Turbomachinery

Concept

Turbomachinery and Compressible Devices — Energy Transfer, Performance, and Thermodynamic Coupling

Scope: rigorous treatment of turbomachinery thermodynamics and fluid dynamics. Includes energy transfer mechanisms, velocity triangles, efficiency formulations, and compressibility effects in compressors, turbines, and nozzles.

1. Fundamentals of Energy Transfer in Turbomachines

1.1 Classification

Turbomachines transfer energy between a fluid and a rotating component (rotor or impeller):

  • Turbines: convert fluid energy → mechanical shaft power.
  • Compressors / Pumps: convert shaft power → fluid energy.

1.2 Control Volume Analysis

Apply the steady-flow energy equation between inlet (1) and outlet (2): W˙s=m˙(h2−h1+u22−u122+g(z2−z1)).\dot{W}_s = \dot{m}(h_2 - h_1 + \frac{u_2^2 - u_1^2}{2} + g(z_2 - z_1)).

For adiabatic flow with negligible potential change: W˙s=m˙(h2−h1).\dot{W}_s = \dot{m}(h_2 - h_1).

1.3 Euler’s Turbomachinery Equation

From angular momentum conservation: W˙s=m˙(u2Vw2−u1Vw1),\dot{W}_s = \dot{m}(u_2V_{w2} - u_1V_{w1}), where:

  • uu: blade speed (tangential)
  • VwV_w: tangential component of absolute velocity.

2. Velocity Triangles and Blade Kinematics

2.1 Velocity Decomposition

At blade inlet and outlet: V=U+Vr,\mathbf{V} = \mathbf{U} + \mathbf{V_r}, where:

  • V\mathbf{V}: absolute velocity
  • U\mathbf{U}: blade velocity
  • Vr\mathbf{V_r}: relative velocity (in rotating frame)

2.2 Velocity Triangles

Visual representation of V,U,VrV, U, V_r at rotor inlet and outlet defines blade angles: tan⁡β1=Vr1,axVr1,tan,tan⁡β2=Vr2,axVr2,tan.\tan β_1 = \frac{V_{r1,ax}}{V_{r1,tan}}, \quad \tan β_2 = \frac{V_{r2,ax}}{V_{r2,tan}}.

2.3 Power and Specific Work

Specific work input/output: w=u(Vw2−Vw1).w = u(V_{w2} - V_{w1}). Sign convention:

  • w>0w > 0: compressor (work input)
  • w<0w < 0: turbine (work output)

3. Thermodynamic Relations

3.1 Isentropic and Polytropic Relations

For compressible adiabatic flow: T2/T1=(p2/p1)(γ−1)/γ.T_2/T_1 = (p_2/p_1)^{(γ-1)/γ}. For non-isentropic (polytropic) flow: pVn=constant,T2/T1=(p2/p1)(n−1)/n.pV^n = \text{constant}, \quad T_2/T_1 = (p_2/p_1)^{(n-1)/n}.

Polytropic efficiency: ηp=γ−1γnn−1.η_p = \frac{γ-1}{γ} \frac{n}{n-1}.

3.2 Enthalpy–Entropy Diagram (h–s)

Flow through a turbomachine appears as:

  • Vertical line: isentropic (ideal)
  • Curved line: actual (losses → Δs > 0)

4. Compressor Thermodynamics

4.1 Isentropic Efficiency

ηc=(h2s−h1)(h2−h1)=T2s−T1T2−T1.η_c = \frac{(h_{2s} - h_1)}{(h_2 - h_1)} = \frac{T_{2s} - T_1}{T_2 - T_1}.

For perfect gas: T2s/T1=(p2/p1)(γ−1)/γ.T_{2s}/T_1 = (p_2/p_1)^{(γ-1)/γ}.

4.2 Polytropic Efficiency and Stage Analysis

For multi-stage compressors: ηp=ln⁡(p2/p1)ln⁡(T2/T1)γ−1γ.η_p = \frac{\ln(p_2/p_1)}{\ln(T_2/T_1)} \frac{γ-1}{γ}.

4.3 Compressor Work and Power

W˙=m˙cp(T2−T1).\dot{W} = \dot{m} c_p (T_2 - T_1).

4.4 Performance Maps

Plot of pressure ratio vs. mass flow for constant speed lines.

  • Surge line: unstable oscillations.
  • Choke line: sonic flow limit.

5. Turbine Thermodynamics

5.1 Isentropic Efficiency

ηt=(h2−h1)(h2s−h1)=T2−T1T2s−T1.η_t = \frac{(h_2 - h_1)}{(h_{2s} - h_1)} = \frac{T_2 - T_1}{T_{2s} - T_1}.

For isentropic expansion: T2s/T1=(p2/p1)(γ−1)/γ.T_{2s}/T_1 = (p_2/p_1)^{(γ-1)/γ}.

5.2 Polytropic Expansion

T2/T1=(p2/p1)(n−1)/n,ηt=γ−1γn−1n.T_2/T_1 = (p_2/p_1)^{(n-1)/n}, \quad η_t = \frac{γ-1}{γ} \frac{n-1}{n}.

5.3 Specific Work Output

wt=cp(T1−T2).w_t = c_p(T_1 - T_2).

6. Stage and Blade Design Principles

6.1 Axial Flow Machines

  • Flow primarily along axis.
  • Work proportional to change in tangential velocity component.

w=u(Vw2−Vw1).w = u(V_{w2} - V_{w1}).

Degree of reaction (fraction of static enthalpy rise in rotor): R=h2−h3h1−h3.R = \frac{h_2 - h_3}{h_1 - h_3}.

6.2 Radial Flow Machines

For centrifugal compressors: w=u2Vw2−u1Vw1≈u2Vw2.w = u_2V_{w2} - u_1V_{w1} ≈ u_2V_{w2}.

Slip factor (due to non-ideal exit flow): σ=1−πZcosβ21−cosβ2,Z=number of blades.σ = 1 - \frac{π}{Z} \sqrt{\frac{cosβ_2}{1 - cosβ_2}}, \quad Z = \text{number of blades}.

Actual work: w=σu2Vw2.w = σu_2V_{w2}.

7. Compressibility and Shock Effects

7.1 Transonic Flow in Compressors

At high Mach numbers, local shocks form on blade surfaces → increase in entropy.

Normal shock relations govern local losses: Δs=cpln⁡T2T1−Rln⁡p2p1.Δs = c_p \ln\frac{T_2}{T_1} - R \ln\frac{p_2}{p_1}.

7.2 Choking and Mach-Limited Operation

Mass flow rate in blade passages is limited by sonic conditions at throat regions: m˙max=A∗p0γRT0(2γ+1)(γ+1)/2(γ−1).\dot{m}_{max} = A^* p_0 \sqrt{\frac{γ}{RT_0}} \left(\frac{2}{γ+1}\right)^{(γ+1)/2(γ-1)}.

8. Loss Mechanisms and Efficiency

8.1 Sources of Loss

MechanismDescription
Profile lossFriction and boundary layer on blade surfaces
Tip leakageFlow over blade tips in turbomachines
Secondary flow3D flow in blade passages causing mixing
Shock lossesCompression shocks in transonic blades
Clearance lossesLeakage between rotor and casing

8.2 Total-to-Total Efficiency

ηtt=(h02−h01)ideal(h02−h01)actual.η_{tt} = \frac{(h_{02} - h_{01})_{ideal}}{(h_{02} - h_{01})_{actual}}.

8.3 Total Pressure Ratio

p02p01=[1+ηttγ−12M12]γ/(γ−1).\frac{p_{02}}{p_{01}} = \left[1 + η_{tt}\frac{γ-1}{2}M_1^2\right]^{γ/(γ-1)}.

9. Diffusers and Nozzles

9.1 Converging–Diverging Nozzle

Isentropic flow relations: AA∗=1M[2γ+1(1+γ−12M2)](γ+1)/2(γ−1).\frac{A}{A^*} = \frac{1}{M}\left[\frac{2}{γ+1}(1 + \frac{γ-1}{2}M^2)\right]^{(γ+1)/2(γ-1)}.

Mass flux at choked condition: G∗=p0γRT0(2γ+1)(γ+1)/2(γ−1).G^* = p_0 \sqrt{\frac{γ}{RT_0}} \left(\frac{2}{γ+1}\right)^{(γ+1)/2(γ-1)}.

9.2 Diffuser Design

For deceleration of subsonic flow: p2/p1=(T2/T1)γ/(γ−1).p_2/p_1 = (T_2/T_1)^{γ/(γ-1)}.

In supersonic flow, compression must occur through controlled shocks or wave systems.

10. Performance Maps and Operating Regions

Performance maps show:

  • Pressure ratio vs. mass flow at various speeds.
  • Efficiency contours overlayed.
RegionFlow Behavior
StableAttached flow, predictable
SurgeFlow reversal, oscillation
ChokeSonic limitation, reduced efficiency

11. Exergy and Irreversibility in Turbomachinery

Entropy generation: σs=Q˙T+∑im˙i(sout,i−sin,i)V.σ_s = \frac{\dot{Q}}{T} + \sum_i \frac{\dot{m}_i (s_{out,i} - s_{in,i})}{V}.

Exergy destruction: E˙D=T0m˙(s2−s1).\dot{E}_D = T_0 \dot{m} (s_2 - s_1).

Efficiency in exergy form: ηex=1−T0(s2−s1)h2−h1.η_{ex} = 1 - \frac{T_0 (s_2 - s_1)}{h_2 - h_1}.

Losses primarily arise from viscous dissipation, mixing, and shock formation.

12. Advanced Topics

12.1 Multistage Axial Compressors

  • Stage loading: ψ=Δh0/U2ψ = Δh_0/U^2
  • Flow coefficient: φ=Vax/Uφ = V_{ax}/U

Design optimization aims for balanced ψ,φψ, φ minimizing losses.

12.2 Turbine Blade Cooling

At high-temperature operation (>1500 K):

  • Film cooling: injection through discrete holes.
  • Internal convection channels.
  • Thermal barrier coatings.

12.3 Unsteady Blade Row Interaction

Wake and potential field interactions between stator and rotor rows cause periodic loading and efficiency reduction.

13. Summary of Key Relations

ConceptEquationNotes
Euler’s equationw=u(Vw2−Vw1)w = u(V_{w2}-V_{w1})Basis of turbomachinery theory
Compressor efficiencyηc=(T2s−T1)/(T2−T1)η_c = (T_{2s}-T_1)/(T_2-T_1)Isentropic efficiency
Turbine efficiencyηt=(T2−T1)/(T2s−T1)η_t = (T_2-T_1)/(T_{2s}-T_1)Expansion performance
Slip factorσ=1−(π/Z)cosβ2/(1−cosβ2)σ = 1 - (π/Z)\sqrt{cosβ_2/(1 - cosβ_2)}Centrifugal correction
Exergy lossED=T0Δsm˙E_D = T_0Δs\dot{m}Irreversibility quantification
  • Fluid_Dynamics/09_Compressible_and_Supersonic_Flow.md — compressible flow and choking conditions.
  • Thermodynamics/03_First_and_Second_Laws.md — energy and entropy foundations.
  • Heat_Transfer/Convective_Mechanisms.md — blade cooling and heat transfer.
  • Aero_Thermodynamics/Propulsion_Systems.md — integration of turbomachinery into jet engines.