Turbo Machinery Dynamics Part 6 pdf

and where η i = 0, (6.16) Using a thickness vector and midsurface coordinates, then (6.17) where x i , y i , and z i are the average values of the coordinates at the two surfaces. Then (6.18) where are unit orthogonal vectors with displacements in global axes x, y, z. u i , v i , w i are displacements at the midsurface nodes, and b, a are rotations about , providing a total of 5° of freedom at each node. In matrix notation, {U} = [N]{a}, where {a} is a column vector and [N] is obtained by expanding Eq. (6.18). With the displacements available, element properties, strains, and stresses need to be defined. From Fig. 6.40 the strain components are (6.19) where Hence {U′} = [N]{a′} and {e′} = [B]{a′} where [B] = [L][N]. Corresponding stresses in matrix form are {s′} = [D]{e′} (6.20) where elasticity matrix [D] is (6.21) []D E = −               − − − 1 10000 01000 00 0 0 00 0 0 00 0 0 2 1 2 1 2 1 2 ν ν ν κ ν κ [] , , ,, ,, ,, {}L x y yx zx zy U u v w = ′ ′ ′′ ′′ ′′               ′ = ′ ′ ′           00 00 0 0 0 and [] []{} , , ,, ,, ,, ′ =               = ′ ′ ′ + ′ ′ + ′ ′ + ′               = ′ ′ ′ ′′ ′′ ′′ ′ ′ ′′ ′′ ′′ ε ε ε γ γ γ x y xy xx yx x y yx xx yx u v uv wu wv LU ˜˜ VV ii12 and ˜˜ VV ii12 and u v w N u v w N t VV i i i i i i i ii i i i           =           +−       == ∑∑ 1 8 12 1 8 2 ζ α β [ ˜˜ ] x y z N x y z NV i i i i i ii i           =           + == ∑∑ (, ) ˜ ξη ζ mid 1 8 3 1 8 2 N io =+ − 1 2 11 2 ()( ) ξη FAN AND COMPRESSOR AIRFOILS 183 where E and ν are Young’s modulus and Poisson’s ratio. Factor κ approximates displace- ment due to shear. Properties of elements call for integration over their volume, and take the form of (6.22) where [S] is a function of the global coordinates x, y, z, and [S] = [B] T [D][B], with strain defined by Thus, [B] is defined in terms of displacement derivatives in the local cartesian coordinates x′, y′, z′, and {a} e is the displacement field. Integration of the element in the curvilinear coordinates can be performed after transformations from the local to the global system and then to the curvilinear x, h, z coordinates. Equation (6.13) relates global displacements u, v, W to the curvilinear coordinates. Derivatives of these displacements with respect to x, h, z may be obtained by the jacobian matrix, so derivatives of displacements in global coordinates are given by [U d ] global = [J] −1 [U d ] curvilinear . Matrix terms for the derivatives of the displacements in the curvilinear coor- dinates may then be written. Also, components of the jacobian matrix in curvilinear coor- dinates can be written using Eq. (6.8). Further transformation from global to local cartesian coordinates allows the establishment of strains. The direction of the local orthogonal axes can be ascertained by obtaining a vector normal to the surface z = constant by taking a vec- tor product of any two vectors tangent to the surface. Global derivatives of displacements u, v, W are next transformed to local derivatives of local orthogonal displacements to explic- itly obtain displacement derivatives at any x, h, z in the element as also the components of strain. The infinitesimal volume in curvilinear coordinates is dx dy dz = det[J] d ξ d η d ζ So the stiffness matrix is (6.23) and the mass matrix is (6.24) To obtain the equations of motion, the principle of virtual work can be gainfully employed to derive expressions for equilibrium of the body as an assembly of m finite ele- ments. According to this principle, the total internal virtual work done by compatible small [ ] [ ] [ ]det[ ]MNNJddd eT to = −+ ∫∫∫ ρξηζ 11 [ ] [ ] [ ][ ]det[ ]KBDBJddd eT to = −+ ∫∫∫ ξ η ζ 11 {} [ ]{} {} . . ε ==               Ba a a a a e e e 1 2 8 []Sd d d xyz ××× ∫ 184 COMPONENT DESIGN virtual displacements applied to a body must equal the total external virtual work in order to maintain equilibrium of the body. Mathematically the requirement may be expressed by (6.25) Internal virtual work done is denoted by the left side of this equation, and equals the stresses going through the virtual strains corresponding to the virtual displacements . External work is given on the right side. It equals work done by body forces f B , friction force {dU/dt}f F , and inertia force r{d 2 U/dt 2 } going through the virtual work. Virtual dis- placements in global coordinates in x, y, z directions are given by (6.26) Substitution in Eq. (6.20) yields equilibrium requirements. Variation in strain energy V of the elastic continuum and in the potential energy W of the applied loads may then be used in Hamilton’s equation dp = 0, where p = V − W, or (6.27) Simplification yields: (6.28) where Individual element matrices given by Eqs. (6.18) and (6.19) may be used for numerical integration (6.29) (6.30) [ ] [ ] [ ]det[ ]MHHHNNJ e ij k T k n j n i n = === ∑∑∑ ρ 111 [ ] [ ] [ ][ ]det[ ]KHHHBDBJ e ij k T k n j n i n = === ∑∑∑ 111 [ ] [] [] [] [][] [] [][][] {} [ ] MNNdV CNNfdV KBDBdV FNfdV T m T F m T m T B m = = = = ∫ ∑ ∫ ∑ ∫ ∑ ∫ ∑ ρ []{ / }[]{/}[]{}{}M d a dt C da dt K a F B 22 ++= δπ δ ρ δ =+     +−     = ∑ ∫∫ ∑ ∫∫ 1 2 1 2 0 {}[ ][ ][ ]{} {}[ ] [ ]{ / } {}[][]{/} {}[] a B D B a dV a N N d a dt dV aNNdadtfdV aNfdV TT T T TT F TT B 22 { } { }, [ ] [ ]{ }, { } { }U u v w UD=           == = δσ εεδε {}U − ∫ ∑ {} {UdUdtdV T m ρ 22 /} {}[] {} {}{ / } εσ TT B T F mmm dV U f dV U dU dt f dV=− ∫ ∑ ∫ ∑ ∫ ∑ FAN AND COMPRESSOR AIRFOILS 185 where H provides weight coefficients of the gaussian quadrature and n is the order of inte- gration. Reduced integration technique may be used, with the order of integration x, fol- lowed by h, and finally z. Total stiffness and mass matrices, assembled as a summation for all elements in the system, may be obtained using the frontal housekeeping algorithm (Irons, 1970). The eigenvalue problem is solved by a determinant search. See Prob. 6.5 as an illustration of this procedure using a computer program developed by Gupta (1984) for determining natural frequencies of vibration and their modes. 6.13 SWEPT FAN BLADE Fans, propellers, and compressors can benefit from specific advantages of low noise and improved performance by using skewed blades, also known as swept blades. The airfoil is said to have a sweep when tilted within the flow direction, and dihedral when tilted in the direction perpendicular to the flow. In general, airflow through a rotor is three dimensional. To simplify the analysis, it is assumed that the flow takes place in two separate two-dimensional surfaces. In axial flow machines the through flow and the blade-to-blade surfaces are such planes. In a simple case the through flow surface is the meridional plane, while the second surface is axisymmetric. Airfoil theory or experimental cascade data is frequently used to determine the appropriate blade section on the blade-to-blade surface. Flow distribution on through-flow surfaces may be treated with the equation of radial equilibrium. Fan blades are frequently skewed in the circumferential direction, as depicted in Fig. 6.41. For blade angles between 0° and 90° this results in a combination of sweep l and dihedral n. The effect of the skew is to create a force acting in a direction normal to the blade sur- face (Dejc, Trojanovskij, and Fillipov, 1973). The radial component of this force is similar in nature to a distributed body force expressible in terms of swirl velocity distribution in Euler’s turbine equation. But in the case of a high space and chord ratio, as in axial flow fans, this representation is not appropriate. One approach is to eliminate the dihedral (n = 0), thus suppressing the radial component. The swirl imparted to the airflow by axial fans of low-pressure rise is sufficiently small to omit a stator. This omission also has the benefit of reduced noise emission due to inter- action between the rotor and the stator. Two designs of a fan of different flow and pressure coefficients using elementary air- foil theory (Eck, 1972) with a successfully proven range of design points (Carolus and 186 COMPONENT DESIGN FIGURE 6.41 Fan blade with (a) forward skew, (b) circumferential tilt (Beiler and Carolus, 1999). Blade axis (a) (b) r v r q d d w l b b z Scheidel, 1988) will be considered. Design parameters for the blades are: rotor speed = 3000 rpm, hub-to-tip ratio = 0.4, outer fan diameter = 0.305 m, tip clearance = 0.0015 m, and number of blades = 6. Both fans are initially designed using NACA airfoil sections selected from Abbott and von Doenhoff (1959, NACA 0010-65). In the second phase, a series of fans with a systematic pattern of skew in the circumferential direction are devel- oped for each fan. Skew angles range from a constant value from hub to tip and a variable d(r) starting from 0° at the root to ±60° at the tip. The flow field is analyzed using a fully three-dimensional viscous computational fluid dynamics code (Beiler and Carolus, 1999). The code solves Reynolds averaged Navier- Stokes equation in primitive variable form. It employs a finite volume method with a lin- ear profile skew method of up-winding discretization, combined with a physically based advection corrected term. Second-order accuracy is achieved by linking pressure with velocity through a fourth-order pressure redistribution. Turbulence is modeled by the stan- dard k − e procedure, and uses logarithmic wall functions in the end regions. Although eddy viscosity models do not correctly mimic turbulent stress generation due to Coriolis forces associated with the rotating conditions, the k − e procedure yields sufficiently accurate results for the comparative study. Figure 6.42 shows an example of the grid. Boundary conditions at the hub and blade simulate fixed walls in a rotating frame of ref- erence, while the shroud is counterrotating, or stationary, in the fixed frame. Between adja- cent blade segments, the plane is expressed as a periodic boundary. Global parameters such as pressure and flow coefficients are established across relevant control surfaces. Validation of computer results is obtained from measurements on an aerodynamic test facility (Fig. 6.43). Velocity and total pressure distribution is measured in the absolute frame using fast response probes, as also static pressure and flow losses to evaluate fan perfor- mance and characteristics. Probes are mounted on a traversing unit that may be moved radi- ally. The revolution counter also acts as a trigger to allocate a measuring signal appropriate FAN AND COMPRESSOR AIRFOILS 187 FIGURE 6.42 Computational grid with boundary conditions (Beiler and Carolus, 1999). with circumferential position. The drive shaft extends to the driving unit, and includes a torque meter. The aluminum fan rotor, manufactured on a computer numerical control (CNC) milling machine center, is located inside the test unit. A hot film probe with mutu- ally orthogonal sensors measures the velocity distribution in all three directions at measur- ing plane no. 2. Data are acquired through an A/D converter with a maximum sampling frequency of 5 MHz. Analog signals from the pressure probe go through a dc amplifier, while the hot film indications are amplified by an anemometer before being digitized. The sound pressure level of straight and swept fans is measured with a standard acoustic mea- suring kit. Total pressure and velocity field measurements are carried out for a rotor with conven- tional straight blades, with blades skewed in the circumferential direction (d =+30°) and with blades skewed against the circumferential direction (d =−30°). The calculated flow, being solved in the rotating frame of reference, is converted to a fixed frame to facilitate comparison with test results. Figure 6.44 illustrates measured fan performance as reflected by the pressure rise and efficiency for a range of flow coefficients. With blades skewed in 188 COMPONENT DESIGN FIGURE 6.43 Aerodynamic fan test rig (Beiler and Carolus, 1999). FIGURE 6.44 Measured fan performance (Beiler and Carolus, 1999). the direction of rotation, flow separation tends to occur at a lower speed mostly due to deflection toward the hub, and hence the fan works at a higher pressure increase (or decreased flow volume) without flow separation. But the separation is more abrupt and pressure rise is lower than for fans with straight blades. For blades skewed against the direc- tion of rotation, flow separation may be deduced to occur at a higher volume flow rate if the flow is throttled. In Fig. 6.45 lift distribution obtained from the numerical analysis is compared for different skew angles. The straight fan blade exhibits the maximum lift coef- ficient. Sweep decreases blade loading and pressure rise across the rotor. Lift distribution does not increase gradually along the blade span, mostly because of interference with the hub and shroud surfaces. The opposite holds true for a swept-back airfoil. Note that the back- ward sweep is with respect to the hub, but is swept forward with respect to the shroud. Measured pressure and sound pressure level for a straight-bladed fan and a fan with blades swept in the upstream direction are shown in Fig. 6.46 for varying flow rates. The sweep angle increases linearly, starting from the hub with a 5° backward sweep and ending at the shroud with a 55° forward sweep to provide a favorable shape. An increase in pres- sure has a negative impact of a separating flow on the emitted noise. Noise increases sig- nificantly as soon as the flow separates. FAN AND COMPRESSOR AIRFOILS 189 FIGURE 6.45 Analytically predicted lift distribution (Beiler and Carolus, 1999). FIGURE 6.46 Comparison of measured pressure and sound (Beiler and Carolus, 1999). Some interesting conclusions may be drawn from this investigation. Blades with sweep but without dihedral can be treated in the same manner as swept airfoils. For the given com- bination of pressure rise, Mach number, and incorporated shroud, separation at high flow rates in swept-back blades corresponds to poor aerodynamic performance and noise emis- sion. On the other hand, forward-swept blades improve fan performance with a more uni- form outlet flow distribution and reduction in discharge losses. Forward-swept blades appear to have the potential for widespread application. 6.14 DESIGN OF AXIAL COMPRESSOR Simplification of the general layout of an axial compressor may be achieved by optimizing the blade’s flow path and by employing advanced aerodynamic design techniques. Consider the case of a 10.5-MW gas turbine with a high-performance compressor to obtain a 10 percent increase in the rated power and at the same time reduce the number of stages and sharply lower the manufacturing costs. Nuovo Pignone, the manufacturer, modified an existing 17-stage unit to obtain an 11-stage machine with wide chord high strength blades, with minimal changes at the interface with the remaining components and auxiliaries (Benvenuti, 1996). The target of increasing the mass flow by 10 percent and a compression ratio of 14:1 called for the addition of three new front stages with a fixed hub diameter and conically tapered outer case. To limit stresses in the root dovetails, the blades are made of titanium. Stall-related problems at start-up and at reduced operating speed are avoided by providing variable geometry vanes at the inlet and at the following four stator rows. Firing tempera- ture in the combustion system is left unchanged. Principal consequences on the blade design due to constraints imposed at the interface are as follows: • Maintain the diameter of the first-stage hub at the inlet so that the existing compressor end bearing design is not affected. • Exit flow path diameter must be compatible with the present combustor and turbine tran- sition piece. • Ensure that the overall compressor length is not affected for accommodation in the pres- ent base plate and accessories unit. The need to maintain overall length, coupled with fewer stages, makes it possible to increase the airfoil chord while reducing the total number of blades and enhancing the mechanical strength of the rotor. The one-piece rotor is modified to include bolted disk assembly for the first six stages and an integral structure for the remaining stages. The front-end disks with axial entry dovetails are designed for considerably higher centrifugal load, necessitating the use of titanium. High-strength 17-4 PH steel is used on the first three stages and 13 percent chromium steel on the rest of the shaft. A picture of the new rotor is shown in Fig. 6.47. A major impact of changes in the flow path is the increase in first-stage hub and tip diameters. This difference results in substantially higher blade peripheral speed with increased work and pressure ratio capability per stage without a corresponding increase in the aerodynamic loading coefficient. Peripheral speed at the hub experiences an average increase of 25 percent, resulting in 56 percent higher specific work without increasing the load at the hub. The number of stages theoretically required is then 17/1.56 = 10.9. The larger blade exit annulus diameter required the exit diffuser to be appropriately shaped to match existing downstream components. The diffuser’s contour calls for a constant outer wall diameter and a curved decreasing diameter inner wall, designed with the aid of a 190 COMPONENT DESIGN three-dimensional viscous flow analysis code. Bleeding from the fourth stage provides cooling and buffering for low-pressure components. Stage seven bleed is used only during engine start, and is closed during operation. The number of blades for each row is selected to achieve the right level of solidity to maintain diffusion factors below 0.5. The inlet swirl angle is set to provide tip relative Mach number between 1.15 and 1.20. Multiple circular arc airfoils have proved to be ade- quate in limiting shock losses in the supersonic flow region. The subsonic stages in the rear are designed with standard NACA65 series airfoils for the rotating blades and stator vanes. At the final stage a single-row exit guide with reduced number of vanes turn the airflow by 40°. The design flow coefficient is increased by 20 percent to reduce the rotor blade exit swirl. The potential penalty associated with the higher exit axial Mach number is offset by the increased length of the diffuser. Low aspect ratio blades provide increased mechanical strength, but the three-dimensional shapes have complex steady-state stress patterns and vibration mode-shapes. Detailed finite element models are built to evaluate secondary stresses and to compensate them with appropriate airfoil section stacking. Special attention is required in the dynamic analysis to predict natural frequencies accurately and to interpret the high-order complex mode shapes typical of thin, wide chord airfoils. Higher-order modes with one or more mode lines run- ning almost parallel to the edges are of concern due to the associated higher level of vibra- tory stress. Nodal lines of the first 2-stripes mode at 2769 Hz for the first stage blade are shown in Fig. 6.48. During engine tests the compressor discharge valve requires adjustment as the shaft accelerates to reproduce the regular startup curve. All variable stator rows are continuously modified to acquire knowledge of stall safety margins during the start. Blade dynamic stresses are also monitored during this phase to detect shortcomings. Performance map is checked at corrected speeds from 85 to 110 percent of the design speed. For each speed, the pressure range, from the turbine’s no-load lineup to an upper limit set by the appearance of marked increase in the dynamic pressure transducer signals, indicates approach to a stall. Actual surge points are checked at the end of the tests to avoid premature internal instru- mentation failure. FAN AND COMPRESSOR AIRFOILS 191 FIGURE 6.47 Compressor rotor during assembly (Benvenuti, 1996). Measured natural frequencies in operation are determined to be within 5 percent of the predicted values. Unexpected resonance due to the coincidence of blade natural frequencies and passing frequencies is not encountered. The 90 to 100 percent speed range indicated in Fig. 6.48 represents normal compressor operation on a typical two-shaft gas turbine equipped with variable inlet nozzles. The vertical bars indicate dynamic stress amplitudes, and may be used to establish material high-cycle fatigue limit provided on the same scale in the diagram. Dynamic stress levels do not exceed 25 percent of the high-cycle fatigue endurance limit in the low-speed range. Stresses related to the two-stripe modes are low, and the mode is not within the resonance range of a known excitation source. Stress amplitudes at resonance with low-order harmonics arising from inlet distortion and strut wake are not considerable, and confirm the quality of flow at blade inlet. Similar dynamic stress patterns are observed from upstream and downstream blade passing frequencies on subsequent stage blades. Interstage pressure and temperature measurements made with stator leading edge instrumentation are used to correct flow mismatches and to bring overall performance to design target. By selecting an appropriate test point matrix, it is possible to determine the complete characteristic lines of each stage from choke to near-stall conditions. Stage work coefficient and efficiency are calculated between consecutive measurement stations. When the flow coefficients corresponding to design conditions are not at peak maximum effi- ciency in the front-end stages with variable stators, restaggering the vanes can shift the flow coefficients and improve overall performance. As an example, second stage work and efficiency curves show an improvement of 2 percent at design setting after correction of mismatch in the vane angles (Fig. 6.49). With performance targets finally achieved, 192 COMPONENT DESIGN FIGURE 6.48 Campbell diagram for stage-1 rotating blade (Benvenuti, 1996). 6 5 4 3 2 1 0 0 20 40 60 80 100 120 Rotor speed - % Frequency (Hz × 1000) 3rd flex 1st complex 1st 2-stripes 1st tors. 1st flex 1× 2× 3× 4× 5× 6× 7 × (struts) 14× (struts × 2) 34× (IGV, stat. 1) High-cycle fatigue limit Min. speed 2nd flex [...]... (Hz) Mode 62 percent 68 percent 68 percent 69 percent 66 percent 69 percent 69 percent 100 percent 412.09 1148.45 1227. 86 1770.72 434.5 (+5.44 percent) 1251.7 (+9.03 percent) 1251.7 (+1.94 percent) 1800.0 (+1 .65 percent) 1F 2F 1T 2S TABLE 6. 4 Forced Response Computation Results Aero speed (rpm) Mode Modal force (N/kg) Aero Q-factor Max displacement peak to peak (mm) 66 percent 69 percent 69 percent... TABLE 6. 5 Blade Mistuning Pattern Blade no Mistuning (dn) Blade no Mistuning (dn) Blade no Mistuning (dn) 1 2 3 4 5 6 7 8 9 10 0.05704 0.01207 0.0 467 0 −0.01502 0.05 969 −0.03324 −0.00078 −0.0 168 8 0.00242 −0.02747 11 12 13 14 15 16 17 18 19 20 −0.0 363 1 −0.03570 −0.0 363 1 −0.0 363 1 0.00242 0.04934 0.04479 0.03030 0.00242 0.01734 21 22 23 24 25 26 27 28 29 — 0.02919 −0.00328 0.000 86 −0.0 365 4 −0.0 363 1 −0.0 166 5... of root-to-tip ratios rroot/rtip rtip (m) N (rps) 0.40 0.45 0.50 0.55 0 .60 0.2283 0.2343 0.24 16 0.2505 0. 261 5 2 26. 6 220.8 214.1 2 06. 5 197.8 Using values calculated at the root-to-tip ratio of 0.5, tip radius of 0.24 16 m and rotor speed of 214.1 rps, or 128 46 rpm, are selected The corresponding tip speed is: (2 × p × 0.24 16 × 128 46/ 60) = 325.01 m/s Speed of sound a and Mach number at the rotor tip Mflow1... radius Outlet root radius 0.24 16 m 0.1208 m 0.2010 m 0. 161 4 m Problem 6. 2 In Prob 6. 1 determine the number of stages required to obtain the stated compression Solution Rise in the stagnation temperature in the compressor is 475 .6 − 290 = 185 .6 K Increase in stage temperature rise may be estimated from the blade mean speed U = 2πrmean N /60 = 2 × π × 0.1812 × 128 46/ 60 = 243. 76 m/s If the first stage sees... AIRFOILS Design mass flow 0. 86 0.94 0.84 0.92 0.82 0.90 0.80 0.88 0.78 Work coefficient Isentropic efficiency 0. 96 0. 76 0. 86 Work 0.84 0.50 FIGURE 6. 49 0.52 Efficiency 0.54 0. 56 Flow coefficient 0.58 0.74 0 .60 Measured performance curves of 2nd stage (Benvenuti, 19 96) application of the new compressor design achieves an increased power output of 1 MW over the present design 6. 15 INCREASED POWER BY ZERO... 20)/(0.98 × 243. 76) ∆vwhirl = = 84. 56 m/s where the factor l for work done is assumed to be 0.98 in a preliminary estimate At stage 1 vwhirl1 = 0, hence vwhirl2 = 84. 56 m/s FAN AND COMPRESSOR AIRFOILS 213 Then tan β1 = 243. 76/ 170 = 1.4339 or β2 = 55.11° tan β2 = (vtip − vwhirl2 )/vaxial1 = (243. 76 − 84. 56) /170 = 0.9 364 or β2 = 43.12° tan α 2 = vwhirl2 /vaxial1 = 84. 56/ 170 = 0.4974 or α 2 = 26. 45° The velocity... outside the contact at C′ max max σ hµ = c4 µσ c (6. 37) 210 COMPONENT DESIGN where c4 is to be determined from fitting Peak tensile hoop stress in the disk will vary with rotor speed, as in the blade, by the expression max σ h = c5ω + c6ω 2 (6. 38) During unloading, Fb and Fw in Fig 6. 66 are reduced, as also the radial pull on the disk at PP′ (Fig 6. 63) The material above PP′ in the disk retracts radially... 27.47 K A temperature rise of 27.47 K per stage then suggests (475 .6 − 290)/∆Tos = 185 .6/ 27.47 = 6. 76, or 7, stages are needed Then the average temperature rise per stage will be 185 .6/ 7 = 26. 5 K Since the first and last stages see a smaller increase, say 20 K, the remaining five stages may be expected to see a rise of about 27 .6 K Problem 6. 3 Prepare the velocity diagrams for the first two stages of the... 475 .6 K If it is assumed that the air exiting the last stage of the compressor has no swirl and an exit velocity of 170 m/s, static temperature, pressure, and density at exit are T2 = 475 .6 − 170 2 /(2 × 1.01 × 1000) = 461 .3 K p2 = p02 × (T2 /T02 )γ / (γ −1) = 4.80 × ( 461 .3/475 .6) 3.5 = 4.31 bar ρ2 = p2 /(RT2 ) = (100 × 4.31)/(0.287 × 461 .3) = 3.2 56 kg/m 3 Area of annulus at exit is A2 = 25/(3.2 56 ×... are given in Table 6. 4 for all resonant conditions FIGURE 6. 53 Test fan configuration (Breard et al., 2000) FAN AND COMPRESSOR AIRFOILS FIGURE 6. 54 et al., 2000) Steady-state Mach number contours at 85 percent span (Breard FIGURE 6. 55 Structural analysis mesh of fan and disk (Breard et al., 2000) 199 200 COMPONENT DESIGN FIGURE 6. 56 Campbell diagram (Breard et al., 2000) TABLE 6. 3 Blade Natural Frequencies . Design mass flow Work Efficiency 0.74 0. 76 0.78 0.80 0.82 0.84 0. 86 0.84 0.50 0.52 0.54 0. 56 0.58 0 .60 0. 86 0.88 0.90 0.92 0.94 0. 96 Flow coefficient FIGURE 6. 50 LM2500+ gas turbine high-pressure. Aero Q-factor peak to peak (mm) 66 percent 1F 70.0 60 4.0 69 percent 2F/1T 27.5/ 86. 0 44/80 0.14/2.11 69 percent 2F/1T 41.0/87.0 52/98 0.25/2.58 100 percent 2S 60 .0 100 0.37 of interest. The Q. 2000). TABLE 6. 3 Blade Natural Frequencies of Vibration Shaft speed (rpm) Aero speed (rpm) Frequency (Hz) Adjusted (Hz) Mode 62 percent 66 percent 412.09 434.5 (+5.44 percent) 1F 68 percent 69 percent