Unit 4: Centrifugal Compressors

I. Orientation — Governing Principle and Basic Features

A centrifugal compressor is a dynamic-flow machine that continuously raises gas pressure by transferring shaft work through a rotating impeller and then converting much of the resulting kinetic energy into static pressure in a diffuser.

• Governing principle: Angular momentum changes as gas flows radially through the impeller; Euler’s turbomachinery equation relates this change to the specific work transferred.
• Energy conversion:
  • The impeller increases stagnation enthalpy, velocity, and pressure.
  • The diffuser decelerates the gas and converts kinetic energy into static pressure.
• Principal components: The inlet duct, impeller eye, rotating impeller, vaneless or vaned diffuser, collector or volute, and discharge duct form the main flow path.
• Flow convention: Subscripts (1) and (2) denote impeller inlet and exit; a subsequent station (3) may denote diffuser exit.
• Velocity convention: Absolute velocity is (\mathbf C), blade velocity is (\mathbf U), and velocity relative to the blade is (\mathbf W), related by:
  
  TEXT

  C = U + W

• Thermodynamic convention: Stagnation quantities include the gas’s static and kinetic energies; (T_0), (p_0), and (h_0) denote stagnation temperature, pressure, and enthalpy.
• Operating characteristics: A centrifugal stage provides a relatively high pressure ratio for a small mass-flow rate, but its stable range is limited by choking, rotating stall, and surge.

II. Compressor Flow Mechanism — Impeller and Diffuser Action

A. Principle of operation of centrifugal compressor

The compressor operates by accelerating gas in a rotating impeller and subsequently diffusing that high-velocity gas to obtain a useful pressure rise.

• Induction at the eye: Gas enters the impeller approximately axially and is guided into passages between rotating blades; the eye area must keep the inlet relative Mach number and incidence within acceptable limits.
• Centrifugal action: As the impeller rotates, blade forces increase the gas’s tangential momentum while radial motion carries it from the smaller inlet radius (r_1) to the larger exit radius (r_2).
• Impeller pressure rise: Static pressure rises inside the impeller because of centrifugal pressure gradients and passage diffusion, although the gas also leaves with substantial absolute kinetic energy.
• Diffuser action: An increasing flow area reduces the absolute velocity (C), allowing static pressure to rise according to the steady-flow energy equation.
• Collector function: A volute or annular collector receives gas around the diffuser circumference, combines the flow, and directs it toward the discharge duct.
• Continuous-flow process: Unlike a positive-displacement compressor, the machine does not trap a fixed volume; pressure generation depends on aerodynamic momentum exchange.
• Flow capacity: For density (\rho), meridional velocity (C_m), and effective flow area (A), mass flow is:
  
  TEXT

  m_dot = rho A C_m

  
  Here, (\dot m) is mass flow rate in (\mathrm{kg\,s^{-1}}), (\rho) is density in (\mathrm{kg\,m^{-3}}), (A) is effective area in (\mathrm{m^2}), and (C_m) is the meridional velocity component in (\mathrm{m\,s^{-1}}).
• Practical limitation: At low flow, excessive incidence and diffusion can produce separation; at high flow, a passage throat may reach sonic velocity and choke.

III. Energy Transfer and Compression

A. Work done and pressure rise

The impeller work is determined by the change in angular momentum, while the attainable pressure rise depends on how efficiently that work becomes stagnation pressure.

• Euler compressor equation: The specific stagnation-enthalpy rise is:
  
  TEXT

  Y = h02 - h01 = U2 Ctheta2 - U1 Ctheta1

  
  Here, (Y) is specific work in (\mathrm{J\,kg^{-1}}); (h{01}) and (h{02}) are inlet and exit stagnation enthalpies; (U) is blade speed; and (C_\theta) is the tangential component of absolute velocity.
• Blade speed: At radius (r), the peripheral velocity is:
  
  TEXT

  U = omega r = pi D N / 60

  
  Here, (\omega) is angular speed in (\mathrm{rad\,s^{-1}}), (r) is radius, (D) is diameter, and (N) is rotational speed in revolutions per minute.
• No-prewhirl condition: If the inlet flow is axial, (C_{\theta1}=0), so:
  
  TEXT

  Y = U2 Ctheta2

• Radial-blade idealization: For ideal radial blades at exit, (C_{\theta2}=U_2), giving (Y=U2^2). Actual gas slips behind the blade, making (C{\theta2}<U_2); a slip factor (\sigma) accounts for this reduction.
• Temperature rise: For a calorically perfect gas:
  
  TEXT

  Delta T0 = Y / cp

  
  Here, (\Delta T0=T{02}-T_{01}), and (c_p) is specific heat at constant pressure in (\mathrm{J\,kg^{-1}K^{-1}}).
• Pressure ratio: With compressor isentropic efficiency (\etac):
  
  TEXT

  p02 / p01 = [1 + eta_c Y / (cp T01)]^(gamma/(gamma - 1))

  
  Here, (p{02}/p{01}) is the stagnation-pressure ratio, (T{01}) is inlet stagnation temperature, and (\gamma) is the specific-heat ratio.
• Worked example: If (U2=400\,\mathrm{m\,s^{-1}}), (C{\theta2}=320\,\mathrm{m\,s^{-1}}), and there is no prewhirl, then:
  
  TEXT

  Y = 400 x 320 = 128,000 J/kg = 128 kJ/kg

  
  For (c_p=1005\,\mathrm{J\,kg^{-1}K^{-1}}), the stagnation-temperature rise is approximately (127.4\,\mathrm K).
• Power requirement: Neglecting mechanical loss, shaft power transferred to the gas is (\dot W=\dot mY), where (\dot W) is power in watts.

IV. Flow Kinematics at the Rotor

A. Velocity diagrams

Velocity diagrams resolve absolute, blade, and relative velocities at impeller inlet and exit, thereby connecting blade geometry with work and incidence.

• Vector relationship: At every impeller station:
  
  TEXT

  C = U + W

  
  The meridional and tangential components are denoted (Cm) and (C\theta).
• Inlet triangle: For axial entry without prewhirl, (C{\theta1}=0); consequently, the blade sees a relative velocity formed from axial velocity (C{m1}) and the opposing peripheral speed (U_1).
  
  TEXT

  tan(beta1) = Cm1 / (U1 - Ctheta1)

  
  Here, (\beta_1) is the relative-flow angle measured from the tangential direction.
• Incidence: Incidence is the difference between the relative inlet-flow angle and blade metal angle; large positive incidence causes leading-edge separation and loss.
• Exit triangle: For an impeller exit blade angle (\beta2), measured from the tangential direction:
  
  TEXT

  Ctheta2 = U2 - Cm2 cot(beta2)

  
  Here, (C{m2}) is exit meridional velocity and (\beta_2) is the relative exit-flow angle.
• Blade-form comparison:
  1. Radial blades: With (\beta2=90^\circ), the ideal relation gives (C{\theta2}=U_2), producing high theoretical work.
  2. Backward-swept blades: With (\beta2<90^\circ), (C{\theta2}<U_2); work is lower, but efficiency and operating stability are usually improved.
• Absolute exit angle: If (\alpha_2) is measured from the tangential direction:
  
  TEXT

  tan(alpha2) = Cm2 / Ctheta2

• Slip and blockage: A finite blade number reduces whirl below the blade-congruent value, while boundary layers reduce effective passage area and increase actual meridional velocity.

V. Static-Pressure Recovery

A. Diffuser vane design considerations

A diffuser must decelerate non-uniform impeller discharge flow with high pressure recovery while avoiding excessive incidence, separation, choking, and operating-range reduction.

• Diffuser alternatives:
  1. Vaneless diffuser: Offers a broad operating range and tolerates varying inlet angles, but requires greater radial size and usually gives lower pressure recovery.
  2. Vaned diffuser: Provides stronger controlled diffusion and compactness, but is more sensitive to incidence and low-flow instability.
• Inlet matching: The vane leading-edge metal angle should approximately match the diffuser inlet absolute-flow angle (\alpha_2); mismatch creates stagnation, suction-surface acceleration, and separation.
• Leading-edge radius: Placing the vane too close to the impeller intensifies interaction with blade wakes and pressure fluctuations; greater radial clearance improves mixing but increases skin-friction loss.
• Area growth: Passage area must increase gradually. An excessive diffusion rate creates an adverse pressure gradient that causes boundary-layer separation before useful pressure recovery occurs.
• Throat control: The minimum area between adjacent vanes sets the choking limit; at high mass flow, a throat Mach number near unity prevents further corrected-flow increase.
• Vane number and solidity: More vanes increase guidance and diffusion surface but also increase blockage and friction. Solidity is vane chord divided by circumferential pitch.
• Mach-number effects: Transonic impeller discharge may produce shocks at diffuser leading edges; rounded leading edges and suitable sweep reduce shock loss and sensitivity.
• Vaneless-flow relation: Neglecting friction, conservation of angular momentum gives:
  
  TEXT

  r Ctheta = constant

  
  Thus, increasing radius (r) reduces (C_\theta), contributing to velocity reduction and pressure recovery.
• Design compromise: High peak pressure recovery generally narrows the stable operating range, whereas conservative diffusion sacrifices compactness for tolerance of off-design flow.

VI. Inlet Swirl Control

A. Concept of prewhirl

Prewhirl is a deliberate tangential velocity imparted to the gas before it enters the impeller, commonly by adjustable inlet guide vanes.

• Positive prewhirl: Swirl in the direction of impeller rotation makes (C_{\theta1}>0), reduces relative inlet velocity, and lowers the Euler work through the term (-U1C{\theta1}).
• Negative prewhirl: Swirl opposite to rotation makes (C_{\theta1}<0), increasing theoretical work but also increasing relative Mach number, incidence sensitivity, and inlet loss.
• Relative inlet velocity:
  
  TEXT

  W1 = sqrt[Cm1^2 + (U1 - Ctheta1)^2]

  
  Here, (W_1) is the magnitude of relative inlet velocity; the remaining symbols denote inlet velocity components and blade speed.
• Incidence matching: Proper prewhirl aligns (W_1) with the impeller-eye blade angle, reducing leading-edge separation at an intended operating condition.
• Capacity control: Closing variable inlet guide vanes introduces positive prewhirl and reduces work, pressure ratio, and power demand at part load.
• Limitation: Guide-vane turning creates its own wake, friction, and possible separation; excessive prewhirl can reduce compressor pressure rise and move operation toward instability.

VII. Aerodynamic Instability

A. Rotation stall

Rotating stall is a local flow-separation instability in which one or more low-flow cells travel circumferentially through an impeller or diffuser passage row.

• Formation mechanism: At reduced mass flow, incidence and adverse pressure gradients increase until some passages separate and become partially blocked.
• Cell propagation: Blockage diverts flow into a neighboring passage, causing that passage to stall while the previous one may recover; this sequence makes the cell rotate at a fraction of rotor speed.
• Likely locations: Stall may originate at the impeller inlet, inducer, impeller exit, or vaned diffuser, depending on incidence and diffusion loading.
• Observable effects: Rotating pressure fluctuations cause noise, vibration, alternating blade loads, reduced pressure rise, lower efficiency, and possible fatigue damage.
• Distinction from surge:
  1. Rotating stall: Primarily circumferential and local; the time-averaged through-flow can remain forward.
  2. Surge: A system-wide axial oscillation of mass flow and pressure that may include complete flow reversal.
• Detection: Circumferentially distributed fast-response pressure sensors identify a pressure pattern passing successive probes with a measurable phase delay.
• Prevention and recovery: Adequate surge margin, controlled diffuser loading, inlet-guide-vane scheduling, bleed or recycle flow, and prompt movement to higher mass flow reduce stall risk.
• Performance-map significance: Rotating stall commonly develops near the low-flow boundary and may precede surge, although the exact sequence depends on the compressor and connected system.