Unit 6: Elementary Turbine Design

I. Foundations of Turbine Operation

A. Introduction

A gas turbine converts the stagnation enthalpy of a hot, pressurized gas into shaft work through stationary nozzle rows and rotating blade rows.

• Governing principle: Conservation of angular momentum gives the Euler turbine equation; shaft work results from a reduction in the gas’s tangential momentum.
• Energy conversion:
  • The stator accelerates and directs the flow by converting enthalpy into kinetic energy.
  • The rotor extracts energy by turning and usually expanding the flow.
• Stage definition: One stator row followed by one rotor row forms a turbine stage; several stages may be arranged in series.
• Station convention: Station 1 denotes rotor inlet and station 2 denotes rotor exit.
• Velocity convention:
  • (C): absolute gas velocity, m/s.
  • (W): velocity relative to the rotor, m/s.
  • (U=\omega r): blade speed, m/s.
  • (Cx): axial component; (C\theta): tangential or whirl component.
• Thermodynamic quantities: (h) is static enthalpy, (h_0=h+C^2/2) is stagnation enthalpy, and (s) is entropy.
• Ideal assumptions: Preliminary analysis commonly assumes steady, adiabatic, one-dimensional flow with uniform mean-radius properties and negligible potential-energy change.

II. Principal Turbine Types — Location of Expansion

A. Impulse and reaction turbine

Impulse and reaction turbines differ primarily in how the stage pressure and static-enthalpy drops are divided between stator and rotor.

1. Impulse turbine:

   • Expansion location: Ideally, the entire stage pressure drop occurs in the stator nozzle; rotor static pressure remains approximately constant.
   • Rotor action: Work is obtained by changing the direction and tangential component of a high-speed jet.
   • Degree of reaction: An ideal impulse stage has (R=0).
   • Velocity behavior: Relative velocity ideally retains its magnitude through the rotor, although friction causes (W_2<W_1).
   • Practical feature: Large stator acceleration can produce high rotor-inlet velocity and significant aerodynamic loss.

2. Reaction turbine:

   • Expansion location: Pressure and static enthalpy decrease in both stator and rotor passages.
   • Rotor action: The rotor behaves as a moving nozzle, producing torque through both impulse turning and relative-flow acceleration.
   • Degree of reaction: (0<R<1); a Parsons stage conventionally has (R=0.5).
   • Practical feature: Expansion distributed between rows permits lower velocity per row but introduces rotor pressure loading and stronger clearance sensitivity.

• Explicit contrast: Impulse staging concentrates acceleration before the rotor, whereas reaction staging shares acceleration between stationary and moving passages.

III. Distribution of Enthalpy Drop

A. Compounding of turbine

Compounding divides a large turbine pressure or velocity change among multiple blade rows so that blade speed, Mach number, and losses remain acceptable.

1. Velocity compounding—Curtis arrangement:

   • Construction: One nozzle row supplies several rotor rows separated by fixed redirecting blades.
   • Pressure distribution: Most pressure drop occurs in the first nozzle; successive rotors extract the resulting kinetic energy.
   • Use: It reduces required wheel speed but incurs extra turning and friction losses in intermediate rows.

2. Pressure compounding—Rateau arrangement:

   • Construction: Several impulse stages are placed in series, each containing a nozzle and rotor.
   • Pressure distribution: Total pressure drop is divided among the nozzle rows.
   • Use: Lower stage velocity improves efficiency but increases turbine length, blade count, and cost.

3. Pressure-velocity compounding:

   • Construction: Pressure is divided among groups, while each group uses multiple velocity-compounded rotor rows.
   • Trade-off: It combines moderate rotational speed with fewer pressure stages, but its aerodynamic arrangement is more complex.

• Selection basis: The designer balances rotational speed, stage loading, turbine diameter, efficiency, mechanical stress, and manufacturing constraints.

IV. Aerodynamic Stage Analysis

A. Efficiency of turbine

Turbine efficiency measures how effectively the available isentropic enthalpy drop becomes useful mechanical work.

• Total-to-total efficiency: Appropriate when outlet kinetic energy remains useful to a following component.

η_tt = (h_01 - h_02) / (h_01 - h_02s)

Here, (\eta{tt}) is total-to-total efficiency; (h{01}) and (h{02}) are actual inlet and outlet stagnation enthalpies; (h{02s}) is the isentropic outlet stagnation enthalpy at the actual outlet total pressure.

• Total-to-static efficiency: Appropriate when exhaust kinetic energy is treated as a loss.

η_ts = w / (h_01 - h_2s)

Here, (\eta{ts}) is total-to-static efficiency, (w) is specific shaft work, and (h{2s}) is isentropic outlet static enthalpy at the specified outlet static pressure.

• Loss sources: Profile friction, secondary flow, tip leakage, trailing-edge mixing, shocks, cooling flow, disc windage, and residual exit kinetic energy reduce efficiency.
• Interpretation: Because total-to-static efficiency penalizes unrecovered exit velocity, it is generally lower than total-to-total efficiency for the same stage.

B. Velocity diagrams

Velocity diagrams graphically relate absolute flow, blade motion, and rotor-relative flow at rotor inlet and exit.

• Vector relation:

C = U + W

Here, (C) is absolute velocity, (U) is blade velocity, and (W) is relative velocity; all are vectors.

• Inlet triangle: (C1) is resolved into (C{x1}) and (C_{\theta1}), while (W_1=C_1-U).
• Exit triangle: Blade geometry determines the relative exit direction; adding (U) to (W_2) gives (C_2).
• Angles: (\alpha) commonly denotes the absolute-flow angle and (\beta) the relative-flow angle, both measured from the axial direction.
• Design objective: The stator sets rotor incidence through (\alpha_1), while rotor camber and stagger turn the relative flow toward (\beta_2).
• Exit swirl: A large (C_{\theta2}) represents kinetic energy not fully extracted; many designs seek small exit whirl unless another stage can use it.

C. Work and efficiency

Stage work follows directly from the change in absolute whirl velocity across the rotor.

• Euler turbine equation:

w = U_1 C_θ1 - U_2 C_θ2

Here, (w) is specific work in J/kg; (U_1,U2) are inlet and exit blade speeds; and (C{\theta1},C_{\theta2}) are absolute whirl components.

• Constant-radius stage:

w = U(C_θ1 - C_θ2)
P = ṁw

Here, (U) is mean blade speed, (P) is power in watts, and (\dot m) is mass flow rate in kg/s.

• Blade or diagram efficiency for an impulse rotor:

η_b = U(C_θ1 - C_θ2) / (C_1²/2)

Here, (\eta_b) compares rotor work with inlet absolute kinetic energy.

• Worked example: If (U=300) m/s, (C{\theta1}=450) m/s, and (C{\theta2}=50) m/s, then

w = 300(450 - 50) = 120,000 J/kg = 120 kJ/kg

At (\dot m=20) kg/s, the ideal aerodynamic power is (P=2.4) MW.

D. Degree of reaction

Degree of reaction specifies the fraction of stage static-enthalpy drop occurring in the rotor.

• Definition:

R = (h_1 - h_2)_rotor / (Δh_static)_stage

Here, (R) is degree of reaction, and (h_1-h_2) is the rotor static-enthalpy decrease.

• Velocity form: For constant radius and approximately constant axial velocity,

R = 1 - (C_θ1 + C_θ2)/(2U)

Here, the velocity symbols retain their earlier meanings.

• Interpretation:
  • (R=0): ideal impulse rotor.
  • (R=0.5): equal stator and rotor static-enthalpy drops, often giving similar blade shapes when velocity triangles are symmetric.
  • (R>0.5): a greater fraction of expansion occurs in the rotor.
• Design effect: Increasing reaction changes incidence, pressure loading, relative Mach number, leakage sensitivity, and axial thrust.

V. Multistage Turbine Systems

A. Turbine multi staging and stage performance

Multistaging divides the required total enthalpy drop among successive stages while maintaining efficient velocity levels and acceptable loading.

• Stage loading coefficient:

ψ = w/U²

Here, (\psi) is the nondimensional stage loading, (w) is stage specific work, and (U) is blade speed.

• Flow coefficient:

φ = C_x/U

Here, (\phi) is the flow coefficient and (C_x) is representative axial velocity.

• Stage matching: The outlet pressure, temperature, swirl, and mass flow of one stage become the inlet conditions of the next.
• Annulus variation: As density decreases during expansion, annulus area generally increases to pass the same mass flow; later stages therefore use longer blades.
• Performance map: Efficiency depends strongly on (\psi), (\phi), reaction, Reynolds number, and Mach number.
• Off-design behavior: Changes in shaft speed or mass flow alter incidence and velocity triangles, potentially causing separation, choking, excessive exit swirl, or reduced work.
• Reheat effect: Irreversibility in an early stage raises the temperature entering later stages; consequently, overall turbine efficiency is not obtained by simply averaging individual stage efficiencies.

VI. Practical Design Boundaries

A. Factors limiting turbine design

Turbine geometry and operating conditions are restricted by interacting aerodynamic, thermal, mechanical, and manufacturing limits.

• Centrifugal stress: Blade-root and disc stresses increase approximately with material density and (U^2), limiting rotational speed and radius.
• Gas temperature: Turbine-entry temperature may exceed the uncooled material capability, producing creep, oxidation, and thermal fatigue.
• Aerodynamic loading: Excessive pressure gradients cause boundary-layer separation, while high Mach numbers introduce shocks and choking.
• Vibration: Blade-passing excitation must not coincide with blade or disc natural frequencies; Campbell diagrams identify dangerous resonances.
• Tip clearance: Clearance prevents rubbing during thermal growth, but leakage over the tip reduces work and efficiency.
• Blade height: Very short blades suffer proportionally high end-wall and clearance losses; very long blades face stress and vibration constraints.
• Materials and manufacture: Single-crystal alloys, internal passages, coatings, tolerances, inspection requirements, cost, and service life constrain feasible shapes.
• System constraints: Required mass flow, power, shaft speed, combustor exit profile, exhaust pressure, and cooling-air availability determine the design envelope.

B. Cooling of turbine blade

Blade cooling permits operation above the allowable bulk-metal temperature by reducing heat transfer to the alloy and removing absorbed heat.

• Internal convection: Compressor bleed air flows through serpentine passages; ribs and pin fins increase internal surface area and turbulence.
• Impingement cooling: Jets strike the inner blade wall, producing high local heat-transfer coefficients near leading edges.
• Film cooling: Air discharged through surface holes forms a protective layer between hot gas and metal; poor injection can mix rapidly and increase aerodynamic loss.
• Trailing-edge cooling: Narrow slots or pin-fin passages cool the thin trailing region before discharging air into the wake.
• Thermal-barrier coating: A ceramic layer reduces metal temperature by providing thermal resistance, while a metallic bond coat improves adhesion and oxidation protection.
• Cooling effectiveness:

ε = (T_g - T_m)/(T_g - T_c)

Here, (\varepsilon) is cooling effectiveness, (T_g) is mainstream gas temperature, (T_m) is blade-metal temperature, and (T_c) is coolant temperature.

• Cooling penalty: Bleeding compressor air reduces core mass flow and compressor work recovery; coolant mixing also causes pressure and aerodynamic losses.
• Design balance: Coolant quantity and hole placement must satisfy metal-temperature and life limits without causing excessive efficiency loss, thermal gradients, or structural weakening.