Turbomachinery Class 8a

Axial Flow Turbomachine Design Three-dimensional flow through machine very complex Decompose problem into series of two-dimensional problems Blade-to-blade [Cascade] – (x, y=) Plane cascade (no spanwise effect) Stream surface (span effect modeled within 2D construct Through flow [Meanline] – (r, x) Blades modeled as thin or actuator discs Secondary flow [normal to mainstream] – (r, ) Design problem largely treated as inviscid analysis with viscous effects accounted for through empirical and semi-empirical methods

Airfoil/Cascade Design Cascade - array of airfoils providing forces that change flow vectors. Requirements: Produce required forces Low total pressure loss Wide range of low loss incidence: operate at off-design points Stable exit angles Turning should be produced so that losses are minimum Airfoils [compressors] typically selected from family or “series” of airfoils

Geometry of a Rectilinear Cascade Geometry Parameters Solidity Stagger angle: angle between chord line and leading edge front of cascade Camber / Turning Maximum thickness Leading edge / trailing edge thickness / radius Uncovered turning

Airfoil Nomenclature Chord: c or b = xTE-xLE; straight line connecting leading edge and trailing edge Camber line: locus of points halfway between upper and lower surface, as measured perpendicular to mean camber line itself Camber: maximum distance between mean camber lineand chord line Thickness t(x), tmax Angle of attack: , angle between freestream velocity and chord line

Geometry of a Rectilinear Cascade

Geometry of a Rectilinear Cascade

Compressor Airfoil Design Compressor Airfoil Series - geometric families Circular arc (CA): for high Mach number flows mean camber line is circular arc with max camber and max thickness at 50%c 65-series for moderate subsonic Mach numbers mean camber line is a parabola with max camber at 50%c and max thickness at 40%c 400-series for low Mach numbers max camber at 40%c and max thickness at 30%c thickness distribution NACA SP-36 p.198 or Abbot and Von Doenhoff

Compressor Airfoil Design Insert NACA [Nat’l Advisory Committee on Aeronautics] file here

Compressor Airfoil Design Compressor Airfoil Series - geometric families Initially developed from wing data Used through 1970's Large bodies of cascade data NASA, P&W, UTRC, DFVLR, NGTE, ONERA Loss & flow turning = f (incidence, Mach no., area ratio & geometry) Experimental Data Verifies Design Codes

Airfoil/Cascade Performance Shear layer development over solid surface

Airfoil/Cascade Performance Friction effects can adversely affect cascade performance

Airfoil/Cascade Performance Boundary layer thickness assessment parameters: * = displacement thickness  = momentum thickness H = shape factor = * /  (1< H < 2.2) Equal mass =

Airfoil/Cascade Performance Boundary layer thickness assessment parameters: * = displacement thickness Howell correlation

Airfoil - Cascade Comparisons Isolated Wing Lift-Drag Curve Observations Cd dependence on Mach is small until critical Mcr Cl dependence on Mach is strong Shocks & shock boundary layer interaction lead to flow separation, lift loss, drag rise

Airfoil - Cascade Comparisons Transonic flow shocks M<1 M>1

Airfoil - Cascade Comparisons Plane Cascade Lift-Drag Curve Lift  Circulation ( )  turning  deflection Drag  total pressure change  loss Dixon – Howell [1942] What observations can you make about these curves?

Loss “Bucket”

Minimum loss incidence range Choke Stall

Limits of Compressor Cascade Operation Stall Analogous to wing stall High positive incidence, separation from suction side Loss & Deviation (difference between flow and exit camber angles) rising rapidly, work & efficiency fall Choke Pressure surface separation due to negative incidence Actual choke: not enough area at throat to pass mass flow Choke margin decreases as flow becomes more axial (M=const)

Generic Airfoil Velocity Distribution

Airfoil/Cascade Design Airfoil shapes constructed to have desirable surface pressure distributions and boundary layer characteristics Airfoil shapes must also meet structural criteria Low pressure, high Mach number [suction side] Adverse gradient T.E. Kutta condition High pressure, low Mach number [pressure side] % chord

Design for Max Lift, Min Loss, Max Range & Choke Margin Avoid leading edge sep. bubble Ideal Avoid separation Avoid flow reversal Can also view this in terms of ps/p0?

Design for Max Lift, Min Loss, Max Range & Choke Margin

Airfoil/Cascade Design Cascade Testing: Vital Requirements Periodicity Endwall boundary layer control Uniform flow Accurate pitch-wise Traverse data

Compressor Airfoil/Cascade Design Additional Factors: AVDR - Axial Velocity - Density Ratio - area ratio due to end wall boundary layer growth [Geometry 2D but flow 3D]– Effects deviation [ ] Contraction of streamlines due to boundary layer thickening

Compressor Airfoil/Cascade Performance Background: Boundary layer thickness parameters:  = momentum thickness * = displacement thickness H = shape factor = * /  (1< H < 2.2)

Compressor Airfoil/Cascade Performance Loss Analysis - Lieblein's Dfactor [Diffusion Factor] Momentum Integral Equation describes the growth of boundary layer thickness along the suction surface: where: V = relative velocity at edge of boundary layer x = distance along airfoil surface - Boundary layer eqns. - Integrate y: 0 to  Boundary layer PDE integrated over y from 0 to 

Compressor Airfoil/Cascade Performance Loss Analysis - Lieblein's Dfactor [Diffusion Factor] Momentum Integral Equation describes the growth of boundary layer thickness along the suction surface: where: V = relative velocity at edge of boundary layer x = distance along airfoil surface - Boundary layer eqns. - Integrate y: 0 to  Boundary layer PDE integrated over y from 0 to 

Compressor Airfoil/Cascade Performance Vmaximum Lieblien's insight: Correlation of cascade pressure distribution data for constant radius: Velocities are in Relative Frame [V=W]. Solidity =b/s. The 2 is empirical [from cascade data]. Now to connect Df to loss……... V(x) Vsurf V2

Wake Momentum Thickness vs. Calculated Diffusion Empirical relation [Leiblien] connecting /c (or loss) to Df

Compressor Airfoil/Cascade Performance Uses of Dfactor Today - Preliminary design surge margin limit for known clearance & aspect ratio - Low speed loss prediction in mean line systems 0 < Dfactor < 0.7 - Given loss [ /c ], now find loss coefficient [  ]

Example A compressor rotor with the following conditions: U=200 mps Cx1=Cx2=150 mps 2=35 degs.  = 1 Calculate W1, W2, Df

Loss Coefficient Directly Related to Wake Momentum Thickness Ratio, /c

Relation Between /c and  [Extra] Empirical expression

What is the Impact of D-factor on Airfoil Shape Carter’s Rule for compressor cascades For HWK 6.3, since  =1 Metal angle decreases to achieve design exit angle goals

Carter’s Rule for Compressors

Compressor Airfoil/Cascade Performance Compressor Airfoil Performance Dfactor sets Wake Thickness Wake Thickness and # Wakes sets Loss Coefficient Loss Coefficient and Work Coefficient sets Efficiency

Turbomachinery Class 8b

Overview of Loss Analysis - Lieblein's Dfactor Correlation of cascade data [Velocities in Relative Frame] or de Haller [ 0.72<W2/W1<1] Momentum thickness [ ] correlated to Dfactor [0 < Dfactor < .7] Loss coefficient related to cascade wake momentum thickness Efficiency related to loss coefficient

Compressor Airfoil/Cascade Performance High deflection compressor airfoil [reducing 2 with fixed 1] means reducing V2. Early diffusion based design method by de Haller called for 0.72<V2/V1<1 to avoid high losses. Lieblien [NACA] related diffusion to velocity gradient and solidity Vmaximum V(x) Vsurf V2

Compressor Airfoil/Cascade Performance Repeating stage, repeating row [mirror image airfoil]

Compressor Airfoil/Cascade Performance Dfactor analysis for repeating row stage

Airfoil/Cascade Performance Airfoil Sections Now Designed to Pressure Distributions Compressible Potential flow code for local M<1.1 (no shocks) Integral Boundary Layer codes for performance prediction (viscous effects) Wake Mixing (Stewart’s Control Volume) Analysis to Obtain Relative P0 Loss Navier-Stokes codes for studies near separation (viscous & inviscid flow solved at once)

Compressor Airfoil/Cascade Design Controlled Diffusion Airfoils (CDA @P&W, CTA @ GE) Higher peak Mach no., tapered dV/dx More camber in front half of chord Elliptical Leading edges Stall range assured by gradual initial acceleration Not optimum at end walls

Turbine Airfoil/Cascade Design Turbine Airfoil Design Historically more Analytical than Compressors Accelerating Flow...Probably the Reason Boundary layer separation easier to avoid Design to Pressure Distributions Correct for deviation effects Zweifel Load Coefficient & Convergence Desirable Pressure Distributions

Turbine Rotor 2  3 Constant Cx, no exit swirl [3=0] High loading stage [E<<0, turbine] Lower weight, lower stage efficiency  lower R

Turbine Rotor 2  3 For repeating row design and no exit swirl E=2[R-1]

Cascade Design Problem

Cascade Design Problem Desirable pressure distributions Pressure side Exit Pressure Suction side Good Poor- too much diffusion

What is the Impact of Deviation on Airfoil Shape Carter’s Rule for turbine cascades Metal angle decreased to achieve design exit angle goals Effect for turbine is much smaller for turbines due to thinner boundary layers

Zweifel Coefficient Derivation Solidity play important role in turbine efficiency: (1) spacing small, fluid gets maximum turning force with large wall friction forces; (2) spacing large, fluid gets small turning force with small wall friction losses. p01 Area Fideal Area F p2 Solidity issue: Enough airfoils must be used so that F = change in tangential momentum of the fluid.

Zweifel Coefficient Derivation

Turbine Airfoil/Cascade Design [Details] Zweifel Load Coefficient: where bx is the axial chord and h is the streamtube spanwise width and x = bx / s Treating the pressure difference incompressibly:

Turbine Airfoil/Cascade Design [Details] Simplifying: Zweifel Coefficient: 1st estimate of solidity.

Turbine Airfoil/Cascade Design Zweifel Coefficient: 1st estimate of solidity. Zweifel noted that turbine min. loss for z 0.8. Optimum spacing estimated from given inlet and outlet angles.

Turbine Airfoil/Cascade Design Zweifel Coefficient: 1st estimate of solidity. Other Factors: Pressure Distributions Cooling Resonances Disk Stress Weight Cost Some of this covered after meanline analysis

Turbine Airfoil/Cascade Design Turbine Airfoil Design Approach [covered in following charts] Balance Acceleration & Diffusion Use Laminar Boundary layer where possible Diffuse with Turbulent Boundary layer Control Maximum Mach No. & Shocks Control Leading Edge Overspeed Manage Uncovered Turning

Turbine Airfoil/Cascade Design Balance Front End Acceleration With Rear Diffusion Laminar boundary layers on suction surface reduces losses on front, turbulent boundary layers on rear more tolerant to flow separation.

More Blades Increases Solidity [ = b/s] Reduces Force per Blade

Aft Curvature Moves Loading Aft

Turbine Airfoil/Cascade Design High Convergence Eases Cascade Design Like an area ratio: Aan is an annulus area Convergence is Function of Velocity Diagrams, not airfoil shape High Reaction = higher blade convergence

Turbine Airfoil/Cascade Design

Turbine Airfoil/Cascade Design Thin Edges are best for Aerodynamic Performance Trailing Edge Blockage Drives Base Drag TET = Trailing Edge Thickness Stewart’s mixing loss correlation is function of TET and *s.s. , *p.s. Casting capability, stress & cooling set minimum edge thickness High wedge angle eases effect at leading edge Elliptical LE is better Pressure Distribution Improved; Overspeed reduced in both surfaces

Turbine Airfoil/Cascade Design Manage Uncovered Turning Transonic Flow: More uncovered turning lowers base pressure loss (<pex) but higher boundary layer loss. Subsonic Flow: Base pressure  pex Uncovered turning Throat

Consequences of Stagnation Point Location

Turbine Airfoil/Cascade Design Manage Uncovered Turning Uncovered turning is less of an issue for subsonic turbine airfoils

Turbine Airfoil Active Cooling Methods

Thermo & Kinematic View of Compressor Stage Note: U>C

Thermo & Kinematic View of Turbine Stage Note: U<C Cu3 Cu2

Thermo & Kinematic View of Turbine Stage