the disk had corrosion pits with nickel and cadmium plated on the pitted surface. Only one other P & W JT8D engine experienced a failure of the seventh-stage disk during taxi to take off in 1985, but the cause was not established because disk fragments were not recovered. 7.13 EXAMPLE PROBLEMS Problem 7.1 A gas generator turbine disk has a design speed of 10,970 rpm. The geometry and material properties of the disk, similar to that shown in Fig. 7.27, are described in Table 7.4. Material density is 0.29 lb/in 3 Stress at the rim due to the IMPELLER AND BLADED DISK 263 TABLE 7.4 Disk Data Element Inner radius, r i Outer radius, r o Axial length, hE n no. (in) (in) (in) (10 6 lb/in 2 ) – 1 1.625 2.25 4.125 28.0 0.3 2 2.25 2.625 4.125 27.5 0.3 3 2.625 3.25 3.99 27.0 0.3 4 3.25 3.98 3.42 26.5 0.3 5 3.98 4.78 2.5625 26.0 0.3 6 4.78 5.75 1.98 25.5 0.3 7 5.75 6.625 1.62 25.0 0.3 8 6.625 7.1875 1.3025 24.75 0.3 9 7.1875 7.8125 1.125 24.5 0.3 10 7.8125 8.39 1.0625 24.25 0.3 11 8.39 8.81 1.39 24.0 0.3 12 8.81 9.16 1.745 24.0 0.3 FIGURE 7.43 High-magnification view of corrosion pit filled with nickel-cadmium deposit (NTSB/AAR-96/03). attached blades and retainers is calculated to be 24,210 lb/in 2 Determine the radial and tangential stress distribution. Solution The method and equations provided in Secs. 7.7 and 7.9 will be used to calculate the stresses. The calculation sequence is carried out at design speed and at no speed. For the first iteration it will be assumed that the stress at the bore is 40,000 lb/in 2 and the r 2 ∆ value is 90,000 lb/in 2 Using Eqs. (7.25) and (7.26) calculate the Σ and r 2 ∆ values at the inner and outer radii of the first, or inner, ring element. Subtract the inner radius value from the one at the outer radius to determine the difference val- ues dΣ and dr 2 ∆. Then at the outer radius Σ=40,000 − 1559 = 38,441 and r 2 ∆= 90,000 + 3234 = 93,234, from which ∆=93,234/2.25 2 = 18,417. Thus, at the outer radius σ t + σ r = 38,441 and σ t − σ r = 18,417, from which σ t = 28,429 and σ r = 10,012. To account for the difference in axial hub length between two adjacent rings, the condition of equality of radial growth at the interface will be applied. Then ( σ ti − νσ ri ) n = ( σ to − νσ ro ) n−1 or ( σ ti ) n = ( σ to ) n−1 − ν {( σ ro ) n−1 − ( σ ri ) n } Thus, ( σ ti ) 3 = 26,252 lb/in 2 . Together with Eqs. (7.27), (7.28), and (7.29) the proce- dure is repeated for the other elements to obtain stresses at the outer rim for the design speed and at zero speed. The results are shown in Tables 7.5 and 7.6. The value of σ ro obtained at the rim will not agree with the known rim radial stress based on the rim load. Denoting the true stress by σ rim and the stress in the first and the second set of calculations by σ ′ rim and σ ′′ rim , a factor K can be obtained from the expres- sion σ rim = σ ′ rim + K σ ′′ rim . The value of the factor is K = {24210 − (−13959)}/19464 = 1.961 Actual stresses in the disk are found by multiplying all values in the second set by the factor K and adding them to the results of the first calculation set (Table 7.7). The disk stress curves as a function of the radius will be discontinuous at the element inter- faces. Average values of the radial and tangential stresses should be obtained at the 264 COMPONENT DESIGN TABLE 7.5 Disk Sum and Difference Calculations at Design Speed (10,970 rpm) s ri ss ro s ti s to Element no. d Σ dr 2 ∆ (lb/in 2 ) (lb/in 2 ) (lb/in 2 ) (lb/in 2 ) 1 −1559 3234 0 10012 40000 28429 2 −1177 3788 10351 11592 28530 25672 3 −2364 11108 13524 12331 26252 22569 4 −3398 24154 16458 11576 23806 19927 5 −4512 46995 14981 9572 20948 17418 6 −6576 98984 11699 5999 18057 14415 7 −6971 144427 7462 1907 14854 11537 8 −5002 128676 2207 −1116 11627 9557 9 −6036 183124 −1181 −4814 9538 7220 10 −6024 213151 −3680 −8540 7560 4922 11 −4651 185324 −6803 −11433 5444 3164 12 −4049 176082 −11433 −13959 3164 1642 interfaces to obtain a smooth curve through the data points. The results are shown in Table 7.8, and the data are graphically shown in Fig. 7.44. The average tangential stress for the whole disk is determined by multiplying the average tangential stress over each ring element by the area of that element, adding the values for all the M elements and dividing the summation by the sum of the cross sec- tional areas of all the elements. (7.33) The average tangential stress value thus obtained is of considerable interest. It may be used for comparison with experimental data to establish the speed at which the disk may be expected to fail from burst when the tangential stress reaches the limit. A large number of disks have been tested in a spin pit with the express purpose of deter- mining the speed at which failure occurs from burst, thus establishing the overspeed σ σσ ave = ∑− ∑− = + = n M oi n M oi to ti rrh rrh 1 1 2 () () IMPELLER AND BLADED DISK 265 TABLE 7.7 Disk Stress Distribution after Modification by Factor K Element no. s ri (lb/in 2 ) s ro (lb/in 2 ) s ti (lb/in 2 ) s to (lb/in 2 ) 1 0 31,802 118,442 85,082 2 32,878 38,006 85,404 77,700 3 44,341 43,198 79,601 70,145 4 57,653 45,226 74,481 64,719 5 58,531 44,931 68,710 60,502 6 54,916 42,551 63,497 56,306 7 52,924 39,117 59,418 52,769 8 45,289 36,397 54,620 50,487 9 38,538 32,962 51,129 47,887 10 25,195 29,428 45,557 45,397 11 23,441 26,652 43,601 43,523 12 26,652 24,210 43,523 41,915 TABLE 7.6 Disk Sum and Difference Calculations at Rest (0 rpm) s ri s ro s ti s to Element no. d Σ dr 2 ∆ (lb/in 2 ) (lb/in 2 ) (lb/in 2 ) (lb/in 2 ) 1 0 0 0 11,111 40,000 28,889 2 0 0 11,487 13,469 29,002 26,531 3 0 0 15,714 15,740 27,204 24,260 4 0 0 21,007 17,159 25,840 22,841 5 0 0 22,207 18,030 24,355 21,970 6 0 0 22,037 18,639 23,172 21,361 7 0 0 23,182 18,975 22,724 21,025 8 0 0 21,969 19,129 21,923 20,871 9 0 0 20,254 19,263 21,209 20,737 10 0 0 14,724 19,361 19,376 20,639 11 0 0 15,422 19,420 19,458 20,580 12 0 0 19,420 19,464 20,580 20,536 limit of a rotating machine. The failure may be either due to tangential forces across the diameter or due to hoop stresses in the circumferential direction as a conse- quence of excessive radial forces. As the disk overspeeds beyond the point at which the bore begins to yield, stress from the tangential forces increases at a rate less than that in the rest of the disk. The nonyielded material away from the bore then picks up more of the centrifugal loading, causing more material in the proximity of the inner radius to yield. The process spreads toward the rim, and eventually the entire disk yields. The ultimate strength of the material is reached along the full cross sec- tion, and a further increase in speed will theoretically cause a complete fracture on a diametral plane if the material is fully ductile and is not sensitive to notch effects. In practice, burst failures are encountered when the average tangential stress is between 75 and 100 percent of the material’s ultimate tensile strength. In this exam- ple problem, the average tangential stress is calculated to be 64,857 lb/in. 2 266 COMPONENT DESIGN TABLE 7.8 Average Disk Stress Distribution at Element Interface Stress Element no. Radial Height (in) Radial (lb/in 2 ) Tangential (lb/in 2 ) — 1.625 0 11,8442 1 2.25 32,340 85,243 2 2.625 41,173 78,650 3 3.25 50,426 72,313 4 3.98 51,878 66,715 5 4.78 49,923 62,000 6 5.75 47,738 57,862 7 6.625 42,203 53,695 8 7.1875 37,468 50,808 9 7.8125 29,078 46,722 10 8.39 26,434 44,499 11 8.81 26,652 43,523 12 9.16 24,210 41,915 FIGURE 7.44 Calculated average disk stress distribution at element interface. Bursts from high radial stress generally cannot be predicted precisely since the dis- tribution does not tend to follow a pattern. This causes the radial direction stresses to vary nonlinearly at high speeds. Growth in the radial direction causes the centrifugal forces in the disk and at the rim also to increase until failure takes place. To avoid this situation, a more desirable approach is to limit peak values of radial stress to the aver- age tangential stress in the disk. REFERENCES Amedick, V., and Simon, H., “Numerical simulation of flow through the rotor of a radial inflow tur- bine,” ASME Paper # 97-GT-90, New York, 1997. Bladie, R., Jonker, J. B., and Van den Braembussche, R. A., “Finite element calculations and experi- mental verification of the unsteady potential flow in a centrifugal pump,” International Journal of Numerical Methods in Fluids 19(12), 1994. Cairo, R. R., and Sargent, K. A., “Twin web disk—A step beyond convention,” ASME Paper # 98- GT-505, New York, 1998. Childs, D., “Fluid-structure interaction forces at pump impeller-shroud surfaces for rotor dynamic cal- culations,” Transactions, 111:216–233, ASME, New York, 1989. Dambach, R., Hodson, H. P., and Huntsman, I., “An experimental study of tip clearance flow in a radial inflow turbine,” ASME Paper # 98-GT-467, New York, 1998. Fatsis, A., Pierret, S., and Van den Braembussche, R. A., “Three-dimensional unsteady flow and forces in centrifugal compressors with circumferential distortion of the outlet static pressure,” ASME Journal of Turbo-Machinery 119:94–102, 1997. Hiett, G. F., and Johnston, I. H., “Experiments concerning the aerodynamic performance of inward radial flow turbines,” Vol. 178, Proceedings, Institute of Mechanical Engineers, England, Part 3I(ii), 1964. Hillewaert, K., and Van den Braembussche, R. A., “Numerical solution of impeller-volute interaction in centrifugal compressors,” ASME Paper # 98-GT-244, New York, 1998. Japikse, D., “The technology of centrifugal compressors: A design approach and new goals for research,” VKI Lecture Series no. 1987–1, 1987. Justen, F., Ziegler, K. U., and Gallus, H. E., “Experimental investigation of unsteady flow phenomena in a centrifugal compressor vaned diffuser of variable geometry,” ASME Paper # 98-GT-368, New York, 1998. Kenny, D. P., “A comparison of the predicted and measured performance of high pressure ratio cen- trifugal compressor diffusers,” ASME Paper # 72-GT-54, New York, 1972. Kerrebrock, J. L., Aircraft Engines and Gas Turbines, MIT Press, Cambridge, MA, 1992. Miner, S. M., Flack, R. D., and Allaire, P. E., “Two-dimensional flow analysis of a laboratory cen- trifugal pump,” ASME Journal of Turbo-Machinery 114:333–339, 1992. Moore J. J., and Palazzolo, A. B., “Rotor dynamic force prediction of whirling centrifugal impeller shroud passages using computational fluid dynamic techniques,” ASME Paper # 99-GT-334, New York, 1999. Mowill, J., and Strom, S., “An advanced radial component gas turbine,” ASME Journal of Engineering for Power 105:947–952, 1983. Nakazawa, N., Ogita, H., Takahashi, M., Yoshizawa, T., and Mori, Y., “Radial turbine develop- ment for the 100 kW automotive ceramic gas turbine,” ASME Paper # 96-GT-366, New York, 1996. Orth, U., Ebbinh, H., Krain, H., Weber, A., and Hoffmann, B., “Improved compressor exit diffuser for an industrial gas turbine,” ASME Paper # 2001-GT-323, New York, 2001. Rhone, K., and Baumann, K., “Untersuchungen der Stromung am Austritt eines offenen Radialverdichterlaufrades und Vergleich mit der klassichen Jet-Wake Theorie,” VDI Berichte No. 706, 1988. IMPELLER AND BLADED DISK 267 Saravanamuttoo, H. I. H., Rogers, G. F. C., and Cohen, H., Gas Turbine Theory, 5th ed., Prentice-Hall, Harlow, England,1999. Sawyer, T., Gas Turbines, Vols. I–III, International Gas Turbine Institute, Atlanta, ASME, 1982. BIBLIOGRAPHY Casey, M. V., “The effects of Reynolds number on the efficiency of centrifugal compressor stages,” ASME Journal for Engineering Power 107:541–548, 1985. Childs, P. R. N., and Noronha, M. B., “The impact of machining techniques on centrifugal compres- sor impeller performance,” ASME Paper # 97-GT-456, New York, 1997. Frischbier, J., Schulze, G., Zielinski, M., and Ziller, G., “Blade vibrations of a high speed compressor blisk-rotor,” ASME Paper # 96-GT-24, New York, 1996. Hall, R. M., and Armstrong, E. K., “The vibration characteristics of an assembly of interlock shroud turbine blades,” in A. V. Srinivasan (ed.), Structural Dynamics Aspects of Bladed Disk Assemblies, ASME, New York, 1976. MacBain, J. C., Horner, J. E., Stange, W. A., and Ogg, J. S., “Vibration analysis of a spinning disk using image de-rotated holographic interferometry,” Experimental Mechanics, pp. 17–22, SESA, 1978. Mikolajczak, A. A., Snyder, L. E., Arnoldi, R. A., and Stargardter, H., “Advances in fan and com- pressor blade flutter analysis and predictions,” AIAA Journal of Aircraft 12(4):325–332, 1975. National Transportation Safety Board, “Uncontained engine failure/fire, valujet airlines flight 597,” NTSB Report # AAR-96/03, Aircraft Accident Report, Washington, D.C., July 30, 1996. National Transportation Safety Board, “Aircraft Accident Report—Uncontained Engine Failure,” NTSB Report # NTSB/AAR-98/01, Washington, D.C., 1998. Pfeiffer, R., “Blade vibrations of continuously coupled and packed steam turbine LP stages,” in R. E. Kielband, N. F. Rieger (eds.), Vibrations of Blades and Bladed Disk Assemblies, ASME, New York, 1985. Simon, H., and Bulskamper, A., “On the evaluation of Reynolds number and relative surface rough- ness effects on centrifugal compressor performance based in systematic experimental investigations,” ASME Journal for Engineering Power 106:489–498, 1984. Srinivasan, A. V., Lionberger, S. R., and Brown, K. W., “Dynamic analysis of an assembly of shrouded blades using component modes,” ASME Journal of Mechanical Design 100:520–527, 1978. Srinivasan, A. V., “Flutter and resonant vibration characteristics of engine blades,” ASME Paper # 97- GT-533, New York, 1997. Stetson, K. A., and Elkins, J. N., “Optical system for dynamic analysis of rotating structures,” Air Force Aero Propulsion Lab Contract F33615-75-C-2013, AFAPL-TR-77-51, October 1977. Wadell, P., “Strain pattern experimentation,” Engineering and Materials Design, 17(3), 1973. Wiesner, F. J., “A new appraisal of Reynolds number effects on centrifugal compressor performance,” ASME Journal for Engineering Power 101:384–396, 1979. 268 COMPONENT DESIGN TURBINE BLADE AND VANE 8.1 INTRODUCTION Extensive analytical predictions and rig testing notwithstanding, blade failures in engines due to excessive vibrations still occur during final test phases. Even worse is the occurrence of failures after an engine has successfully passed a number of rigorous qualification and certification tests and goes into regular service. The failures may point to a lack of adequate design tools, but may indicate another issue arising from the business perspective. In the ongoing search for efficiency improvement and better thrust-to-weight ratio, engine design- ers and researchers are pursuing higher speeds and temperatures, radical new blade pro- files, higher stage loads, and lower aspect ratios. A more exacting consideration of blade flutter, resonance, cooling, material characteristics, and manufacturing methods is required than hitherto. The calculation of blade resonant frequencies and mode shapes of a bladed disk system calls for the definition of boundary conditions at shroud and dovetail interfaces. The max- imum operating speed of the engine cannot be established unless vulnerable frequencies and response amplitudes at resonance are known to a satisfactory degree of confidence. Research efforts have focused on the need to develop a fundamental understanding of top- ics such as Coulomb and viscous damping, coolant passage turbulators, and unsteady vis- cous flow in turbomachines. Without adequate knowledge of damping, for example, there is no alternative but to guess values for the coefficient in determining the response from a particular excitation. Note that damping may arise from material characteristics, friction, aerodynamic flow, and possibly from impact. Shrouds in the form of a protrusion are used in turbine blades to alleviate problems aris- ing from dynamic motion in the blade. Long and slender turbine blades in the last stages of gas turbines can take advantage of the support provided by mating surfaces of shrouds on adjacent blades to reduce the flexural and twisting motion at the tip. Even longer blades in low-pressure steam turbines may be equipped with some form of dampening mechanism at midspan and tip locations. However, the shroud also imposes penalties in the form of a mass at the tip of the blade, which requires body loads due to the centrifugal force field to be carried by the rest of the airfoil. It also adds to the manufacturing cost of the blade. In a shrouded blade system the protrusion constrains blade motion not only along the contact plane between the shrouds of adjacent blades but also along the normal direction of the plane. In-plane tangential relative motion is mostly two-dimensional. On the other hand, normal relative motion can cause variations in normal contact load and, in extreme cases, separation of the contact interface. Low-cycle fatigue (LCF) failures of turbine blades are of great significance in aircraft engines. Thermal gradients and mechanical stress from centrifugal loading rapidly increase as the engine is started and goes to full speed during aircraft takeoff from the ground. CHAPTER 8 269 Copyright © 2005 by The McGraw-Hill Companies, Inc. Click here for terms of use. Figure 8.1 shows a typical stress and temperature distribution in a freestanding and shrouded turbine blade. The reverse pattern is repeated during landing and engine shutdown. When blade rows in a turbine are operating close to resonance conditions, they are prone to failure from high-cycle material fatigue. In power generation turbines operating at a near constant speed blade resonance can occur during engine startup and shutdown conditions. Excitation is encountered due to two primary reasons: (i) flow path interference between stator and rotor blade rows at nozzle passing frequencies and (ii) manufacturing and assembly errors at per rev- olution harmonics. Interference between the stationary and moving blades is an aerodynamic phenomenon, and is based on potential interaction, wake interaction, and viscous effects. Mechanical excitation from manufacturing errors and mounting of stationary diaphragms can- not be readily simulated using mathematical formulations. Turbine blades and vanes constitute a considerable portion of the total cost of the equip- ment, given the fact that thousands of airfoils are used in any turbomachine. An accurate determination of the operating life of turbine and compressor blades plays a central role in the design of aircraft power plants. The rotating parts must be retired prior to failure, but 270 COMPONENT DESIGN Blade peak stress Metal temperature Allowable stress Stress or temperature 0 20406080100 Base Tip Percent of blade height 0 20 40 60 80 100 Base Tip Allowable stress Metal temperature Blade peak stress Stress or temperature Percent of blade height FIGURE 8.1 Stress and temperature distribution in freestanding (upper) and shrouded (lower) turbine blades. must still possess adequate life to be commercially acceptable to airline operators. Life esti- mation using stress-based theories is a multifaceted technology, and calls for the calcula- tion of mean steady stresses, dynamic stresses, failure surface, load history, and cumulative damage. The influence of mean stress may be described by a number of linear and nonlin- ear relations, for example, Goodman, Soderberg, Gerber, Marin, and Kececioglu. Several cumulative theories for alternating stresses of varying amplitudes are based on the damage accumulated during the load cycles. A common theory due to Palmgren and Miner asserts that damage fraction at any stress level is linearly proportional to the ratio of the number of cycles of operation to the total number of cycles that would produce failure at that stress level. However, the order of application of different stress levels is not recognized, and damage is assumed to accumulate at the same rate at a given stress level without the con- sideration of the history. Experimental evidence indicates that fatigue damage accumulates nonlinearly, depending on the alternating stress level. Nonlinear theories proposed by Marco and Starkey, Corten and others rely on some exponent of the same ratio. A problem sometimes encountered with the application of nonlinear theories is the lack of material data for the exponent at different stress levels. The benefits of reduced fuel consumption and increased power arising from increasing the turbine inlet temperature have been clearly brought out in Chaps. 2 and 3. Despite losses experienced in cooling of blades and vanes, the gains are still considerable. Methods to cool the blades receive serious research attention. Cooling of blades with liquids is difficult because of practical problems associated with the delivery and retrieval of the coolant in the primary cooling system in the forced or free convection modes, or in a closed secondary system. Difficulties are also encountered due to corrosion and deposits in open systems. In closed systems it is difficult to obtain sufficient secondary surface cooling area at the base of the blade. Internal air cooling in the forced convection mode is more practical in engines. Turbine blade metal temperatures may achieve a reduction of 200 to 300°C by channeling 1.5–2.0 percent of the airflow for cooling for each blade row. The blades may be cast with internal cooling passages in the core or forged and drilled with holes of any required shape and size using the electrodischarge, electrochemical, or laser drilling process. Outer surface cooling is achieved by pushing cooling air out of holes in the blade walls. In this process heat is extracted more uniformly from the surface and at the same time provides a layer of cooler air isolating the metal from the hot gases of the main stream. The concept of tran- spiration cooling requires porous blade walls for the cooler internal air to ooze out from the internal blade cavity, but successful application will depend on the availability of appro- priate porosity in the skin material and manufacturing methods. Cooling of the rotating airfoils still represents unusual difficulties from engineering and manufacturing considerations. At elevated gas and metal temperatures oxidation and creep impose limitations on the blade’s capabilities. Unlike rotating airfoils, nozzle vanes do not experience high stress levels. In spite of it, cooling of annulus walls and stator vanes requires special attention. A class of materials referred to as superalloys find application at a higher proportion of their actual melting point than any other group of commercial metallurgical materials. The alloys have made much of the very high temperature engineering technology possible, and are the leading edge materials of gas turbines in the air transport, power generation, and process industries. In turn, gas turbines have been the prime driving force for the develop- ment and subsequent existence of superalloys. The alloys respond to the need for materials with creep and fatigue resistance at high temperatures. Many of the alloys, perhaps 15 to 20 percent, have been developed for utilization in corrosion-resistant applications. Despite the higher cost of superalloys, the economic implications of increased temperature at the tur- bine inlet are overwhelming through increased efficiency and power output by the applica- tion of this group of materials. Figure 8.2 shows bladed disk assemblies for a steam turbine rotor, and Fig. 8.3 provides examples of new steam turbine buckets. TURBINE BLADE AND VANE 271 272 COMPONENT DESIGN FIGURE 8.2 Steam turbine rotor. (Courtesy: General Electric/Toshiba). FIGURE 8.3 Steam turbine buckets. (Courtesy: General Electric/Toshiba). [...]... p2 ( r ) p1 γ −1 γ (8. 17) 297 TURBINE BLADE AND VANE Shifted values Unshifted values 0 .84 5 0 .84 4 4° 0 .84 3 4° 6° p/pt0 (−) 0 .84 2 8 −4 −2 0° 2° 6° 0 .84 1 0 .84 0 0 .83 9 0 .83 8 0 .83 7 0 .83 6 0 .83 5 −6 −4 0. 18 −2 0 2 PHI (deg) 4 −6 Minimum due to the 2nd stator LE 0.175 4 6 Minimum due to the 1st stator wake 0° 0.17 Ma (−) 2 0 PHI (deg) 4° 0.165 0.16 0.155 6° 0.15 2° 0.145 0.14 −6 FIGURE 8. 24 −4 −2 0 2 PHI (deg)... 0 .8 0.5 0.2 0 50 2.0 100 Frequency (kHz) 0.0 150 1.3 1.0 0 .8 0.5 0.2 50 100 Frequency (kHz) 150 Kulite PS x/cax,r = 0 .81 1 .8 Amplitude p (kPa) 1.5 0 2.0 Kulite SS x/cax,r = 0 .81 1 .8 Amplitude p (kPa) 1.3 0.0 150 Amplitude p (kPa) Amplitude p (kPa) 100 Frequency (kHz) Kulite SS x/cax,r = 0.26 1 .8 0.0 1.5 0.2 2.0 0.0 Kulite PS x/cax,r = 0.01 2.0 Amplitude p (kPa) 2.0 1.5 1.3 1.0 0 .8 0.5 0.2 0 FIGURE 8. 20... of thermodynamics Vorticity, defined by ω= ∂v ∂u − ∂x ∂y is negative when shed from the vane suction side and positive from the pressure side 293 TURBINE BLADE AND VANE 1 .8 Amplitude p (kPa) Kulite SS x/cax,r = 0.05 2.0 1 .8 1.5 1.3 1.0 0 .8 0.5 0.2 0.0 0 1.5 1.3 1.0 0 .8 0.5 5 10 15 20 1 .8 0.2 0.0 0 50 1.0 0 .8 0.5 1.5 1.3 1.0 0 .8 0.5 0 50 2.0 100 Frequency (kHz) 150 Kulite PS x/cax,r = 0.36 1 .8 0.2 1.5... Parameter Axial chord, mm Tip radius, mm Inlet hub radius, mm Exit hub radius, mm Aspect ratio at inlet Stagger angle Number of blades Trailing edge diameter, mm Stator Rotor 29 .86 274.00 2 38. 84 2 38. 84 0.71 51.90° 43 1. 18 27.45 274.00 2 38. 00 235.31 1.07 32.71° 64 0.90 is unlikely to affect conditions at midspan As a consequence, a two-dimensional analysis gives a good representation Kulite pressure transducers,... (Fig 8. 8) constrained as given in Eq (8. 11) For each mode i and nodal diameter n (n = 0 to N/2), two identical eigen frequencies are computed, which refer to two possible mode shapes of the bladed disk assembly Static calculations may be readily performed by substituting n = 0 in Eqs (8. 10) and (8. 11), so the static equation of the assembly rotating at Ω becomes [ K (Ω)]{q} = {Po} + {T} + {F(Ω)} (8. 12)... level σ2 is applied first for n/N = 0.5 (OD in Fig 8. 6) and then with the higher stress level σ1 (line EC in Fig 8. 6) until damage occurs at D = 1, then ∑ N = N σ n n 2 n + N σ 1 = 0.5 + 0.946 08 = 1.446 08 FIGURE 8. 6 Fatigue damage for two cases (Rao, Pathak, and Chawla, 1999) (8. 3) 279 TURBINE BLADE AND VANE When the higher stress is applied first, it takes only a few more lower stress cycles... 8. 17 Stage configuration at midspan (Hummel, 2001) 291 TURBINE BLADE AND VANE 2.0 1.5 1.0 y/cax, r 0.5 0.0 −0.5 −1.0 −1.5 −2.0 −2.5 −2 −1 0 x/cax,r 1 2 Vane Blade FIGURE 8. 18 Computation grid entire domain (upper), detail at edges (lower) Figure 8. 19 shows the comparison between measured and calculated pressure fluctuations at the measurement points on the rotating blade surface indicated in Fig 8. 17... Fig 8. 6 using available data for two stress levels s1 and σ2, with σ1 > σ2 First, the specimen is loaded for n/N = 0.5 at stress level σ1 (line O − A), followed by the lower stress level σ2 (line B – C), until damage occurs at D = 1 Then ∑ N = N σ n n 1 n + N σ 2 = 0.5 + 0.05 384 = 0.55 384 (8. 2) On the other hand, if the lower stress level σ2 is applied first for n/N = 0.5 (OD in Fig 8. 6)... Figure 8. 8 shows the pressure and temperature variations Each inlet segment is then connected to an exhaust pipe serving three cylinders, and the pressure pulses are then directly led to the turbocharger turbine This results in considerable differences between the entrance conditions at the two inlets, and hence high dynamic loads may be expected to act on the blades 284 COMPONENT DESIGN FIGURE 8. 9 Resulting... Fig 8. 23) to receive data for 10 different stator positions Three stator positions (4°, 7°, and 9°) are TABLE 8. 2 Turbine Geometry Data Parameter Stators Rotor Tip diameter, mm Hub diameter, mm Radial height, mm Radial gap, mm Aspect ratio Number of airfoils Midspan pitch t, mm Flow angle a Relative flow angle b Rotor speed, rpm 600 490 55 — 0 .88 7 36 47.6 20° 49.3° — 600 490 55 0.4 0.917 41 41 .8 90° . 32,340 85 ,243 2 2.625 41,173 78, 650 3 3.25 50,426 72,313 4 3. 98 51 ,87 8 66,715 5 4. 78 49,923 62,000 6 5.75 47,7 38 57 ,86 2 7 6.625 42,203 53,695 8 7. 187 5 37,4 68 50 ,80 8 9 7 .81 25 29,0 78 46,722 10 8. 39. 40000 284 29 2 −1177 3 788 10351 11592 285 30 25672 3 −2364 111 08 13524 12331 26252 22569 4 −33 98 24154 164 58 11576 2 380 6 19927 5 −4512 46995 14 981 9572 209 48 174 18 6 −6576 989 84 11699 5999 180 57. 31 ,80 2 1 18, 442 85 , 082 2 32 ,87 8 38, 006 85 ,404 77,700 3 44,341 43,1 98 79,601 70,145 4 57,653 45,226 74, 481 64,719 5 58, 531 44,931 68, 710 60,502 6 54,916 42,551 63,497 56,306 7 52,924 39,117 59,418