UNIT 1: FUNDAMENTALS OF TURBOMACHINERY

1.0 Introduction and Classification

Turbomachinery encompasses devices that extract energy from or add energy to a continuously flowing fluid by the dynamic action of rotating blades.

Classification by Energy Transfer

Classification by Flow Path

• Axial Flow: Fluid moves parallel to axis (e.g., axial compressor, Kaplan turbine).

• Radial Flow (Centrifugal): Fluid moves perpendicular to axis (e.g., centrifugal pump/compressor, Pelton wheel).

• Mixed Flow: Combination of radial and axial components (e.g., Francis turbine).

Common Applications

• Power generation (thermal, hydro, nuclear)

• Aviation (jet engines, APUs)

• Process industries (oil & gas, chemical, HVAC)

• Marine propulsion

2.0 Fundamental Thermodynamic and Fluid Dynamic Principles

Steady Flow Energy Equation (SFEE - First Law)

For a turbomachine:

$$ \dot{Q} - \dot{W}_s = \dot{m} \left( h_2 + \frac{C_2^2}{2} + gz_2 - h_1 - \frac{C_1^2}{2} - gz_1 \right) $$

For adiabatic machines ($$\displaystyle \dot{Q}=0 $$), shaft work per unit mass:

$$ w = h_1 - h_2 + \frac{C_1^2 - C_2^2}{2} + g(z_1 - z_2) $$

[!TIP] In most analyses, potential energy change ($g\Delta z$) is negligible.

Second Law & Isentropic Efficiency

• Entropy Generation ($$\displaystyle \Delta S_{gen} > 0 $$): Real processes are irreversible.

• Isentropic Efficiency:

  • Turbine: $$\displaystyle \eta_t = \frac{h_1 - h_2}{h_1 - h_{2s}} $$

  • Compressor/Pump: $$\displaystyle \eta_c = \frac{h_{2s} - h_1}{h_2 - h_1} $$

  where subscript s denotes isentropic state.

Moment of Momentum Theorem (Angular Momentum Principle)

For a control volume:

$$ \sum M_{shaft} = \dot{m} (r_2 C_{\theta 2} - r_1 C_{\theta 1}) $$

This is the fundamental equation for torque transmission.

Euler's Turbine Equation

Derivation: Apply moment of momentum to rotor blades.

$$ \boxed{w = U_2 C_{\theta 2} - U_1 C_{\theta 1}} $$

where $w$ = work done per unit mass, $U$ = blade speed, $$\displaystyle C_\theta $$ = whirl (tangential) component of absolute velocity.

Physical Significance: Energy transfer depends on change in angular momentum of fluid.

For Reaction Turbines (General Euler Equation):

$$ w = U_2 C_{\theta 2} - U_1 C_{\theta 1} + \frac{C_1^2 - C_2^2}{2} $$

The last term accounts for kinetic energy change in stator and rotor.

3.0 Velocity Diagrams and Analysis

Velocity Triangle Components

• Absolute Velocity ($\vec{C}$): Fluid velocity relative to stationary frame.

• Blade Speed ($\vec{U}$): Tangential velocity of blade.

• Relative Velocity ($\vec{V}$): Fluid velocity relative to moving blade ($$\displaystyle \vec{V} = \vec{C} - \vec{U} $$).

Key Components:

• Whirl Velocity ($$\displaystyle C_w $$ or $$\displaystyle C_\theta $$): Tangential component (does work).

• Flow Velocity ($$\displaystyle C_f $$ or $$\displaystyle C_r $$): Axial/radial component (determines mass flow).

Construction of Velocity Triangles

At inlet (1) and outlet (2) of a blade row:

Inlet:  C₁ (α₁) → U₁ → V₁ (β₁)

Outlet: C₂ (α₂) → U₂ → V₂ (β₂)

where $\alpha$ = absolute flow angle, $\beta$ = relative flow angle.

Shockless Entry Condition

For optimal performance (no incidence loss):

$$ \alpha_1 = \beta_1 \quad \text{(at stator outlet/rotor inlet)} $$

$$ \alpha_2 = \beta_2 \quad \text{(at rotor outlet)} $$

Key Formulas from Velocity Diagrams

Velocity Diagrams for Specific Machines

1. Impulse Turbine (Symmetrical Blades):

   • $$\displaystyle C_1 = C_2 $$, $$\displaystyle \alpha_1 = \beta_2 $$, $$\displaystyle \alpha_2 = \beta_1 $$

   • $$\displaystyle U = \frac{C_1 \cos\alpha_1}{2} $$ for max efficiency.

2. Parsons Reaction Turbine (50% Reaction):

   • Symmetrical velocity diagram: $$\displaystyle C_1 = C_2 $$, $$\displaystyle \alpha_1 = \beta_2 $$, $$\displaystyle \alpha_2 = \beta_1 $$

   • Stator and rotor enthalpy drops equal.

3. Centrifugal Compressor:

   • Inlet: Usually axial ($$\displaystyle \alpha_1 \approx 0^\circ $$)

   • Outlet: Radial ($$\displaystyle \alpha_2 = 90^\circ $$), $$\displaystyle C_{\theta 2} = U_2 - C_{r2}\cot\beta_2 $$

4. Axial Flow Compressor:

   • Typically 50% reaction, symmetric triangles.

[!TIP] Always draw velocity triangles to scale for problem-solving. Mark $U$, $C$, $V$, and angles clearly.

4.0 Degree of Reaction

Definition

$$ \boxed{R = \frac{\text{Enthalpy drop in rotor}}{\text{Total enthalpy drop per stage}}} $$

For a stage (stator + rotor).

Expression in Terms of Velocity Components

For a symmetrical impulse stage ($$\displaystyle C_1 = C_2 $$, $$\displaystyle \alpha_1 = \beta_2 $$, $$\displaystyle \alpha_2 = \beta_1 $$):

$$ R = \frac{1}{2} - \frac{(C_{\theta 2} - C_{\theta 1})}{2U} $$

50% Reaction Turbine (Parsons Turbine)

• Characteristics: Symmetrical velocity diagram ($$\displaystyle C_1 = C_2 $$, $$\displaystyle \alpha_1 = \beta_2 $$, $$\displaystyle \alpha_2 = \beta_1 $$).

• Enthalpy Drop: Equal in stator and rotor.

• Blade Angles: $$\displaystyle \alpha_1 = \beta_2 $$, $$\displaystyle \alpha_2 = \beta_1 $$.

• Work Output: $$\displaystyle w = 2 U C_{\theta 1} \cos\alpha_1 $$ (since $$\displaystyle C_{\theta 2} = -C_{\theta 1} $$).

Relation to Blade Geometry

• $$\displaystyle R = 0 $$: Pure impulse (all drop in stator/nozzle).

• $$\displaystyle R = 0.5 $$: Symmetric reaction (Parsons).

• $$\displaystyle R = 1 $$: Pure reaction (all drop in rotor).

5.0 Steam Turbines

Types: Impulse vs Reaction

Compounding (To Reduce Blade Speed)

1. Velocity Compounding (Curtis): Multiple impulse stages in series on same shaft. Uses multiple moving blade rows separated by fixed blades.

2. Pressure Compounding (Rateau): Multiple nozzle rings separated by single moving blade row. Each stage is impulse.

3. Combined Compounding: Pressure + velocity (e.g., 1st stage pressure compounded, later stages velocity compounded).

Parsons Reaction Turbine Stage Analysis (Given: $$\displaystyle D_m $$, $N$, $$\displaystyle C_2 $$, $$\displaystyle \beta_2 $$, $\dot{m}$)

Step-by-Step:

1. Calculate blade speed: $$\displaystyle U = \frac{\pi D_m N}{60} $$

2. For 50% reaction, symmetrical diagram: $$\displaystyle C_1 = C_2 $$, $$\displaystyle \alpha_1 = \beta_2 $$, $$\displaystyle \alpha_2 = \beta_1 $$.

3. From outlet triangle: $$\displaystyle C_{\theta 2} = C_2 \sin\alpha_2 = C_2 \sin\beta_1 $$ (but $$\displaystyle \beta_1 $$ unknown).

4. Use Euler: $$\displaystyle w = U(C_{\theta 1} - C_{\theta 2}) $$. For symmetric, $$\displaystyle C_{\theta 1} = -C_{\theta 2} $$ → $$\displaystyle w = 2U C_{\theta 1} $$.

5. From inlet triangle: $$\displaystyle C_{\theta 1} = C_1 \cos\alpha_1 $$. But $$\displaystyle C_1 = C_2 $$ and $$\displaystyle \alpha_1 = \beta_2 $$.

6. From outlet triangle geometry: $$\displaystyle C_{\theta 2} = U - C_{f2} \cot\beta_2 $$, where $$\displaystyle C_{f2} = C_2 \cos\alpha_2 = C_2 \cos\beta_1 $$.

7. Solve simultaneously for $$\displaystyle C_{\theta 1} $$, $$\displaystyle \alpha_1 $$, $$\displaystyle \beta_1 $$.

8. Tangential Force: $$\displaystyle F_t = \dot{m}(C_{\theta 1} + C_{\theta 2}) = 2\dot{m} C_{\theta 1} $$ (symmetric).

9. Power Developed: $$\displaystyle P = \dot{m} w = 2 \dot{m} U C_{\theta 1} $$.

[!TIP] In Parsons stage, always assume $$\displaystyle C_1 = C_2 $$ and $$\displaystyle \alpha_1 = \beta_2 $$, $$\displaystyle \alpha_2 = \beta_1 $$ unless specified otherwise.

Losses in Steam Turbines

Efficiencies

• Blade/Stage Efficiency: $$\displaystyle \eta_b = \frac{\text{Work output}}{\text{Energy supplied to stage}} = \frac{\dot{m} w}{\dot{m} \Delta h_{nozzle}} $$

• Nozzle Efficiency: $$\displaystyle \eta_n = \frac{C_1^2/2}{h_0 - h_1} $$ (actual vs ideal kinetic energy)

• Overall Efficiency: $$\displaystyle \eta_o = \eta_b \times \eta_n \times \dots $$ (product of stage efficiencies)

Reheat Factor

• Definition: $$\displaystyle RF = \frac{\text{Total isentropic enthalpy drop (multi-stage)}}{\text{Sum of stage isentropic drops}} $$

• Significance: $$\displaystyle RF > 1 $$ for multi-stage turbines due to improved reheat (steam reheated between stages, average temperature of heat addition increases → higher cycle efficiency).

• Cause: In multi-stage expansion, steam is reheated in the boiler between high-pressure and low-pressure sections, increasing average temperature of heat addition.

Governing of Steam Turbines (Maintain Constant Speed under Variable Load)

1. Throttle Governing: throttle valve controls steam pressure to all nozzles. Simple, but inefficient at part-load.

2. Nozzle Governing: Groups of nozzles are turned on/off selectively. Better part-load efficiency.

3. By-pass Governing: Steam by-passed to condenser or later stages. Used for large load swings.

6.0 Hydraulic Turbines

Classification

• Impulse: Pelton wheel (high head, low flow).

• Reaction: Francis (medium head/flow), Kaplan (low head, high flow).

Pelton Wheel

Components: Nozzle, Runner (bucket), Casing.

Velocity Diagram for Bucket:

• Jet velocity: $$\displaystyle V_j = C_v \sqrt{2gH} $$ ($$\displaystyle C_v $$ = velocity coefficient)

• Bucket speed: $$\displaystyle u = K_u \sqrt{2gH} $$ ($$\displaystyle K_u $$ = speed ratio, typically 0.45–0.5)

• Relative velocity: $$\displaystyle V = V_j - u $$ (for simple bucket, no friction)

• After deflection by ~165°, $$\displaystyle V_2 \approx V_1 = V_j - u $$

• Power Output: $$\displaystyle P = \dot{m} u (V_j - u) \times \eta_m $$ (mechanical efficiency)

• Hydraulic Efficiency (Ideal): $$\displaystyle \eta_h = \frac{2u(V_j - u)}{V_j^2} = \frac{2K_u(1 - K_u)}{1} $$

  Max at $$\displaystyle K_u = 0.5 $$ → $$\displaystyle \eta_h,max = 0.5 $$ (50%)? Wait! Correction: For ideal bucket with 180° deflection, $$\displaystyle \eta_h = \frac{2u(V_j - u)}{V_j^2} $$. Max when $$\displaystyle u = V_j/2 $$ → $$\displaystyle \eta_h,max = 0.5 $$. But actual buckets have 165° deflection and friction, so max efficiency ~0.9–0.95.

Effect of Side Clearance Angle: Loss of jet energy through clearance reduces efficiency. Optimal bucket shape minimizes this.

Design Calculations:

1. Jet diameter from discharge: $$\displaystyle Q = A_j V_j = \frac{\pi d_j^2}{4} V_j $$

2. Runner diameter: $$\displaystyle u = \frac{\pi D_m N}{60} = K_u \sqrt{2gH} $$

3. Nozzle diameter: from $Q$ and $$\displaystyle V_j $$.

Draft Tube

Purpose:

1. Recover kinetic energy from turbine exit.

2. Create negative pressure (suction head) → increases net head.

3. Provides safe discharge location.

Types:

Analysis: Pressure recovery coefficient $$\displaystyle K_p = \frac{p_{exit} - p_{atm}}{\frac{1}{2}\rho V_2^2} $$. Draft tube efficiency $$\displaystyle \eta_d = \frac{gH_d}{\frac{V_2^2}{2}} $$, where $$\displaystyle H_d $$ = static head recovered.

Specific Speed of Turbine ($$\displaystyle N_s $$)

$$ \boxed{N_s = \frac{N \sqrt{P}}{H^{5/4}}} $$

• $N$ = RPM, $P$ = power (kW), $H$ = net head (m).

• Derivation: From similarity laws: $$\displaystyle P \propto ND^3 \rho H $$, $$\displaystyle Q \propto ND^2 H^{1/2} $$, $$\displaystyle H \propto N^2 D^2 $$. Eliminate $D$.

• Significance in Selection:

  • $$\displaystyle N_s < 20 $$: Pelton (impulse)

  • $$\displaystyle 20 < N_s < 300 $$: Francis (reaction)

  • $$\displaystyle N_s > 300 $$: Kaplan (reaction, axial flow)

Cavitation

• Phenomenon: Formation and collapse of vapor bubbles in low-pressure regions (e.g., draft tube, blade suction side).

• Cause: Local pressure $$\displaystyle < $$ vapor pressure of fluid.

• Effects: Erosion, noise, vibration, efficiency drop, damage to material.

• Prevention:

  • Thoma's Cavitation Factor: $$\displaystyle \sigma = \frac{H_{atm} - H_{sv}}{H} $$, where $$\displaystyle H_{sv} $$ = vapor pressure head. For safe operation, $$\displaystyle \sigma > \sigma_{critical} $$.

  • NPSH: Net Positive Suction Head available must exceed NPSH required.

Efficiencies

• Hydraulic Efficiency: $$\displaystyle \eta_h = \frac{\text{Power delivered to runner}}{\text{Water power}} = \frac{P_r}{\rho g Q H} $$

• Mechanical Efficiency: $$\displaystyle \eta_m = \frac{\text{Shaft power}}{\text{Power delivered to runner}} = \frac{P_s}{P_r} $$

• Overall Efficiency: $$\displaystyle \eta_o = \eta_h \times \eta_m $$

7.0 Compressors

Types

• Centrifugal: Radial outflow, single/multi-stage.

• Axial: Multiple stages, high flow, moderate pressure rise per stage.

• Mixed Flow: Diagonal outflow.

Centrifugal Compressor

Main Components: Impeller, Diffuser (vaned/vaneless), Volute. Vector Diagram:

• Inlet: Usually axial ($$\displaystyle \alpha_1 \approx 0^\circ $$), $$\displaystyle C_1 \approx C_{a1} $$.

• Outlet: Radial ($$\displaystyle \alpha_2 = 90^\circ $$), $$\displaystyle C_{\theta 2} = U_2 - C_{r2}\cot\beta_2 $$. Pressure Rise Mechanism:

1. In impeller: $$\displaystyle dh = U dC_\theta $$ (Euler's equation).

2. In diffuser: $$\displaystyle C_2 $$ → $$\displaystyle C_3 $$ (kinetic energy → pressure).

Slip Factor ($\sigma$):

$$ \sigma = \frac{C_{\theta 2, actual}}{C_{\theta 2, ideal}} = \frac{U_2 - C_{r2}\cot\beta_2}{U_2} $$

• Causes: Boundary layer, curvature, finite number of blades.

• Effect: Reduces pressure ratio and efficiency. $$\displaystyle C_{\theta 2,actual} = \sigma U_2 $$.

Axial Flow Compressor

• Vector Diagram: Similar to axial turbine but reversed.

• Typical Reaction: 50% (symmetric velocity triangles).

• Stage Work: $$\displaystyle w = U (C_{\theta 1} - C_{\theta 2}) $$

• Pressure Rise: Calculated from energy equation and isentropic relation.

Surging and Choking

Polytropic Efficiency ($$\displaystyle \eta_p $$)

• Definition: Efficiency for an infinitesimal pressure rise process.

$$ \eta_p = \frac{dh_{s}}{dh} = \frac{\text{Isentropic enthalpy rise}}{\text{Actual enthalpy rise}} $$

• Advantage: Independent of pressure ratio; useful for multi-stage compression.

• Relation to Isentropic Efficiency ($$\displaystyle \eta_s $$):

$$ \eta_s = \frac{r_p^{(\gamma-1)/\gamma} - 1}{r_p^{(\gamma-1)/(\gamma \eta_p)} - 1} $$

where $$\displaystyle r_p $$ = pressure ratio.

Comparison: Axial vs Centrifugal Compressors

8.0 Pumps

Centrifugal Pump

Main Parts: Impeller, Casing (volute/diffuser), Suction & delivery pipes, stuffing box. Working Principle: Impeller imparts kinetic energy → diffuser/volute converts to pressure. Manometric Head:

$$ \boxed{H_m = \frac{p_2 - p_1}{\rho g} + \frac{V_2^2 - V_1^2}{2g} + (z_2 - z_1)} $$

Losses:

• Volumetric (leakage)

• Hydraulic (friction, recirculation)

• Mechanical (bearing, seal, disc friction)

• Overall Efficiency: $$\displaystyle \eta_o = \eta_v \times \eta_h \times \eta_m $$

Reciprocating Pump

Advantages over Centrifugal:

• High pressure (no theoretical limit)

• Positive displacement (constant flow regardless of pressure)

• High volumetric efficiency at high pressure

• No priming needed (self-priming)

• Can handle viscous fluids, slurries. Disadvantages: Pulsating flow, high maintenance, lower flow rate.

Positive Displacement Pumps

Definition: Traps and displaces a fixed volume per cycle. Types:

• Reciprocating: Piston, plunger, diaphragm.

• Rotary: Gear, vane, lobe, screw.

Specific Speed of Pump ($$\displaystyle N_s $$)

$$ \boxed{N_s = \frac{N \sqrt{Q}}{H^{3/4}}} $$

• $N$ = RPM, $Q$ = flow rate (m³/s), $H$ = head (m).

• Significance: Predicts pump type and performance.

  • $$\displaystyle N_s < 500 $$: Radial flow (centrifugal)

  • $$\displaystyle 500 < N_s < 5000 $$: Mixed flow

  • $$\displaystyle N_s > 5000 $$: Axial flow

9.0 Fluid and Power Transmission Devices

Fluid Coupling

Construction: Pump (impeller), Turbine (runner), Casing (filled with oil). Working: Momentum transfer between pump and runner via fluid. Slip: $$\displaystyle s = \frac{N_t - N_p}{N_t} $$ ($$\displaystyle N_t $$ = turbine speed, $$\displaystyle N_p $$ = pump speed). Efficiency: $$\displaystyle \eta = s $$ (since $$\displaystyle P_{out} = s P_{in} $$). Velocity Diagram:

Torque Converter

Construction: Pump, Turbine, Stator (reactor) mounted on casing. Working: Stator redirects fluid returning from turbine to pump, providing torque multiplication at low speed ratios ($$\displaystyle \omega_t/\omega_p < 1 $$). Torque Multiplication Factor: $$\displaystyle T_{mult} = \frac{T_{turbine}}{T_{pump}} > 1 $$ at low $$\displaystyle \omega_t/\omega_p $$. Diagram:

Hydraulic Intensifier

Purpose: Increase pressure of fluid (low pressure → high pressure). Construction: Two rams of different diameters in a cylinder, large cylinder end connected to low-pressure source, small end delivers high pressure. Working: Force $F$ constant → $$\displaystyle P_1 A_1 = P_2 A_2 $$ → $$\displaystyle P_2 = P_1 (A_1/A_2) $$. Applications: Hydraulic presses, lifts.

Power Transmitting Devices in Industry

10.0 Dimensional Analysis and Similarity

Buckingham Pi Theorem

For $$\displaystyle \eta = f(\rho, \mu, \omega, D, Q) $$:

• Variables $$\displaystyle n = 5 $$, Fundamental dimensions $$\displaystyle k = 3 $$ (M, L, T).

• Number of Pi groups: $$\displaystyle n - k = 2 $$.

• Choose repeating variables: $\rho, \omega, D$.

• Pi groups:

$$ \Pi_1 = \eta \quad (\text{dependent}) $$

$$ \Pi_2 = \frac{\rho \omega D^2}{\mu} = Re \quad (\text{Reynolds number}) $$

$$ \Pi_3 = \frac{Q}{\omega D^3} = \phi \quad (\text{flow coefficient}) $$

• Final form: $$\displaystyle \eta = \phi(Re, \phi) $$.

Key Dimensionless Parameters

Scaling Laws: For similar machines:

$$ \phi_1 = \phi_2, \quad \psi_1 = \psi_2, \quad \lambda_1 = \lambda_2 \quad \text{if } Re \text{ large enough (fully turbulent)}. $$

11.0 Blade Efficiency Optimization and Losses

Blade Efficiency for Impulse Turbine

General expression (with blade speed $u$, nozzle exit velocity $$\displaystyle C_1 $$, angle $$\displaystyle \alpha_1 $$):

$$ \eta_b = \frac{2u C_1 \cos\alpha_1 - u^2}{C_1^2} $$

Condition for Maximum Efficiency:

$$ u = \frac{C_1 \cos\alpha_1}{2} $$

and for symmetrical blades with no friction: $$\displaystyle \alpha_1 = \beta_1 $$, $$\displaystyle \alpha_2 = \beta_2 $$.

Effect of Losses

1. Nozzle Loss ($$\displaystyle \eta_n < 1 $$): Reduces $$\displaystyle C_1 $$ → lowers $$\displaystyle \eta_b $$.

2. Blade Friction: Reduces relative velocity $$\displaystyle V_2 $$, increases $$\displaystyle \beta_2 $$ → increases leaving loss ($$\displaystyle C_2^2/2 $$).

3. Overall: Both losses reduce actual $$\displaystyle \eta_b $$ below ideal value.

12.0 Short Note Topics (Frequent in Exams)

Centrifugal Blower

• Definition: Machine delivering large volume of air at low pressure rise (typically < 0.3 bar).

• Difference from Compressor: Lower pressure ratio, higher flow, often no diffuser (just volute).

• Applications: Ventilation, dust removal, cooling, combustion air supply.

Hydraulic Intensifier

• Construction: Cylinder with two rams of different diameters. Large ram connected to low-pressure source, small ram delivers high pressure.

• Working: $$\displaystyle P_1 A_1 = P_2 A_2 $$ → $$\displaystyle P_2 = P_1 (A_1/A_2) $$. Force $F$ constant.

• Applications: Hydraulic presses, lifts, testing equipment.

Draft Tube

• Purpose: Recover kinetic energy, create suction head.

• Types with Sketches:

  1. Conical:

     DiagramSEARCH: conical draft tube
     – simple, prone to separation.

  2. Elbow:

     DiagramSEARCH: elbow draft tube
     – for vertical shafts, space saving.

  3. Moody:

     DiagramSEARCH: Moody draft tube
     – curved, minimizes separation, high efficiency.

• Analysis: Efficiency $$\displaystyle \eta_d = \frac{gH_d}{V_2^2/2} $$, where $$\displaystyle H_d $$ = static head recovered.

Application of Power Transmitting Devices at Industry Level

• Couplings: Connect shafts, accommodate misalignment (flexible), transmit torque (rigid).

• Gearboxes: Change speed/torque, reverse direction (machine tools, conveyors).

• Belt Drives: HVAC systems, agricultural equipment (low cost, quiet).

• Chain Drives: Motorcycles, bicycles, conveyors (positive drive).

• Torque Converters: Automotive automatic transmissions, heavy machinery (torque multiplication).

• Fluid Couplings: Crushers, mills, fans (soft start, overload protection).

13.0 Additional Concepts

Slip Factor in Centrifugal Machines

$$ \sigma = \frac{C_{\theta 2, actual}}{C_{\theta 2, ideal}} = \frac{U_2 - C_{r2}\cot\beta_2}{U_2} $$

• Causes: Boundary layer growth, curvature of flow, finite blade number (Stodola's slip factor: $$\displaystyle \sigma = 1 - \frac{\pi}{z} \frac{\sin\beta_2}{1 + \sin\beta_2} $$).

• Effect: Reduces $$\displaystyle C_{\theta 2} $$ → lowers pressure ratio and efficiency. Must be accounted for in design.

Surging in Axial Compressors

• Characteristic Curve: Pressure ratio vs. mass flow shows a "hump" at low flow.

• Surge Line: Boundary between stable and unstable operation.

• Prevention: Variable stator vanes (change incidence), bleed valves (dump excess air), multi-shaft design.

Choking in Compressors

• When: At high mass flow, Mach number reaches 1 at minimum area (throat of blade passage or diffuser).

• Where: Typically at rotor exit or diffuser throat.

• Implication: Mass flow rate becomes constant regardless of further decrease in downstream pressure. Sets maximum flow limit.

Polytropic vs Isentropic Efficiency

[!TIP] For multi-stage compressors with intercooling, polytropic efficiency is more meaningful as it remains nearly constant per stage.