainley-Mathieson loss parameter is Y = (Total Inlet Pressure) - (Total Exit Pressure) (Total Exit Pressure) - (Static Exit Pressure) The total pressures are referred to relative and absolute velocities in the rotor and stator cases respectively. total loss coefficient for a blade row, Following Reference 1, the Y^,, is the sum of profile losses, Y-nt secondary losses, Y^-,, and tip losses, Yv . Profile losses for stator or rotor blading of arbitrary reaction are empirical functions of solidity, deflection angle, and thickness ratio. Profile losses are easily determined for blades of arbitrary reaction using the methods of Reference 1 with Figures 2a and 2b, which are reproductions of Figures ks. In the present case attention is and 4b of Reference 1. directed only to the special cases of nozzles and impulse rotor blades. In this case the value Yp is determined from Figures 2a or 2b, as T appropriate, for standard thickness — = .20. The profile loss estimate for the nozzle or impulse rotor blade in question is then simply The secondary loss equation is developed from relations given in Reference 1. From Equation From Appendix there is Cx = 2(s/c) I, (k) of Reference 1, there is (tanP x - tan|3 y )cosPm (3) and (C v n ) D'sec = Y sec cos 5 P /cos 2 f3 (-) v m c 7 (4) where the vector mean angle, Pm = tan" Here |3 1 [| (tan P x is the gas angle -relative to as appropriate - - tan |3 y )] (5) moving rotor or stationary nozzle and x and y refer to inlet and exit conditions ' 7- Combining 2, respectively. Tsec " 3, tan P x-tan P ^f " and h, there results y (6. ) The coefficient X is an empirically derived coefficient which accounts for the effect of acceleration on secondary losses. Figure 17 of Reference 1 correlates \ as a function of hub-to-tip radius ratio and (Ay/ Ax ) p . Since these data were presumably gather for low sub- sonic velocities, it is assumed (P^./Ax ) gas velocity. 2 2 = (W /w Based on the foregoing assumption, ) where W is relative , (W appropriate acceleration parameter independent of M. x 2 /W ) should be an y The assumed variation of X with acceleration and hub-to-tip ratio is shown as Figure 3. It should be noted that experimental points of the original figure presented considerable scatter, whereas Figure 3 represents a best fit of the data. The tip clearance loss equation is developed from ^ Cjf . i 2t/i 2 h / of Reference 1, page 13. ( 7 ) c Equation (7) is attributed to Carter, was developed on a theoretical basis for small deflections. and However, Reference 1 points out satisfactory agreement between (7) and tip effects at large deflections, hence the extension seems justified in the absence of a better theory. Similar to (k) above, there is also KV *r fc) ^B. COS^ y (8) Upon combining (3), _ 2( ^ (l), and (8), there results ^=o^ n For single stage stators, B (y K . e gt) ) s f (tan cos Pm h = and (9) simplifies to !if% (10) cos Bm Similarly, for impulse rotors with B = R r = 3, there results 8(^)sin 2 R = (YK ) B (9) •> (11) The secondary loss relation (6) may also be simplified. In the case of impulse rotors (Y = ) (6) becomes 2 l6x sin R (12) sec R In the case of nozzles , Figure 3 indicates that for any practical stator, X s = .OO55. (Y sec ) s = Accordingly,, M-oo55)x sln (6) h becomes = cosPm .022 ^LhL (13) cosR m The stator coefficients, calculated in II-3, tend to be excessively optimistic if (13) is used. Furthermore, the theoretical development based on Reference 1 and leading to (13) has so far made no allowance for effects of blade height on the magnitude of stator secondary losses. However, the data of Figure 15b of Reference 1, as well as Reference 9 a ^d earlier tests, confirms that stator secondary losses are inversely proportional to blade height, while solidity, aspect ratio, and chord have negligible effect on stator losses. In order to resolve the apparent cont rad.it ion, (13 ) has been modified by the coefficient 6/h, chosen empirically based on test results presented in II-3- Hence, the best estimate of stator secondary losses is felt to be The total losses of a blading row are determined by addition of the profile, secondary, and tip losses calculated above. Y Y " T + Y P sec + Y That is fW) K The Ainley-Mathieson loss parameter may be converted to a pressure coefficient or velocity coefficient to simplify subsequent calculations . One may write Y = (note P" becomes P' C Px — " P " - P P"/P x " L. = - ' P 1 y y - " 1 - I (16) " P /P y y in the case of stationary blading). p"V' /P x" 1 = 1 +Y [i - (1 + Z^^ Then ^_ (1T) -7/r-i ) ] The pressure coefficient is seen to be a function of the relative exit Mach number, M , as well as the loss coefficient and gas properties. The conversion to a velocity coefficient is made using -10- which is developed in Appendix III of Reference lk. Solving for \|r, there results t = [i + - —1-5 (i - c/nr 2 (is) Equations (17) and (18) were programmed on a 7090 computer and the results tabulated in Table I for 7 = 1.^-0 and 7 = 1.25- The results are also plotted for 7 = 1.U0 and appear as Figures k and 3. 5- Comparison of Loss Prediction Theory With Available Test Results The loss prediction theory of II-2 was compared with test results for 7 rotors and 5 stators of experimental turbines. These represent the currently available results that have been published with sufficient information to provide a comparison between predicted and experimental loss coefficients for blading rows. The comparisons covered a moderately wide spectrum of Y, NL, degree of reaction, deflection angle, solidity, aspect ratio and other parameters. While the volume of test data is quite limited, agreement between theory and experiment is reasonably good. The average deviation between predicted and experimentally determined coefficients is about one percentage point. The precision of both predicted and experimental coefficients is probably no better than plus or minus one point, for reasons discussed below. The comparative results are shown in Table II. It was necessary to apply certain assumptions and approxi- mations in compiling the data of Table II. with reasons therefore, are: The more important of these, ) -11- (a) The N.A.S.A. rotors of Reference 5 through 9 were assumed to have tip clearance Ah/h = .02 or Ah = .042", since clearance was not specified. The errors should be small, as Y < t Ym in all K cases. (b) Geometric and velocity characteristics are deter- mined for mean radius conditions, deemed representative of the stage. This assumes that three- dimensional effects are minor. (c The effects of Reynolds Number variation are neglected. Reference 1 assumes (Re) = 2 x 10 , while the experimental turbines operated at (Re) only 2 to 3 times this value. (d) The rotor pressure coefficients were not quoted explicitly in References 5 through 9- The co- efficients were deduced from (c P)H = K ?k 7 ~ P 2 (0 p)s [(l-r, T) T^rj-i z,i-rz= +1,0^7 ]*r which is developed in Appendix is exact, p°. o B. While this relation some interpolation was required to estimate and Pu'/P ' ^ rom ^he published turbine maps. stator coefficients were deduced from Reference k. (i) The 9- Comparison of Typical Rotor Velocity Coefficient Estimates Obtained by Different Methods Figure 6 displays the estimated variation of rotor velocity coefficient, as a function of deflection, for typical impulse blading. -12- The subsonic reference curve is taken from Reference 12, and repre- sents a mean obtained from a large body of experimental results. The two supersonic curves were obtained following the method of Tl~2, for a typical geometry. interesting features. The comparison illustrates several The magnitude of the rotor coefficient estimate differs relatively little between the subsonic and supersonic curves. This is not too surprising in view of the comparisons shown in II-l. However, the shape of the supersonic curves at high deflections appears somewhat suspicious, as subsonic experience would anticipate a more rapid deterioration of velocity coefficient. Since there is insuffic- ient test data to confirm or deny the suspicions, one would probably be well advised to use the more conservative estimate, or perhaps a mean. One also notes the increase of \|/t, with M, for fixed losses, Y. This trend is also seen in Figure 5. 5. Detailed Analysis of the Effect of Rotor Profile Geometry on Losses Reference 13 reports extensive two-dimensional cascade tests of rotor profiles. These tests compared conventional constant- curvature steam turbine profiles to several experimental profiles designed to minimize flow separation in supersonic rotor channels. The inlet Mach number was 1.9 and deflection angle was 1^0 . Schlieren photographs and total pressure measurements indicated that the resulting "shockless" profiles essentially eliminated shock-induced separation from the suction surface. The over-all velocity coefficient correspondingly increased from .929 to .952. -13- Reference 2 compared the conventional and shockless profiles in a test turbine, which also operated with several combinations of rotor "lap" and flare. Exact comparison between cascade and turbine tests is difficult due to the limited number of configurations reported. Furthermore, the rotor inlet Mach number was about 1.6 instead of the design value of 1-9- The reduced Mach number resulted from off design nozzle pressure ratio, the reasons for which are discussed in Section II-6. The maximum rotor velocity coefficient achieved in the turbine was .872. This peak performance was obtained using zero lap, but sufficient flare to approximately compensate for actual losses and approach the design exit flow conditions. In another test, the shockless profile increased total head efficiency about 2-l/2 percentage points, under otherwise identical conditions, while static efficiency changed little if any. These results will be discussed in more detail presently. The shortcomings of the conventional, constant -curvature passages are thoroughly explored on two-dimensional theoretical grounds in Reference 13. Essentially, it is demonstrated that constant-curva- ture passages cannot transform a uniform incident flow into a uniform curved flow. Rather, from considerations of a characteristics net, it is evident that the pressure on either surface undergoes a continuous zig-zag pattern of abrupt increases and decreases throughout the length of the curved channel. The pressure rises are expected to react with the thickening boundary layer and produce separation from the suction surface. Cascade tests demonstrated the predicted separation for the conventional profiles. -14- A second and complementary fault of the conventional profiles is the likelihood of excessive channel width caused by too large a spacing between blades. In supersonic rotor channels, the wider the channel, the larger becomes the uncompensated convex surface acceler- ation before the compression waves become incident. Hence, an excessively wide spacing produces an increasingly severe pattern of pressure fluctuations. Schlieren photographs also showed a definite tendency toward focusing of the compression "fan" if the spacing were too wide. and flow breakdown. The abrupt pressure rise triggered separation It should be noted that supersonic rotor profiles are determined from channel flow considerations, and solidity is not an appropriate design variable. The "shockless" profiles of Reference 13 were designed to eliminate the foregoing objections. was mentioned above. Their success in cascade tests An example design is discussed and constructed in Section IV-5 and Figure 15. Essential features of the design scheme are: (a) Dual curvature, having an entry or transition curvature half as great as the central or mid- chord curvature. The transition curvature extends to a position corresponding to intersections of the first characteristics originating from the opposite surface. Throughout the central region a uniform free vortex flow is theoretically established in planes parallel to the machine axis. A similar transition at the channel exit reconverts the flow -15- to uniform parallel exit conditions. (b) A sufficiently small channel width to reduce to acceptable levels the initial, uncompensated acceleration along the convex surface. The maximum recommended width is that prescribed by a transition curvature having (JX_) > cos -out tran (c) (sln -l 1 wz ) (!) lx A channel width sufficiently small to preclude deceleration to sonic velocity along the concave surface. This requirement is generally less stringent than (b), unless the incident Mach number is near unity. A comparison of cascade tests of the two profiles is seen in Figure 7> where the data were collected from Figures 7 and 19 of Reference 13B, Aside from the obvious general superiority of profile a closer analysis yields an insight into the nature of the compar- ative losses. The general spanwise variation of losses indicates the effects of secondary flows, wherein the end wall boundary layers are "centrifuged" toward mid-span by secondary vortices I3.7 of Reference 12 for complete discussion). (see Art. A fair estimate of purely profile losses is probably obtained from the curve peaks, near root (and tip). The greatly increased losses in the central span region point out the predominance of secondary losses — in qualitative agreement with the estimates of Table II for the rotor -16- of Reference 2. The spanwise uniform superiority of profile B may- be explained by a sharp decrease in profile losses with secondary losses essentially the same in both cases. Hence one concludes that the improvement shown by profile B is due to a reduction in profile losses caused by virtual elimination of shock-induced suction surface separation. The ratio of losses between the profiles, in cascade, is ( ya/ y b) Cascade = "^ • - ^-It obtained from Figure 5 using the reported 37 velocity coefficients. From Equation III-2-2, static efficiency is directly proportional to (l + tp) parameters assumed fixed. for impulse turbines with all other Thus, the relative improvement in r\ due to profile B, if duplicated in the turbine, would be a factor of only i-±-^2 1#012> __ 1 + .929 The reported data of Reference 2 allows at least a rough comparison of losses obtained using profiles A and B in the test turbine. Total efficiencies were .610 and .635 respectively, with only the rotor profiles changed. From Equation II-J-1, (Cp)g/(Cp)^ = 1.06 for stator velocity coefficient and total pressure ratio fixed. Also, using Figure k for My * i.k, together with an estimated Y * .8k A from Table II, one may show (Y /Y_)_ A improvement from, say Y about .013 increase in . ~ 1.18. Using Figure 5, the ,8k to Y = .dh/l.lB = .71 is equivalent to \|r R . Because tj is proportional to (l + \|0, it is understandable that the static efficiency improvement, due to the improved rotor profile, was indeed negligible in the test results. Several important observations may be made from the results of the cascade and turbine tests of the two profiles. It is obvious -17- that the design features of the shockless profile B prove beneficial in all cases. The relative improvement of profile B is considerably less in the turbine, however, under this single set of test conditions. This result, together with a velocity coefficient reduction from tc ca(i e = -952 to i = .872, R (a) may be at least partially explained by: the likelihood of moderate flow separation in the turbine rotor, while virtually none occurred in cascade tests of profile B (b) the degrading effect of tip losses, not present in cascade, which also are proportionally more detrimental to rotor profile (c) off -design operation of the rotor, which operated at (d) B. =1.6 instead M,, of 1-9- other three-dimensional effects, such as radial pressure gradients, which would be expected to degrade the performance of moving blades. In addition to the effects of profile changes, Reference 2 also illustrates another important facet of rotor losses and the necessity for their accurate estimation during design. series, an improvement from T] = .50 to .52 and \|/ R = .Qkk to .872 was accomplished by a slight increase in the rotor flare. used the "shockless" profile. on an estimated \|/ R = . In one test Both tests In the first case the flare was based 91 and, due to the magnitude of the actual losses, the exit swirl angle (3r was constrained by continuity to be considerably less than its design value. The increased flare approx- imated actual losses and produced the increased exit swirl and efficiency. -18- It is evident that an accurate estimate of the actual velocity- coefficient may be considerably more important than the questionable ability to increase it a few percentage points. 6. Analysis of the Effects of Rotor Incidence on the Supersonic Flow Through Turbine Blading On the basis of experiment, Reference Ik concludes that rotor blading with non-zero edge thickness, in supersonic flow having a subsonic axial component, is constrained by continuity to operate at a unique incidence relative to the oncoming gas flow. This incidence determines the gas direction relative to the rotor. Likewise, the absolute discharge angle of the stator, as well as effective stator exit area, is determined uniquely and is virtually independent of the stator geometry. The rotor incidence effect, if uncompensated, can drastically reduce the stator discharge area, pressure ratio, and Mach number. The stator pressure ratio of the test turbine of Reference 2 was reduced to between l/'2 and 2/3 of design value, for example, due to existence of rotor incidence of k.5 It is evident • care must be taken to estimate the rotor incidence and thus determine the required blading geometry to minimize losses and obtain the desired pressure and velocity distributions. References 2 and l4 set forth a theory for determining the unique incidence, i s Pg - P^ This theory will be stated briefly, and its validity examined, in view of the importance of the incidence. The incidence is postulated to be that for which the expan- sion, corresponding to a Prandtl-Meyer turn of (3 2 ~ equal to the geometrically available expansion ratio Pv (s ^s exac "tly cos £*-t)/scos f3 , -19- There is assumed to be a slender wedge leading edge with attached shock, and negligible total pressure loss accompanying the expansion. Hence, the problem as postulated resolves into simultaneous solution of the continuity and Pr and tl -Meyer relations. The Prandtl -Meyer relation in this case simply v + (3 = constant or v (% 5 = v (Mw ) 2 ) + 02 -P5) = v (%) + ± CO A convenient form of the one -dimensional, steady continuity equation may be developed form p VA = ( )(M — RT n/tRT) A constant = (2) Hence, PMA constant (3) n/T But T = T" -I ETWk rTTETMj v and ana P = r . : Zzi (1 + v~ 2 2 ' O ^_ 777-1 Since T" is constant through a constant diameter rotor, P"MwA (1 7 ~ L ML 2)7+1/2(7-1) 1 = constant (+) 2 Examining stations (see Figure 9) 2 and 3, just upstream and downstream of the leading edge, one writes A5 = A2 V M^ 1 + P," M„ 1 3 + 2^ My,, 2 Z^M„ 2 2 7+1/2(7-1) (5) . . -20- From the blading geometry, there is also A cos TA2 - (3, t/s ^ \ = COS Po Using C p = P,"/ P p" and 5 coml:> i nin g ^ cP (5) and (6), there results <*> ^J) (rT Equations (l) and (7) may he solved for two unknowns. Typically, one knows My- 2f and Po. P*, and t/s and wishes to find M^ 2 3 The incidence is thus determined for a specified incident Mach number and rotor blade profile. Reference lk illustrates the theory for an example rotor having P, = 70 and 1VU =1.9. Reference 2 compared estimated and experimentally deduced incidence, with fair agreement, for and M^ = 1.6l. See Figure 8. P-z = 70° Isentropic conditions are assumed throughout Upon close scrutiny, however, the above theory may be shown to become absurd as the rotor blade angle, p,, is reduced somewhat. The anomaly is easily demonstrated by a simple numerical example Suppose the (relative) incident flow occurs at M^ =1.6. The corresponding Prandtl-Meyer angle for 7 = l.k is Vp = lk.9°- Further suppose a fixed incidence of any reasonable magnitude, say 3 . Now using Equations (l) and (7), calculate the allowable edge thickness t/s, for various P*. C p = 1.0. Assume isentropic conditions, . -21- By (l), v,(M Mr,, = I.7O3. Equation =17-9° which yields ) (7) a corresponding becomes 3 t/s = cos p, - cos (P, + 3 = cos P For P, = 70°., t/s = .027. - 5 Ox 1. 1 + .2(1.703) 2 ) 1 + .2(1.60) 2 703 1.075 cos 5 + 3 (8) ) However, upon reducing P* it is seen that t/s decreases monotonically until t/s = for P, = 52° • The same conclusions are reached regardless of the selected numerical values. In particular, P if C 5 p it is evident that breakdown occurs at even larger < 1.0 or M» > 1.6. The physical explanation of the breakdown of the theory is that, at sufficiently small blade angles, the turned and accelerated Prandtl -Meyer flow (assumed) fails to encounter the requisite geometric expansion necessary to satisfy continuity. In addition to the fore- going arguments, it is also evident that the isentropic, irrotational Prandtl -Meyer flow assumptions cannot hold in an actual turbine rotor. In article 9-10 of Reference 12, Vavra has pointed out the inherent danger in applying two dimensional analysis to such three dimensional flows Aside from the merits of the foregoing theory, rotor incidence remains a physical reality with very important implications. Hence, a more satisfactory explanation of its nature must be sought out. One must agree the continuity relation (7) is valid. Having decided the Prandtl -Meyer relation is not applicable in this case, there remains one more unknown than available equations. -22- Expanding the Mach number terra of (7) in a binomial series, it may be shown that A+^M^ y^-i) -s where e = .? Substituting in = t/s ^ 2 _ i} + s(e2) + - 1. r ( j), cos there results - I 5 From _ i + € (9) y+1 2 + e( ~2~^2 - !) cos P 2 (9) CP it is evident that for moderate incident Mach numbers, the magnitude of the incidence is relatively insensitive to small differences between Equation and M^ M^- was compared with the experimentally deduced (9) incidence of Reference For design speed conditions, one is able 2. Assuming to deduce Cp/l + 1.9e)= •84. .Qk < C p < 1.00 . < ; e < .10 e > 0, one then concludes that . As stated earlier, a second valid equation in the unknowns is apparently lacking. Since it seems probable interest, one might assume t - „ « cos p^ - = e cos (3 e « 1 in cases of and rewrite Equation (9) as Q /,^\ (10) l£ Cp The theoretical incidence of Equation (10 ) and of Reference lk was compared with the experimental incidence of Reference comparison is shown in Figure 8. 2. There is fair agreement, at the The • -23- design point, between the theoretical estimates for isentropic conditions. Furthermore, Equation (10) is seen to agree with experiment for C p = .84 at design speed and . 90 at zero speed. This trend seems reasonable and agrees with the expectations of Higher relative losses at design speed the authors of Reference 2. are attributed to the reduced Mach number with correspondingly stronger, and possibly detached, shocks. By examining the design speed case further, one may deduce the apparent proportion of total rotor loss, occurring at the leading edge, which results in the 4.5° observed incidence. losses were reported as sponds to (YrjOp = 1.12. \|/ = .872. Over-all rotor Using Figure 5, this loss corre- Similarly, from Figure k the apparent lead- ing edge loss, Cp = .84, corresponds to (Yttp)-d = -25- Hence, the apparent ratio of leading edge to total rotor loss is .25/1.12 = 2/9As noted above, this apparent ratio represents an upper limit of actual leading edge losses and, in effect, incorporates the effect of any flow acceleration, e, which may occur at the rotor entrance. From Equation (10), for a given blade (or gas) angle and incidence, leading edge losses would be expected to increase with t/s. The foregoing considerations are utilized to estimate incidence in the design problem of IV- 5 The result of a single experiment hardly constitutes a reliable design rule. However, there is provided a reasonably con- sistent account of the interaction between gas angles, blade angles, and rotor geometry. As mentioned previously, care must be taken to -2k- accurately estimate incidence and preclude serious rotor-stator mismatching with resulting Losses. One test cited in Reference 2 clearly indicates the criticality of the problem. having negligible edge thickness -were run. Rotor blades During the course of the test, the nozzle pressure ratio decreased from l6/l to I'^j'L, corresponding to the accumulation of only a thin carbon deposit on the rotor blades. Nevertheless, the buildup caused the incidence to evidently increase some 2 observed effect. in order to produce the III. A RATIONAL TECHNIQUE FOR DESIGN POINT SELECTION FOR SINGLE-STAGE IMPULSE TURBINES 1. General Discussion Given a design goal in terms of RPM and power required, expansion ratio, working fluid, and thermodynamic inlet properties, one is faced with the task of selecting a design point to satisfy the requirements. Literally an infinite number of design points, in terms of blade speed and gas angles, can meet the requirements. Hence, the designer must choose the most desirable combination of high efficiency, compact size and weight, and tolerable stress limits. An obvious approach is a trial and error process. This results in laborious calculations of debatable final success in determining the "best" design point. A single-stage turbine, especially one with a high expansion ratio, will tend to have sizeable is determined leaving losses. Hence, efficiency primarily by blade to isentropic jet ratio as seen in Figure 1, while the blade speed is limited by material stress considerations. Experience and the mission requirements generally dictate desirable blade materials and allowable stresses. Thus one might be well advised to restrict the infinite number of possible design points to only those which satisfy some specified blade root stress. One could then proceed to decide upon a desirable size, weight, and efficiency combination to meet design requirements. The following method is one way of logically deciding upon the "best" design point. It requires neither machine calculations -25- ) The technique is applicable to single- nor infinite human patience. stage impulse turbines, totally independent of Mach number levels. The loss coefficients which enter may be estimated by any means what- soever, and are not related necessarily to loss prediction methods of II-2. The nomenclature of the development is defined in Appendix A. The method is applied to a design problem in IV. 2. Static Efficiency as a Function of Velocity Coefficients and Rotor Deflection In terms of Figure 9> Euler's Turbine Equation may be expressed as = T) 2 JL C u (W 2 sin f3 + W 4 sin PjJ 2 (l o For symmetrical impulse blading of the present case, P Equation (l ) 2 W^ and = P^ = p = \|r R W 2 may be rewritten, r\ = lO R 2 -iL (1 + y C 2 o " X1 P, Wo K2 "2 sin or 2 By definition, \|r = Vp /C T] = , 2_ £2 C C o (1 + * ) sin P (2) o and (2) may be expressed 2^ s 2 (1 + \|r R ) g sin p (3) £sin P (4) From the geometry of Figure 9> = <Jl - S 2 cos2p - -26- -27- Hence (3) may tj is a \|r-o expressed "be 2 = 2\|z o s (1 O + sin p known function of both pendent of £ and (3 but weakly dependent on , S^cos^P - 8(s/l - r\ £ in general. , (3 Ssin 0) - through CU. \|/ (5) is inde- Equation (5) allows calculation of efficiency for selected design point coordinates 6 , P. While Equation (5) might be considered to "organize" the search for a suitable design point, it places no restriction on the blade stresses. Incidentally, Equation (5) also illustrates the relative importance of \|f fl as compared to tp> point improvement in improvement in \Jr For example, a one percentage should be roughly equivalent to a four point \|f R , since » \|/ « \|/ 1. Limiting Stress Requirement 3. For a selected blading material of a specified taper ratio, including constant cross section blading as a special case, one may show that blade root stress is proportional to the product of radial If bending stresses are neglected, acceleration and blade height. a ~ h ¥i R (6) By continuity, the mass flow rate is, o ¥ = density x axial velocity x annular area, or o W The exit density, the blade exit. p]^ p. , cos P Rh (7) depends on pressure and temperature at The pressure may be assumed independent of the vari- ables and equal to exhaust pressure, provided the axial component of -28- The temperature will vary somewhat with velocity is subsonic. T] = T) (6jP)- If ° ne neglects this variation, Equation (7) becomes W ~ W^ cos P Rh (8) 1 For fixed power output, W . Since Wi, t T] (8) = t-oW„. Equation X\ d becomes i ~ ^ R W 2 cos p Rh (9) Substituting, 1 h ~ ^R W 2^ cos P into (6), there results U 2 a ~ ^ i n+B W 2 cos P ( 10 ) One notes that U/R, the angular velocity, is fixed in the present case. independent of Further, letting W* = % V^ ~ 8 S , by assuming V, and/6*, Equation (10) becomes T|i|r R cos P or o"H^r S cos P = B = constant (ll) The parameter B has stress units and may be viewed as an invariant property of the particular design specifications and working fluid. Experience indicates that B is indeed virtually constant over quite -29- a sizeable range of $ as illustrated in IV. P variation, , If one now selects a specific working stress objective, there results D = — = R as a design cos P 5 T)\|f a a' (12) Thus, for a specified stress, the parameter D is a unique constant. Furthermore, any design point satisfying (12) automatically satisfies the imposed stress requirement, c~. Combination of Performance and Stress Requirements k. By combining the results of Equation (5) and (12) 2^ s 2 + fo) n sin (l S ( sll ^cos^P - J2 Ssin P) = - . Vr> 6 COS P (13) or S ( cos 2 P -F n/1 Ssin P) - I" 2t s Ol T + V R ) sin P cos P R' (ik) For convenience, one may define parameters 2^ s 2 F" = 2 i \|r {Jl R (l + R \|/ - ) ^cos^P GAa) sin P cos P - Ssin p) (lite) Note that F" is a universal function of Since \)r„ = i|r R ( S>P) and D is a known constant (lh) . \|/ $ and P and could be tabulated. is assumed independent of P, while (for a specified 3"), one may evaluate Equation There remains an unlimited number of admissible which are known to satisfy the imposed stress limit. the selection of the most attractive design point decision. S } S , P coordinates The problem is P that represents a final . -30- As an aid in this decision one might examine some variable additional relationships For example, from (13), = ,(S,p) \|r R 6 D cos p (15) The rotor relative Mach number variation may be examined by noting, W2 *» 2 6 n//RT n//RT 2 Furthermore, if Vp = ^ S C V2 is 2 independent of P, as previously assumed, it follows that To is likewise constant. \^2 Hence one notes, ~ S (16) Variation in radius is simply determined from U ~ R ~ 9 (17) Reference 15 gives a useful empirical estimate for single-stage rocketturbine weight as m,lb. = 70jt (R,ft.) 2 = 1-53 (R,in. ) 2 (18) Finally, since a ~ Rh ~ 9h, for constant RPM, blade height variation may be deduced from — = constant (19) a 5 . Procedure for Application Given power and RPM requirements, together with available expansion ratio and turbine inlet conditions for a specified working -31- fluid, one may proceed in an orderly manner to select the "best" design point, as follows: (A) Based on experience and operational objectives, one makes a tentative selection of blade material and one or more values of allowable centrifugal blade root stress. (B) A trial design point is selected in terms of an arbitrary velocity triangle. As a minimum, one evaluates, for the trial design point: W a B P % % h e *os S A single calculation of this type should suffice provided the calculated stress is within, say + 50 percent of the selected values, cf. Equation (lII-4-ll). (C) Calculate the invariant B (d) Calculate D corresponding to selected a - - Equation (lIlA-12). (E) Plot F' against (3 (for a reasonable regime of P) - Equation III-4-l4a. lated in step (b). Consider Consider t \|/ R constant as calcu- s = ^(P.M^ ) ~ V (P), R using constant M^. as calculated in step (b). The -32- curves F' theN represent constant stress lines. See Figure 10 for an example. (F) Plot F" against $, for selected S = constant II-4-lVb. - Equation Intersections of the constant S and constant a curves determine unlimited compatible ( S , a) 3, design point possibilities. (G) For any desired , $, a) , calculate: - Equation (III-4-15) ¥^ - Equation (III-4-16) 9 - Equation (lII-lf-4) R - Equation (III-4-17) U - Equation (lII-U-17) m - Equation (III-4-18) h - Equation (lII-U-19) T) (H) ( £, As an additional aid in visualizing the variations, plot t\ against (3 The addition of for a = constant. £ = constant and 9 = constant points provides for ready- visualization of the variation in t\, h along the constant stress curve. (i) From the plot of (H)_, Mw , U, m, 9, R, and See Figure 11. one may then select the most attractive combination of features to fix the design point (£>, 3, a). Estimate t\, for the selected design point. the estimates above. Mw } 9, R, U, m, and h Repeat (b) to verify One notes that the approximations introduced into the technique are now discarded. A second iteration of (C) through (i) would appear unnecessary unless (i) indicates intolerable lack of agreement between the rough and smooth design point estimates. IV. 1. Specifications and. ILLUSTRATIVE DESIGN PROBLEM General Design Philosophy An example of a preliminary design is presented to illustrate the application of previously discussed methods of loss prediction and design point selection. The specifications and gas properties arbitrarily chosen are those which might correspond to turbodrive requirements of a large liquid propellant rocket using reasonably energetic chemical propellants. Assumed output requirements are: 28,800 SHP at 15,000 RPM. Assumed gas properties are: molecular weight, On = 12.0 specific heat ratio, y = 1.25 constant pressure heat specific * = c lb/mole .827 Btu/lb-°R P total inlet pressure P' = ^50 psia o exhaust static presPl = sure 30 psia gas constant, R = 128.8 ft -lb/lb- R total inlet temperature, T' = 2200°R The turbine is to be of the single-stage, impulse type to minimize axial thrust on the rotor bearings. A reasonable compromise between static efficiency, size and weight, and simplicity is to be achieved. -33- -34- 2. Design Point Selection The design point selection method described in Section III A single, arbitrary trial design point serves is to be followed. as a basis for selection of the final design point. The available isentropic energy is, ^is V - cp n (^ - )r " 1/7] - 827 x 220 ° [1 ^ - o y20] - 76 ° tst The isentropic jet velocity, C = o N/2gJ£H B ^IS = n/5 x 10 4 x 760 (^) 7 " 1/7 1] 6l60 ft/sec = The isentropic Mach number, = (M_) jq J JL [ = JQ [15'^ - 1] = 2.40 The subsequent steps correspond to those of Section III-5. STEP A Suppose consideration be given to blade root stresses of Most suitable blading materials 30,000, 37,500, and 45,000 psia. are known to have densities of about O.3O lb/in^. STEP B For the first trial arbitrarily select a p From Figure 1, the expected efficiency is . 60 < blade speed is, U = .3 x 6l60 = 1840 ft/sec. _ IBM 2it x 60 x 15000 x _ lUa „ T) = 65 < , .67- U/C = .3. The mean The mean blade radius is 2 -55- (l) Determination of Stator Velocity Coefficient, \|/ g — The mass flow rate is 550 o SHP x = fj% t]AH 28800 x .707 .64 x 760 ^ is The annulus area, o A2 2rt Rh = 2 o £ = H - =~ 2 ax, \|/ RT 2 s C Q cosa 2 WRT 2 li 2 ^s P2 C o cosa2 2irR For an estimated T 2 = T ' - « \|r —— 2 AH \|r s ° .95. = 2200 - .95 cP 2 I^2_ = 1570°R .827 and lb 42— x 1 h 2 _ a ft ~ lb o 00 1^^x1370 R 128.8 ,__ .95 x 30 ±° x 6l60 in^ 2- = =1.12" cos 65 x 2rt x 14.1 in sec approximate annulus height. The nozzle secondary losses are calculated from Equation (II-2-14), fv x \ \ sec/s _ ill ,. i 2 sln2a P cos a m where Q^ 1 tan" 1 (| tan C^) = tan" = (^J2l) Thus sec s _13 sin 2 65° 1.12 cos 47° m lk = 47 ° 9 - 56- Stator profile losses are estimated from Figure 2a to be Total stator losses, (Y = T )S (Y sec ) + (Y s ) » q (2) = Vax 2 P„ 'p = - = C ^s o . = 6° 00 COS 65 ° tan" UQli 1 Vp sin (-^—Tj v OLo - U 2 ax, 2 w 2 Vax 2 cos P 2 = = ^390 cos 54.7° S = w 2 /v2 = 4390/6000 = .752 9 = u/v2 18^8/6000 = .308 T T = 2 ^ISl - « = 2200 - M vP g = — = -— C = 1328 827 = 2.29 72.0 n/1328 -J/gRTp % . 6000 = — .97^ Cp Revised „ 5^0 = 2535 2 ^^ = ? ^ ciiic 1 ^55 tan = W2 = = it59£ 1.675 2620 72.0 n/t 2 Determination of rotor velocity coefficient, \|/-n Rotor secondary losses are, using Equation (II-2-12), and estimating ( X = .03 Y sec)R " from Figure - W sin % 3, = -^ sin 2 5^7° 06. See Figure 9 - _-] , ) v = . 97^« ft/sec -97^ x 6l60 = 6000 V cos a 2 2 = -. (3) = \|r Determination of some velocity triangle components V2 = ) From Table I-d, .20. for M2 » 2.2, the velocity coefficient estimate is (Y - Og Tip losses, assuming Ati/h = .03, from Equation (lI-2-ll) - — -37- (YT 8-0 = ) K R 2 — sin P d h = 8.0 x .03 sin Profile losses are estimated from Figure 2b, Estimated total rotor losses are = (Ym)-o \|f 54-7° - -16 (Yp) R ~ .12 -60. With an estimated rotor exit Mach number M^ velocity coefficient from Table I-d is 2 s 1.6, the estimated R = .904. The rotor coefficient compares with an estimated from the subsonic reference curve of Figure 6 for (4) \|; R = 54-7° |3 = .898 • Completion of velocity triangle and performance calculations = %W2 P4 - P2 = W4 V „ = V^ = ^ . ax, 2 work output, = cos Static efficiency, 2290 — = p, 4 4 ^Vj^ V= 337° 54-7° W, cos = '9° k x 459 ° + (Wi, -1/W^ HjJ-) ^ = gj cos p u - U) - cos T) (— 2680 -1/2290 v ) . „ _ 3^3 UW2 (l+^ R )sinp 2 = i848x 4?90 x 1.904 sin 54-7 o gJ 2.5 x 10 4 17 = ; -^— AHIS = — 504 W = = 50l+ Btu /^- .663 760 This estimate appears reasonable compared with Figure 1. (5) Temperatures, annulus (rotor blade) heights, and rotor blade stress Since total relative temperature is constant through the rotor of constant mean radius, — . - 3 8- V w2 To 2 = * Tk + = * 2gJC p W 42 2gJC p = T>" Since w, VV - ^ W2 2 + T (1 V> - M„ = 4 _\ i \f = Check work output, c p AT' W* 720 2680 , ^39° ,^oO + = 1528 2\ /-, ig% 2 1413°R = T, T = r k uTlfx^ (1 -** 1.45 n/T,, 2 n o (2200 - 1589) = 505 Btu/lb. = .827 This constitutes a check for consistency of the velocity and temperature distributions The mass flow, 550 QWP bH^ p W 77S — = = V 2 q 8oo —J70T = ! 40.4 lb/sec 504 o The annulus, or rotor leading edge, height is ho W R To = 2jtRP 2 vax,2 ho d 40.4 x 128.8 x 1328 - 1.024" = 2* x 14.1 x 30 x 2535 o _ h WRT 4 4 2 * RP V 4 ax,4 = h. " 2 BL T I- 2% = 1-024 x ^ 1528 x -JL = 1.207" "** The blade root stress is calculated assuming a blade of height h. and density . 30 lb/ in*. The centrifugal stresses only are considered, as bending stresses are expected to represent a small fraction of total root stress. > -39- -A The root force, F - ma = (pA c lb ~ sec ) F F/A = a £i- g 12 ^ h, - p C ft/sec 2 ) wuw n 2 /2it 15000 (** ) 60 R z± ^t *_ g 12 (3 QQ0Q *)* 60 and R = 14.1" 1.207" h^ = for h l+ p ac p = 2 co '12 ft g (5L. = 1910 — psi h],R a = 32600 psi , Summary of step (b) results (6) = .663 \|/ = .904 W = kO.k lb/sec /& = 54-7° a = 32600 psi 1^ = 1.675 R = 14.1" M^ = 1.45 h = 1.207" 9 = .308 .97^ s = .732 T) ^s = STEP C = B 0fT}t-R R S cos P a 8250 psi , the invariant STEP D h ) B [1) -7TT 1 a^ D^ 8250 B = = = = ) -275 30000 D^ .220 = .I83 STEP E Constant stress lines represented by F' ^ x > = p- a|f are plotted in Figure 10. \|f^ R = \Ir_ R (S x . ' Mwl ) ; For this calculation, = 11^(3, 1.^-5)' t v ' R is assumed. i]r = s ^ R (l+ ^ R )sinPcosp .97^- = Constant, and The total rotor losses are -kO~ esimated following step (B-3) to be <Vr = < Y A + Y sec The velocity coefficient, \|/ + ( Vr Steps + ' 12 M^. =1.45. A simple working = tr,(P) constructed from Table I-d is a convenience in ty (E) ^ 72 = R , is then easily calculated, as in Step (B-3) as a function of variable B with plot of = through (h). STEP F £t constant are plotted in Figure Curves of F" {&) for The range . 10. 52 < 6 < .76 represents practical limits of interest and increments of .02 are satisfactory. STEPS G AND H From intersections of S= constant and Figure 10, Equation (III-4-15) was solved for a = constant curves of Plots of T}(B)_ r\. a - const, are shown for the selected stresses in Figure 11. While not expected to be quantitatively precise, Figure 11 condenses the design variables into an easily digested presentation. only on 0, £ , B, Since T), M^ , R, U, hi depend the selection of a design point to fulfill a specified mission is considerably simplified. In the present case, one can write, (Eq. (Eq. II-4-16), III-4-4), Mv 9 ~ 1.675 -J- 2 . = © ( S,P), 732 = 2.29 S plotted for convenience in the regime of interest. (Eq. III-4-17), R » lil —%s = 45.8 9 . ' -41- (Eq. U « III-U-17), —— 1848 6000 9 = .308 (Eq. HI-4-18), m & (Eq. III-4-19), h. (U5.8 9) 1.53 « —— "k 2 = 3220 9 2 lb , . x 1.207 = l.lU x 10" 5 i 32600 o STEP I A working stress must be selected. The blade temperatures may be estimated, for a (turbulent) recovery factor of about to 9j . be, 2 .9 --%^ 2gJC — p < T + blade - 2 T x,n 2 = + .9 x t-^22 1328 J ' 4.13 x 10 4 = . i747° — R 1 = 1287°F. J ! From Figure 11, one notes that efficiencies are reasonably good for with relatively little gain to be achieved at higher stresses. = 3750O, ij Experience would indicate a - 37500 to be a logical choice for a rocket turbine application of short running time. Ample safety factor should be available for a blade material such as Inconel 703C a "t this temperature In Figure 11 for c = 37500, the regime, . 60 < S < .68 appears A rough idea of the trade-off between fuel consumption most practical. and turbine weight may be quickly made if desired. paring points at $-,= For example, com- g„ = .62 shows the latter turbine would .66 and weigh approximately An = 3220 (9^ - o|) = 3220 (.k2 2 - -37 2 = 127 lb. more ) The corresponding reduction in fuel consumption would be o aw o o = ¥ dP - w-i L = — 530/778 SET ah is 1 — — ,1 - { ^ n2 v ; = .707 * 28800 76o . { —1 — .761+ - 1 -; .794 1.34 lb/sec . = , 42- The choice of S= .62 would appear advantageous only if the burning time exceeds An = -o ZW — 127L- n 95, sec, = l.jk provided the objective is simply minimizing the sum of turbine weight and total fuel consumed. In the light of foregoing considerations a choice was made, &= .6k, selecting: = (3 with the assurance a « 37500. 60 From Figure 11 and Step H, the additional estimates are immediately available i) W ~ -395 ~ .78 ~ 3h %~ lb/sec 1A6 ~ 18.1" h^ ~ 1.08" R U ~ 2370 ft/sec m ~ 500 lb. The design point selection technique has allowed a logical and systematic choice of operating design point. It appears that a considerable improvement over the original arbitrary choice has been However, detailed design calculations must follow to verify achieved. the choice and remove the inherent approximations. 3„ Detailed Design Point Calculations The absolute gas angle required is 0t 2 = cos" 1 ( S cos P 2 ) = cos- 1 (.6k cos 60°) = 71-35° 9 -43- a2 9 = 1.0 sin and S cos P 2 = sin 71-55° - -64 sin 60° = - Recalculation of stator velocity coefficient, (a) In the manner of Step (B-l), * 2200 - .96 -^ 2 392 s must be changed to incorp- i|f orate the revised Q^ and nozzle height. T? \|f . For an estimated * .96, ty 1355°R = 827 3U x 128.8 x 1355 d annulus height. For am = tan (Y v -1 c ) *s Total stator losses, 224 L -tot's (Y +n+ = .28. For M .06 (Y p ) P's cos 55. P 92 approximate .92"., x 18.1 2ir (| tan 71-35°) = 55-9° sin 2 71.35 ,13 sec = (.96 x 6l60 cos 71.35°) 30 a 2.2, Table I-d gives is = -965. It is evident that both (Cp) s and unknown, value of Mo it is assumed \|r fl - estimated to be 2.2. depend on the, as yet \|/_ In subsequent calculations = .965 to determine the actual Mo. The value of Cp will then be revised, where necessary, to coincide with the newly calculated Mo. Determination of some velocity triangle components (b) V = .965 x 6l60 = 3950 ft/sec vax 2 = 595 ° GOS 71-35 w2 = sv2 U = ^s T = 2200 - u = .64 x 5950 = 3810 2330 378 6l60 = %, 72.0 3810 2640 n/13^5 = 2640 1.44 d = 5950 x .392 = 2330 .965 2 76 ° = 13l+5°R .827 V2 d o9 a 2 = 1900 = 5950 2640 __ 2 25 . — 2 ) -kk- Estimate (c ^ — W2 ^R -** ^-p- = M, Tl —91 • » 72.0 Vtl. x 38IO " _ sin 2 60° = 06 .48 -W 81x^60° .36 W)r== (Y_„) p (Yp) w R From Table I-d, for M^ k An estimated = \|/ . 4 y t^R = ( —> = ~ 1.28, = \|/ R *R 2 .03 sin 60° = .18 -^6 .871 6. Completion of velocity triangle and performance calculations (d) Wr^" = .871 x 38IO = 3320 ft/sec V^ = ^1655^ + (3320 cos 60° V , SIX 2330 - 2 a = V \|r SIX x\ ^ M- - cos" 1 , — = .871 x 1900 = — 1655 _ ^ £i 1655 - 18. (i^rr) C '1740' = 1740 R = 6o x 2330 2jt x 15000 mean radius, Work output, 4. i i= T7 Vv W = x 12 = 8" 17. — ! x I.871 sin 60° x 381O ~ = JT^ 2.5 x 10 4 2330 — — 575 m^'W ^ _. ,_. ~^ ° -707 x 28800 w = s " __,. 575 Btu/lb. = gA ,_ , , lb/sec The calculated efficiency appears reasonable but perhaps slightly optimistic, compared with Figure 1. (e) ) 88l is obtained for comparison from the subsonic reference curve of Figure tt N \|/p (Y k ) R = 8 x and estimated .12 , Tj, 72.0^ 1^13 Determination of rotor velocity coefficient, ) , ^ « 1^1 3 (Step B-5 . with = 1.28, . , Temperatures and total head efficiency T ^ T2+ 2gt a-0=^+ V=T, 4 h + j£2gJC p ^^ 2 = 1^29 y + llk ° k 4.13 x 10^ , = 1502°R x. 2^1=1429^ -45- "\J = m C p .827 (2200 = W4 2 _ ^ AT' 72.0 ^ 72.0 nTt^ 4. P^' = ^ = P^ (1 + W _ = aF~ IS ^ 3320 _ >y 1^29 72.0 n/ 1429 2 30 1#2 2 + .125 x .64 (1 575 .827 x 2200 [1 - . — _ 72.0 ^ % F± Btu/lb 1502) = 577 - (2§Z5"poy 2 5,0 ) = 38.5 psia ^ o iq = 1+50 Nozzle Flow Analysis and Profile Layout A detailed analysis of the stator losses and required geometric layout is necessary. Pertinent assumptions are: (a) flow within the nozzles (stations 0-1 (b) all stator losses occur in and downstream of the exit plane, between stations 1 and beyond station 2. by the coefficient ) is considered isentropic, and a uniform flow is established 2, The losses between 1 and 2 are represented \|/ = .965? (c) continuity is satisfied between 1 and 2, (d) tangential momentum is conserved between 1 and (e) the flow is adiabatic, Tp' = T,'. 2. The continuity equation is developed analogous to that expressed by Equation (II-6-7) except using total temperatures and pressures. cos a, The resulting relation is = r t (-)\ cP +l r M2 _£ Ml r 1 + - 125 { 1 + .125 ^-5 mA 1—) MQ 2 COS CH -46- From previous calculations, a g = 71.35 M , = The pressure 2.25. coefficient, C P = ^L P = 30 (1 + .125 x 2.25 2 )^° = 7T5 ^0 o' Hence, there is cos °1 t (j) s = + .125 M-,^) (1 + - 06l 3 Tangential momentum conservation gives M2 N/fgRT sin a, 1 Tt = M, >/fgR sin a, = Vp sin V-, __.._ „, sin a, tl 1/ (!) Mi _ = „ rr-3r Mg \/?gT Oi^ T 1/ 1/ ' 2 sinCU or sin %=^ 1 \ / V ^ ! + Ml 2 sin a 2 Ml -V1+.125 x 2.25* Zli M 2 2 1 + ^l + .125Mf sln 2.25 = 2 sin a, I.67O = + .125 M-l2 n/1 ±- (2) M1 For a nozzle with no trailing edge thickness, simultaneous solution of (l) and (2) gives M1 = 2.150, Q^ = 77.2° . For structural integrity a nozzle having 10$ "blockage," or t/S cos Qip = .10, is selected. , cos a, = n .0682 Equation (1 (l) then becomes 2 )^' +.125 '— Mi —J M 5 ± (3) l Solution of (2) and (3) gives Q^ = 75-3°^ \ = 2.178 for the required nozzle exit conditions consistent with the desired (velocity triangle of Figure 12) discharge flow. The blade angle turns out to be about the largest practically acceptable value. If desired, the magnitude ^o . -hi- of could be decreased by providing a slight annular expansion d-^ between stator and rotor. The type of nozzle profile is probably not critical. A "sharp-corner" nozzle of the type discussed in Reference l6 was chosen somewhat arbitrarily. The sharp-corner nozzle, in general, provides for the shortest possible nozzle length capable of a uniform discharge Its properties become much more favorable in cases having larger flow. nozzle pressure ratios and lower discharge angles than the present example Rather than using a full characteristics network, the approximate nozzle contour required was obtained by calculating the required throat angle, over-all area ratio, and the approximate orientation of the limiting characteristics. — simply where v-,, v-, The angle of turn at the throat is the Pr and tl -Meyer property angle corre- = v-,(Mt) sponding to the exit. — and (y, ) fE£ tan" 1 = v ===1 7*9 p i78' M Z^. (MX 2 ^ > - tan" - 1) the required angle. Reference 17, problem 4.20), Pq' a ~i ( = A t y' For M 1 = ( PT~ ^ 7+1/27 j [( 7-1 7 ^) Pi " ±/7 W ( -i] 2 / 7+1/2(7-1) 21 78, Pq7 pq = (1 + -125 x 2.178 P 5-0 ) = ^F^l The required area ratio is obtained from one -dimensional flow theory. (e.g. 1 10.22 The relation is, — -48- and Al (I0.22)- = 90 UJo /m oo*20 ^8 (l0.22^ A+. t _ tf-5 2<20 9 1 The schematic nozzle layout is shown in Figure 13. The limiting characteristic of the throat fan, ab, stands at approximately 19»6 + ^ = fi - + sin" 17.9 [— 1 -] =-17. 9 + sin -1 (y_7g^) The limiting reflected characteristic, be, stands at approximately • u = sin" ( £) = 27- 4 Thus the minimum nozzle length is . 2.I70 wt 2° f + :o tan 19. 6° tan 27.4 j- - (2.8 + 4.25) v t = 7.05 w t However, due to the required truncation of the nozzle at — - d-^ = ik.'J wt = 8.38 w + Hence, t tan 14.7° minimum throat to discharge dimension, parallel to nozzle axis, is ^ the dimension bg must be no less than eb + bg = 2.8 + 8.4 w.|- 11.2 w^. = . The resulting nozzle profile layout is shown in Figure l4a. Since the edge thickness, t, is l/lO the total width, l/9 x 2.20 w t = .244 w t Total width is w = 2.444. w t 'The spacing 2.444 w+ ,<w — between blades is s = w z+ — = 9«o5 The number of ^G = cos q;^ cos „^ 75-3 blades Z = EjtR _ 2it x 17.8 = 11.59 For a selected throat diment = . . — . s sion w-j- . S 9* = .290", 65 W-^ W-j- Z s = 40 blades. The nozzle cascade is illustrated in Figure l4b. The height of the nozzles (and annulus), may be more pre- cisely calculated, o _ WRTp h. 2 PC VQV c. ax, = 2*R 35.4 x 128.8 x 1345 30 x 1900 x 2* x 17.8 B ^ng!" = h \ d. The total throat area is Z s hw t = 40 x . 961 x .290 = 11.14 in 2 , -49- With the nozzle dimensions fully determined, the calculated stator coefficient of Ill(a) should he checked. For a thickness ratio t/c = .21 and solidity c/s = 1.U+, Figure 2a and Equation II-2-1 gives Pi (Y p ) q = x .05 = .053- - — Si2 The secondary losses are (Y„ Q J C '?• ^ = .272, where the blade angle rather than the gas cos 62. 3° .961 angle is employed to provide the most conservative estimate. Total -1±2_ losses (Y-j- -j.) s .33 exceed the estimated .28 of previous calculations, = Hence, from Table I-d, the estimated .965 may be about .004 too = \|/ optimistic, but recalculation of IV- 3 seems unnecessary. 5- Rotor Flow Analysis and Profile Layout A leading edge thickness ratio, t/s, must be selected first so that incidence may be determined. expected solidity c/s a; 2.5, and a desired edge thickness of about .020", there is t/s = t/c c/s » .020 x 2.5 = .050. • The incidence, Pg (H-6-10), cos (EU J = — + S For a chord of about 1", an - £^, may be determined from Equation —Lp—§ „ From considerations discussed in . II-6, it is assumed that about one-quarter of the rotor losses occur around the leading edge - For - Table I-c, Hence (Y I^) R = (Cp)^ p, = cos -1 between stations 2 and (Y T )R = Y = .165) = C p (M = l.kk, (.050 + ^~- = COS ) = 52.8° .165 = 3. there results from -90 and the incidence is 7. 2°. .90 The assumed pressure loss has increased the incidence by about 3-7° from the isentropic case. It is of interest that, were the blade angle, £,, equal to 60°, the relative incident gas angle would be about 66.1° and the nozzle pressure ratio would be greatly reduced from the desired 15 to 1 value. -50- The rotor profile was laid out following the scheme of Reference 13 as summarized in Section II-5 of the present report. As discussed in Section II-6, the rotor entrance Mach number, M^ assumed equal to M w = l.kk. , is may exceed this value slightly, as It suggested by the discussion of II-6. By Equation II-5-1, — — tran r + r.„ n > (^) tran > ( ) 2_v ^tran cos = \± (sin" ^ 5 (^n" 1 jjL) ^os r ° ut = - f cos " out " T± < S r out + r ± 2(1 = .720 - -720) ^ ± + ) ~ 2Q , p. '3 2b Theoretically a channel width-to-radius ratio of .326 would just insure tangency of Mach waves with the convex surface, while allow- ing an excessively large Prandtl -Meyer acceleration on this surface. Hence, the authors recommend at least a 20 to 30^ reduction in ' ,W Accordingly, a design value, V arbitrary construction scale, let ( r o)tran = 1.11, (r. ) + ran in the central section, (r = = (xr-)-t- T , QY1 ) M tran =1.0, w tran ) = ^M tran For an w cen = .22, Since the curvature is to be doubled «89« M cen (r -22 was selected. (=r-), = -50, (r out ) cen = .61, (r^) cen = -39- The profile is constructed, to arbitrary scale of 1.10" channel width, in Figure 15. leading edge consists of a 10 The transition curvature is 15° and the wedge. This is slightly less than the maximum angle consistent with an attached shock. The edges could prob- ably be rounded at a slight sacrifice in performance. From construction, -51- =2.88 and the solidity, c/s the thickness ratio t/c = .l60. For a selected chord of l", thickness at the shoulder chamber is approximately .020" and the scale of Figure 15 is 5.74/1. The number of blades required is calculated from Z . 2»R E J*R_ = __ 2* x 17.8 S S c , 2g blades =====: _ X c ' The rotor losses may be recalculated more precisely for com- parison with (Ym) = .66 obtained in IV-3. R Figure 2b and Equation (lI-2-l) gives For t/c = .16 and s/c ^— (YtJt, = 1 v K = H ^ R (Y M .6k, = = 1.22) From IV-k, h 2 = h* = \|/ . , \j/ R B = = cjti^^ R The .868 From Table I-d, , From Equation (III-2-)), for rj = .7^8. 961", and . J_ 96liii29x = 1.172" 13^5 1910 Rh^ = 1910 x 17-8 x 1.172 = 398OO psi S cos p = 398OO x Total weight, mi=3220 \ 2 6. .64. the final efficiency estimate is _, ^ 1-= h,=h P ^ 2T 2 % From IV-2, a = = compared to the previous estimate of .871. the final estimates of .35, .20 revised best estimate of rotor loss is (Ym) R \|r x .13 = .104. = = .7I+8 x .868 x .6h cos 60° = 827O 3220 (-392) 2 = ^95 lb. Discussion of Resulting Design The major numerical results are compiled for comparison with estimates of IV-2. 52- .7^8 T) B 8270 6o° psi O W 35-4 Lb/sec. P a 39800 psi % = 1.1+4 R 17.8" %k = 1.22 : *s = .961 9 *R = .868 s = 7I+8 agrees well with that The calculated efficiency of indicated in Figure 1 for U/C Q = .378. . •392 .64 Hence, a confident prediction of efficiencies well above .7 offers encouragement for improvement over the previously demonstrated performances of supersonic turbines. The Design Point Selection Technique, leading to Figure 11, proves to be of great assistance in presenting a thumbnail sketch of available design choices. The slight variation from the predicted values of IV-2 was obviously due to the reductions in both from the approximations assumed. \|/ R and \|r s Of particular interest is the observa- tion that the "invariant" B changed only .24$ from the value based on the original arbitrary design point. Perhaps the most questionable area of the problem was that of estimating rotor incidence. The theoretical basis is certainly open to speculation, as discussed in II-6. . V. CONCLUSIONS AND DISCUSSION This investigation indicates that well designed supersonic - rotor single-stage impulse turbines are capable of rather impressive performance. These machines appear to be an attractive compromise between simplicity and high efficiency. Utilizing recent test results, an effort is made to better understand the losses and flow peculiarities of supersonic stages. Many questions are encountered which cannot be decisively resolved without more tests of the quality previously reported by the Aeronautical Research Council of Great Britain. Analysis of supersonic turbine tests indicate that no appreciable losses need be attributed solely to Mach number effects, pro- vided certain design criteria are followed. The major rotor blade design objective is the prevention of shock-induced flow separation. Profiles have been recently developed by the Aeronautical Research Council which virtually eliminate separation. In supersonic flow these blades have much lower profile losses than the conventional constant- curvature blades commonly used in subsonic rotors. Carefully designed profiles yield considerably greater improvements in total efficiency than in static efficiency. Properly chosen rotor blade flare, accurately reflecting actual losses, has a considerable effect on static efficiency. Since Mach number level need have no direct effect on blade losses, an extension of the Ainley-Mathieson loss prediction procedures to the supersonic case is proposed. Some straightforward modifications to the basic procedure are adopted. The extension to supersonic flow appears tentatively justified by the meager amount of available test results -53- . -5U- The existence of a unique rotor incidence, in supersonic flow through blades of finite edge thickness, profoundly influences pressure and velocity distributions throughout the stage. The incidence results from continuity requirements which demand a turning, and probably an increase in velocity, at the rotor entrance. The explanation of the phenomenon presented in Reference Ik and based on Prandtl -Meyer corner flow is shown to be unsatisfactory. An alterative explanation is developed based on the assumption of negligible velocity increase. More experimental work is required to yield a clear under- standing of this important phenomenon. A technique is developed to assist the designer in selecting the "optimum" design point for a single-stage impulse turbine having specified RPM, power, and blade root stress. This method is independ- ent of Mach number level and the method of estimating blade losses. It greatly simplifies the decision by promptly resolving design point selection criteria into a readily digested form. The preliminary design of a large rocket-turbine is developed as a numerical illustration of the foregoing procedures. Blade speeds, but not necessarily stresses, are typically very high in order to reduce leaving losses. The results indicate static efficiencies exceeding .7 may reasonably be anticipated from well designed turbines of this type . VI. REFERENCES 1. Ainley, D. G., and Mathieson, G. C. R. An Examination of the Flow and Pressure Losses in Blade Rows of Axial-Flow Turbines. A.R.C. R. and M. 2891, 1955- 2. Johnston,, I. H. , and Dransfield, D. C. The Test Performance of Highly Loaded Turbine Stages Designed for High Pressure Ratio. A.R.C. R. and M. 3242, 1962. 3. Moffitt, T. P. Design and Experimental Investigation of a Single-Stage Turbine With a Rotor Entering Relative Mach Number NACA RM E58F20a, 1958. of 2. 4. Stewart, W. L. , Whitney, W. J., and Miser, J. W. Use of Effective Momentum Thickness in Describing Turbine Rotor-Blade Losses. NACA RM E56B29, 1956. 5. Stewart, W. L. , Wong, R. Y. , and Evans, D. G. Design and Experimental Investigation of Transonic Turbine With Slight Negative Reaction Across Rotor Hub. NACA RM E53L29a, 1954. 6. Investigation Wong, R. Y., Monroe, D. E., and Wintucky, W. T. of Effect of Increased Diffusion of Rotor-Blade Suction-Surface Velocity on Performance of Transonic Turbine. NACA RM E54F03, 1954. 7. Whitney, W. J., Monroe, D. E., and Wong, R. Y. Investigation of Transonic Turbine Designed for Zero Diffusion of Suction-Surface Velocity. NACA RM E54F23, 1954. 8. Investigation of Whitney, W. J., Wong, R. Y. , and Monroe, D. E. a Transonic Turbine Designed for a Maximum Rotor-Blade SuctionNACA RM E54G27, 1954. Surface Relative Mach Number of 1.57- 9. Effect of Stator and Rotor Wong, R. Y., and Monroe, D. E. NASA MEMO Aspect Ratio on Transonic -Turbine Performance. 2-11-59E, 1959- 10. Effect of a Miser, J. W. , Stewart, W. L. , and Wong, R. Y. Reduction in Stator Solidity on Performance of a Transonic Turbine. NACA RM E55L09a, 1956. 11. Analytical Investigation of Multistage -Turbine Stewart, W. L. Efficiency Characteristics in Terms of Work and Speed Requirements. NACA RM E57K22b, 1958. 12. Aero-Thermodynamics and Flow in Turbomachines Vavra, M. H. John Wiley and Sons, New York, i960. -55- -56- 13. Stratford, B. S., and Sansome, G. E. Theory and Tunnel Tests of Rotor Blades for Supersonic Turbines. A.R.C. R. and M. 3275, 1962. Ik. Stratford, B. S., and Sansome, sonic Turbine Nozzles. A.R.C. G. R. The Performance of Superand M. 3273, 1962. E. 15. Stewart, ¥. L. , Evans, D. G., Whitney, W. J. A Method for Determining Design Characteristics for Rocket Turbodrive Applications. NACA RM E57K25a, 1958. 16. Edelman, G. M. The Design, Development, and Testing of TwoDimensional Sharp -Cornered Supersonic Nozzles. Rep. No. 22, M.I.T., May 1, 19J+8. 17. Shapiro, A. M. The Dynamics and Thermodynamics of Compressible Fluid Flow, Vol. 1, The Ronald Press Company, New York, 1953« APPENDICES -57- APPENDIX A NOMENCLATURE A - area B - function defined by Equation Ill-ij—11 c - chord - constant pressure specific heat Cp - pressure coefficient; ratio of actual to isentropic total pressure C - isentropic jet velocity corresponding to stage pressure ratio D - function defined by Equation III-A-12 F - force F' - function defined by Equation III-U-l4a F" - function defined by Equation III-4-l4b g - gravitational constant H - enthalpy h - blade or annul us height c XT Ah/h- rotor tip clearance ratio i - rotor J - mechanical equivalent of heat, 778 Btu/lb. M - Mach number m - mas s N - RPM P - pressure R - mean blade radius R - gas constant incidence angle -58- -59- s - blade spacing T - temperature t - blade edge thickness U - mean blade speed V - absolute gas velocity W - gas velocity measured relative to moving rotor - mass flow rate - specific work w - channel width Y = total inlet pressure - total exit pressure total exit pressure - static exit pressure o W \fj ^ Ainley _ Mathieson loss parameter. Z - number of blades GREEK LETTERS a absolute gas angle - relative gas angle / - ratio of specific heats s « w 2/v 2 Mw - e Mw 2 5 Mw 2 o = u/v2 X - secondary loss acceleration parameter of Figure - wall angle - velocity coefficient; ratio of actual to isentropic velocity r\ - efficiency (static) a - rotor blade centrifugal root stress p - density \J/ J. -6o- t - maximum blade thickness v - Pr and tl -Meyer angle sin" the Mach angle angular velocity - co (tt)j SUPERSCRIPTS AND SUBSCRIPTS total gas properties referenced to absolute gas velocity 11 - total relative gas properties, refernoed to relative gas velocity - stator inlet - stator exit plane o - downstream of stator, after mixing ^ - just after rotor leading edge - downstream of rotor, after mixing - profile A of Section II-5 q 1 k A «y ' B axial component profile B of Section II-5 cross section c central or mid-chord rotor channel cen inner or convex rotor profile surface l -rg - isentropic yr - rotor tip loss rotor leading edge LE J4 m out - mean rotor profile curvature vector mean angle outer or concave rotor profile surface -61- p - profile loss d - rotor o - stator secondary J loss sec m -j- - total loss, or total head efficiency - nozzle throat tran tt - rotor entrance transition channel - corresponding to relative velocity - upstream, or inlet, position downstream, or exit, position APPENDIX B DEVELOPMENT OF EQUATION II-3-1 By reference to a T-S diagram one may easily show (e.g. Appendix B of Ref. h) that (c ? } T. With Ti ' H ) V/V R ~ '/T TC ioo p4'/ p o' _ Ps'/Po'^'/To') 777 "1 " (C ma y now ^ e expressed in terms of ' T]m 777 " 1 and Ph'/p V Vis o' [1 ' ) (V/V) " and " ' 1 Since \ ' • the temperature corresponding to an isentropic pressure decrease top^', there is T^'jg = T Q (P^'/Pq T (V/V) p)s 5 T °' - V V - Vis V V 1711177 ^I^V/V) - = ]" there is 7 "1 ? V/V = (1 - nT ) + n T (pi+'/Po') and finally, (0'P^R P)r V/V (c p )s Ki-V +ti T (V/P -62- , ) (7"1)/ ^ 7/7_1 » « -63- o o o m — H cr> 0> o o >T in o> !ct> o o o o g- m in ce leg l«-* m r» m sr o !* !lM CO CO gr» e- ;<0 (" CO !•*" r>- •o <o CO gr- eg eg * (« =0 m in jo* r~ CO h i CO m CO o o CO p«- f- ;<0 !o> !<o r-- CO ce •o r- in m m — -0 •0 -o -o m m g- in !r» •-I f-H o> in m m i* in m -0 •o -0 r- o ;in !o> o g- CO CO o in I** CO r~ r» !0 r» |h- o CO ph e- r» -0 m i"-* ;<\| CO ^p» in CO -0 <0 •0 m t-l M g- r* CO cr r- r- g-o re <\ •a IO CO gCO •a- ih -h o m o CO r- m CO gCO in eg •O in i* in m in in -0 o m ,ro in in rn in m m m CO -0 m r- m |en o 0- o eg in eg eg in 43 eg in ;r» 'o m 0"> o in «> _H m in o> g- m CO g- eg r- Wi g- •o g- p" •o g- g- ag- CO :p- •o g- g- * o in o> CO r~ g- r- CO •o gg- ci CO g- eg in m o g- -n g- i i o o o 3 3 in ITt CT- in O m — a o >"2 ^ ^- in o eg "J in eg o m m m m o in fM o fM CO •a !r- r- -o -o o g- in g - fn ^ o m in in o c in -0 cr CO m in o m -o e- r- m rin o CO CO in in gg- fO — in CO m o 0> o ft m in 0> CO —* in o fO ^ o o in o cr> o *"" "" rg- eg • * I * -64- 3 ty 3 3 3 O D D d m CT> 00 CO <J> 0" 0- a- 3 >»- J 0> 3 <7> c a- <7> CT- in CO 0> [0> o> CD co o ro C7> CO ico lO^ ro> 0> CO 0^ -»• o o CO 0> |co i0> r0> in r- C\l 0* r- o> <7> -0 Cy (M r- CO vO o> o >r -o c 0> 0> o- 0> o> co •o IT IT o> o> o"> o CO 0* 0> NO •o m -o o o> O CO m <M cr in o- ,r-- 0> 0> t> Ct> (7> CT> ~0 C> m in .0 .0 J1 O CT- J- 0> o (7> in m 0> m .n o r» «r 0> -I (M fNJ ^ —> r^ II £ O • O *V o O • • O a> CT> in in in in 0"> LI 3< 0> r.* a* o <* -J- <\J -)- *T o> 0"> ro 0> (M -H -l O -H -i o> LO o> o> o> iO CO -r r- in 0"> in i0> in >i -3- -» o> ;o> * 0> o CO ro <0 in r^l pn o I-4" -a- -3" !o> o< 0> o O (•- in f^ m o> IM i- in rv; <VJ cr- <7> <J- o- r~i m 0* CT> m o — 1 — —I (NJ — — . 0> — •-! -h LO r\i ,<7> r* eg — C7> CD ro rg ay CM a; oir»oiAoiAoirtomomoinomoir»ouno^noi^o PH ^t ^-. -H ^ » -65- o o o m 1A o> •r o CP eo CO ao pr- m 0> f- p- m o> o m -0 <0 o 1*1 <0 o P<P» •0 a- in jl/N in in <0 ft m in o m CO r- o •c * i o o o •T in 0> o o o in in CP o o o m H m cp in PP o o o in o o o ao in O M a in CM r- »-< o fM -T m o p-> p» 90 m p- CO eo o CO CO ao pr- rreo CD CO o r- in CO CO CO in CO <M in CO oo in pca o o •3" in p» p- — * co CO *- in r- (\J CO co p- fM 00 P- <p r- nj p- r- m If*- o o O fM r- P- m in <0 •o w -0 <0 O m -0 !p» ;p- n CP in r- fM CO CO rr ,<0 cp p- p- .0 r-o >o pr- CO o in o CP IN fM -0 o CM in *0 pi NO CO ;in o >0 < icO •* CP m •0 o 0> in 'a* .0 in in m in in o m oo in fM in in m o m r- I'm m o CO in J- pCO in in m mm CO OS in m oo in in in o in 00 in m in in in pin in m cp m .0 o o cp in pn vO in CO CO cp NT o m m m •J- m in m : CO co rp- fM o o> -H o o in m ppi -3- in CM in in in fM ^^ ^ O -H P- P- -* fM 00 in —I — I -H r-l p^ O P•J- g h O Po- o in in =0 o * 00 O * < I i < < ' -66- rvj co c\j f\l <\t CM — — ~* — ( *»t .-i ~4 — CN. CNj <\J — —I i-f ~H — — 67- X 1 1 1 1 1 1 1 « -p w 1 ft X 1 1 O o o OJ UACO CO [— o\ oaco CO CO OACO OA OA o ft 0) H4H OJ ft o CO CO CO CO LT\ CA OA CA CA CA na ON CO CO CO CO t— t OJ CO CQ CQ I I I — — — r-vo O O O O MA c utn t o CD CQ >H p I I I CO UA CO CO CO CO -4 OAVO CA CA OA G\ OJ -4 O H O O O O VO VO VO VO OJ VO VO VO VO -4 O O O O MA O LfMAO lalaJO O O O O O H O O O OJ OJ H H H OJ H O O O O O CO LT\CO CO CO CO H rAH OJ OJ OJ H VO VO VO VO CO O K> CACQ CA CA CA CA (A fO fO fO tA Q0l tr^ i: I VO VO UA t~CO CO VO Ph >h S K O £• S3 ft k O t^vo VO VO VO CO -4 OJ -4 ir\ ir\ _4 KA CM OJ OJ OJ CO O— O O O O O -4 C~— C— C— c— tr— O O O O O O OJ >H S ft s < H 3 § >H H t- I i i^O §§ H H c OAco OAco CO co EH >H Q — — — ON CA CA CA CA OiaoihOO o o\ o ca ca ca Ph O O ^ ^ O OIlA £r CO t^ VO c— in H X H s pq ca So ^; li En o H O En CO KA J- ft _rj- o Eh O H OJ OJ OJ OJ OJ OJ 3 pq » Q O r<A _Hr o O O O O O ka nA na ka ^t J- J- _=J- fpq KA _d- J- -4 o OJ rH H H H KA HO O O O O O O _h- ft O En UA UA UA UA UA UA UA UA UA UA o o o o o ft ft ft •H -P ft O O OO O O G\ r- [— ot— t-co t- CO O O O O t^- OA t— t- L^-CO CQ $£ _d" LTN VO -4 -4 -4 -4 t— UA UA UA LfA o a CQ OJ >> O H H o H ft ft < ft VO OJ OJ OJ OJ -4 o CO O o O O VO H rH H H H H UA UA UA UA UA UA co O H t — — — — C : t C C- *=i» — C—VO t— t— b— C— o o o o o OJ OJ OJ OJ K^ H H H H H H H H H OJ H H H O H r<A MA r<A r<A H SB O O o MA CO MA MA MA ON H H O H H H OJ O OJ OJ OJ OJ OJ OJ 3h t~- t— I I 1 OA O O O CO O O CO MA oo o O o O O I I H H H H H H CO OJ OJ OJ OJ OJ OJ o H H H H CO OJ OJ OJ OJ O H VO VO H CO H -4 OA -4 -4 -4 UA t>- CO ft t- VO VO VO rH MACO CO -4 H H OJ OJ OJ OJ OJ OJ OJ H A H LT\VO C— CO CA CA OJ OA OA CA r-\ O H H OJ -68- 0.80 0.70 0.60 0.50 >- o y o 0.40 u. b. UJ U < 0.30 0.20 0.10 0.20 0.50 0.30 BLADE VELOCITY _U_ j ISENTROPIC JET VELOCITY Figure 1. ' C Comparison of Theoretical Efficiency Estimates for (Subsonic ) Single Stage Impulse Turbines with Results of Supersonic Turbine Tests. - -6 9 - NOZZLE BLADES (£ x =0 ) 0.12 £ 0.10 VALUES OF GAS OUTLET ANGLE UJ o ao8 u. g 0.06 CO o 0.02 •H O 0.4 0.2 cr as o.6 a. PITCH /CHORD (a) IMPULSE BLADES (ft 0.20 a. O.I > ' i_ ft) 70° / 8 * n is U.I o * * y o 014 u. "UJ m V s s O 1 U.I 65° V 9 £ —™ o o nin 0I ° r "' 60^ 55° OT Py 50° </> o OQ8 J \j.\jo 40° III —J — u. o n r*ft u.uo JT Q. U.U *» . no?* n .. 0.2 0.4 0.6 0.8 1.2 1.0 PITCH /CHORD GO Figure 2. Profile Loss Coefficients for Conventional t/c = .20; Section Blades at Zero Incidence, Re = 2 x 10 5 ; M < 0.6. (From Ref l) . -70- 0.035 0030 0.025 0.020 0.015 0.010 0.005 0.20 0.10 0.30 (Wx /Wy 0.40 p [l+UD/O.D.)] Figure 3. Secondary Loss Acceleration Parameter. 050 0.60 -71- 0.90 0.80 0.70 0.60 050 0.40 PRESSURE COEFFICIENT, Cp Figure k. Pressure Coefficient as Function of M, Y, 7 = l.k. 0.30 -72- Y 0.6 1.0 098 096 0.94 0.92 030 088 086 VELOCITY COEFFICIENT, f Figure 5. Velocity Coefficient as Function of M, Y, 7 = l.k. 0.84 -73- 1.00 t/c c/s 0.98 -h"- 2.5 uua Rhub/RTip = 0.90 ^ <vr --> X=..4 ^-^ 0.92 TR = Ah 0.96 0.94 * .25 *> i«: 0.90 ^5 s \ %* ?o ^"^L \ 0JB8 ,A TURBINE^ KOF REF. 0.86 \ 2 ^JEST ^RESUl 1 \ 0.84 \ SUBSC )NIC CU RVE 0.82 F tEF.^ \ \ \ (REF. 12) 0.80 25 30 35 40 45 fi, £' Figure 6. 50 55 60 65 70 DEGREES Estimate of Rotor Velocity Coefficient Typical Impulse Blade in Supersonic Flow. 75 -74- 0.98 0.96 H Z O 0.94 LlI U_ Ll. LJ O O 0.9?. 8 LU > 0.9 0.88 0.86 0.84 0.20 0.30 0.40 FRACTION OF SPAN Figure 7. Cascade Loss Results of Reference 13. 0.50 0.60 -75- 0.2 0.4 BLADE SPEED Figure 8. 0.6 / 0.8 DESIGN SPEED Comparison of Rotor Incidence From Experiment and Two Theories - For Turbine of Reference 2. -76- STATIONS A _vf3 / _V2 >V _ jr-s^i 02 Figure 9. = ^4 Velocity Tri&ngle Nomenclature. : -77- 5=. 68 0.175 • 6 " a- = 30000 psi RA O X .66^70/ ^ t><; .76 \v 0.170 .58 C\s ^N. w 0165 v\ o-=375 psi X\N VS. > / ' \ s X 0.160 ^*v a= 45000 nsi . >» \N \ v .56 y / FlF" 7 ^cg NX/ 0.155 L ^S 9 s "*"* 4 .54 > s/fr 0.150 P$9 8 =.52 Nfe 0.145 ^s^ 0.140 ^# 0135 44 46 48 50 52 54 /3 Figure 10. F' 56 58 60 62 64 66 -DEG. and F" Functions for Turbine Design Problem. 68 -78- 0.56 44 46 48 50 52 54 fyFigure 11. 56 58 60 62 64 DEG. Constant Stress Efficiency Curves for Turbine Design Problem. 66 68 -79 U=2330 Figure 12. Design Velocity Triangle Mean Radius Conditions. - -8o- AXIAL DIRECTION THROAT n igure 13. Sharp-Corner Nozzle Geometry Considerations (Not to Scale) -8l- SCALE Figure ika. Design Nozzle Profile. Figure lVb. Sharp-Corner Nozzle Cascade •• 1.72/1 -82- SCALE-- 5.74/1 T/C =.160 c /s - 2.88 Figure 15. Design Rotor Profile. thesB954 Investigation of the influence of supers 3 2768 002 08876 7 DUDLEY KNOX LIBRARY

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Download PDF

advertising