UNIT 3: TURBOMACHINERY - EXAM-FOCUSED SHORT NOTES

I. FUNDAMENTAL PRINCIPLES & ANALYSIS

A. Thermodynamic Foundations & Euler's Equation

• First Law for Turbomachines (Steady Flow Energy Equation):

$$h_0 = h + \frac{V^2}{2} + gz$$

Where $$\displaystyle h_0 $$ is stagnation enthalpy. Work done per unit mass: $$\displaystyle \dot{w} = h_{01} - h_{02} $$.

• Second Law & Entropy: Real processes are irreversible, causing entropy generation ($$\displaystyle \Delta s > 0 $$). This defines losses (e.g., friction, shock) which reduce actual work output from the isentropic ideal.

• Euler's Turbine Equation (Fundamental Equation):

$$\dot{W}_u = \dot{m} \cdot U_2 V_{w2} - U_1 V_{w1}$$

Where $$\displaystyle \dot{W}_u $$ = useful shaft work (power), $\dot{m}$ = mass flow, $U$ = blade speed, $$\displaystyle V_w $$ = whirl component of absolute velocity.

**Physical Significance:** It relates the **torque** and **power** developed to the change in **angular momentum** of the fluid. It is the cornerstone for all velocity diagram analysis.

B. Dimensional Analysis & Similitude

• Buckingham Pi Theorem: For a performance parameter (e.g., efficiency $\eta$), we form dimensionless groups (Pi terms).

  Example: $$\displaystyle \eta = f\left( \underbrace{\frac{\rho \omega D^2}{\mu}}_{\text{Re}}, \underbrace{\frac{Q}{\omega D^3}}_{\phi}, \underbrace{\frac{gH}{\omega^2 D^2}}_{\psi} \right) $$

• Key Dimensionless Parameters:

  • Flow Coefficient: $$\displaystyle \phi = \frac{Q}{ND^3} $$ (or $$\displaystyle \frac{V_f}{U} $$)

  • Head Coefficient: $$\displaystyle \psi = \frac{gH}{N^2 D^2} $$ (or $$\displaystyle \frac{\Delta h}{U^2} $$)

  • Power Coefficient: $$\displaystyle \lambda = \frac{P}{\rho N^3 D^5} $$

  • Reynolds Number: $$\displaystyle Re = \frac{\rho ND^2}{\mu} $$ (governs viscous effects).

• Affinity Laws (for Pumps/Fans of same geometry):

  1. $Q \propto N$

  2. $$\displaystyle H \propto N^2 $$

  3. $$\displaystyle P \propto N^3 $$

  [!TIP] Exam Alert: These laws are used for performance prediction when speed changes.

C. Performance Efficiencies

[!NOTE] Polytropic Efficiency: Crucial for multistage compressors because it remains constant along the compression process, unlike isentropic efficiency which varies with pressure ratio.

II. STEAM TURBINES

A. Classification & Degree of Reaction

• Impulse Turbine: No pressure drop in moving blades. All pressure drop occurs in nozzles. High blade speed ratio for max efficiency.

• Reaction Turbine: Pressure drop occurs in both fixed and moving blades (stator & rotor). 50% reaction (Parsons) is most common.

• Compounding: Used to reduce blade speed.

  • Velocity Compounded (Curtis): Multiple blade rows in one stage, same pressure.

  • Pressure Compounded (Rateau): Multiple stages, each with nozzle + blade row.

  • Pressure-Velocity Compounded: Combination of both.

• Degree of Reaction (R):

$$R = \frac{\text{Static enthalpy drop in rotor}}{\text{Stagnation enthalpy drop in stage}}$$

For axial flow: $$\displaystyle R = \frac{1}{2} + \frac{V_{w2} - V_{w1}}{2U} $$.

**Significance of 50% Reaction (Parsons):** Symmetrical velocity diagrams ($$\displaystyle \alpha_1 = \beta_2 $$, $$\displaystyle \beta_1 = \alpha_2 $$), optimal for high efficiency.

B. Velocity Diagrams & Blade Geometry

1. Single-Stage Impulse Turbine (Symmetrical Blades):

• $$\displaystyle V_1 = V_2 $$ (no friction), $$\displaystyle \beta_1 = \beta_2 $$, $$\displaystyle V_{r2} = V_{r1} $$.

• Blade speed ratio for max efficiency: $$\displaystyle \rho = \frac{U}{V_1} = \cos \alpha_1 / 2 $$.

• Shockless (Axi-symmetric) Entry: $$\displaystyle V_{r1} $$ must be relative to blade inlet angle $$\displaystyle \beta_1 $$: $$\displaystyle \tan \beta_1 = \frac{V_{f}}{V_{w1} - U} $$. 2. Single-Stage Reaction Turbine (Parsons, 50% R):

• $$\displaystyle \alpha_1 = \beta_2 $$, $$\displaystyle \beta_1 = \alpha_2 $$.

• $$\displaystyle V_1 = V_2 $$, $$\displaystyle V_{r1} = V_{r2} $$.

• Work done: $$\displaystyle \dot{W} = \dot{m} U (V_{w1} + V_{w2}) $$.

[!DIAGRAM] SEARCH: "steam turbine velocity diagram impulse reaction comparison"

C. Blade Forces, Power & Efficiency

• Tangential Force on Blades: $$\displaystyle F_t = \dot{m} (V_{w1} + V_{w2}) $$ for reaction; $$\displaystyle F_t = \dot{m} V_{w1} $$ for impulse (sym).

• Axial Thrust: $$\displaystyle F_a = \dot{m} (V_{f1} \cot \beta_1 - V_{f2} \cot \beta_2) $$.

• Blade/Diagram Efficiency ($$\displaystyle \eta_b $$):

$$\eta_b = \frac{2 U V_{w1}}{V_1^2} \quad \text{(Impulse, sym. blades)}$$

Optimized: $$\displaystyle \eta_{b,max} = \cos^2 \alpha_1 $$.

• Stage Efficiency: Includes nozzle efficiency $$\displaystyle \eta_n $$: $$\displaystyle \eta_{stage} = \eta_n \cdot \eta_b $$.

D. Multistage Turbines & Reheat Factor

• Reheat Factor (R.F.): $$\displaystyle R.F. = \frac{\text{Actual total enthalpy drop}}{\text{Isentropic enthalpy drop for ideal stages}} $$.

  • Cause: Cumulative effect of blade friction losses in early stages increases steam temperature in later stages, allowing more expansion (higher actual drop).

  • Significance: R.F. > 1. Multistage turbine with same stage efficiency has higher overall efficiency than single-stage because of reheat factor.

• Governing: Methods to maintain constant speed under varying load.

  • Throttle Governing: Control valve at nozzle inlet.

  • Nozzle Governing: Groups of nozzles controlled separately.

  • Bypass Governing: Steam bypassed to later stages.

E. Losses in Steam Turbines

III. HYDRAULIC TURBINES (WATER TURBINES)

A. Fundamentals & Specific Speed

• Heads:

  • Net Head ($H$): $$\displaystyle H = \frac{P_1 - P_2}{\rho g} + \frac{V_1^2 - V_2^2}{2g} + (z_1 - z_2) $$

  • Manometric Head: Head measured by pressure gauge (excludes velocity head).

• Specific Speed ($$\displaystyle N_s $$): The speed at which a geometrically similar turbine would develop 1 kW under 1 m head.

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

**Physical Meaning:** Indicates **type** of turbine suitable for given $H$ and $P$.

*   Low $$\displaystyle N_s $$ (< 30): Pelton (Impulse)

*   Medium $$\displaystyle N_s $$ (30-300): Francis (Reaction)

*   High $$\displaystyle N_s $$ (> 300): Kaplan (Reaction)

B. Pelton Wheel (Impulse)

• Main Parts: Nozzle (with spear valve), Buckets (double-cup, split), Casing.

• Velocity Diagram: Jet velocity $$\displaystyle V_j = C_v \sqrt{2gH} $$. Bucket speed $U$. For double bucket, $$\displaystyle V_{w2} \approx 2U \cos \phi $$ (where $\phi$ = bucket angle, usually $$\displaystyle 165^\circ $$).

• Force & Power:

$$F = \rho Q V_j (1 + \cos \phi) \quad \text{(for double bucket)}$$

$$P = F \cdot U$$

• Hydraulic Efficiency: $$\displaystyle \eta_h = \frac{2U V_j (1 + \cos \phi)}{V_j^2} $$.

  Max when $$\displaystyle U = V_j/2 $$.

• Design Parameters:

  • Jet Ratio (m): $$\displaystyle m = D/d $$ (runner dia / jet dia). Optimum $m \approx 6-10$.

  • Number of Buckets: $Z \approx 15 + 0.5 m$ (to ensure continuous jet).

  • Side Clearance Angle: $$\displaystyle \approx 1.5^\circ - 2^\circ $$ to avoid bucket interference.

C. Reaction Turbines (Francis & Kaplan) & Draft Tube

• Working: Pressure + Velocity change in runner. Requires full casing and draft tube.

• Draft Tube:

  • Necessity: Allows runner to be placed above tailrace by converting kinetic energy to pressure, recovering head.

  • Types: Conical, Elbow, Moody (spreading), Simple Tapered.

  • Maximum Efficiency: $$\displaystyle \eta_{dt} \approx 0.85-0.9 $$.

• Francis vs. Kaplan:

  • Francis: Medium head, adjustable guide vanes, fixed blades.

  • Kaplan: Low head, adjustable runner blades, propeller-type.

D. Cavitation

• Phenomenon: Formation and collapse of vapor bubbles in regions of low pressure (below vapor pressure).

• Causes: High $NPSH$ required ($$\displaystyle NPSH_r $$) > $NPSH$ available ($$\displaystyle NPSH_a $$).

$$NPSH_a = \frac{P_{suction}}{\rho g} + \frac{V^2}{2g} - \frac{P_v}{\rho g}$$

• Effects: Pitting, noise, vibration, performance drop.

• Cavitation Parameter ($\sigma$): $$\displaystyle \sigma = \frac{NPSH_a}{H} $$.

• Prevention: Ensure $$\displaystyle NPSH_a > NPSH_r + \text{margin} $$; use strong materials; optimize inlet design.

IV. CENTRIFUGAL & AXIAL FLOW MACHINES

A. Centrifugal Pumps

• Main Parts: Impeller (open/semi-open/closed), Casing (volute/annular), Suction & delivery pipes, stuffing box.

• Velocity Triangles (Inlet & Outlet):

  • Inlet (Eye): $$\displaystyle V_{f1} = \frac{Q}{\pi D_1 b_1} $$, $$\displaystyle V_{w1} \approx 0 $$ (ideal radial flow).

  • Outlet: $$\displaystyle V_{w2} = U_2 - V_{f2} \cot \beta_2 $$.

• Theoretical Head (Euler's Eq.):

$$H_{th} = \frac{U_2 V_{w2}}{g} = \frac{U_2^2}{g} \left(1 - \frac{V_{f2}}{U_2} \cot \beta_2\right)$$

• Manometric Head ($$\displaystyle H_m $$): $$\displaystyle H_m = H_{th} \cdot \eta_h - \text{losses} $$.

$$H_m = \frac{P_2 - P_1}{\rho g} + \frac{V_2^2 - V_1^2}{2g}$$

• Performance Curves: $H$ vs $Q$ (drooping), $\eta$ vs $Q$ (peak at BEP), $P$ vs $Q$ (increasing).

• Cavitation in Pumps: Same as turbines. NPSH Required ($$\displaystyle NPSH_r $$) is provided by manufacturer; must have $$\displaystyle NPSH_a > NPSH_r $$.

B. Positive Displacement Pumps

• Principle: Traps fixed volume and forces it into discharge.

• Types: Reciprocating (piston/plunger), Rotary (gear, vane, lobe).

• Characteristics: Constant discharge (independent of pressure), high pressure, pulsating flow, positive suction head.

C. Centrifugal Compressors

• Vector Diagram: Similar to pump but with diffuser (vaned/vaneless) to convert velocity head to pressure.

• Work Done: $$\displaystyle \dot{W} = \dot{m} U_2 V_{w2} $$ (with $$\displaystyle V_{w2} < U_2 $$ due to backward-curved blades).

• Pressure Rise: $$\displaystyle \frac{P_2}{P_1} = \left[1 + \frac{\eta_p (\gamma-1)}{2\gamma} M^2 \left(\frac{U_2}{C_s}\right)^2\right]^{\frac{\gamma}{\gamma-1}} $$ where $M$ = Mach number.

• Limits:

  • Choking: $$\displaystyle V_1 = \text{speed of sound} $$ at inlet (sonic condition). Max mass flow.

  • Surging: Flow reversal at low flow, high pressure. Stable operating range is between choke and surge.

D. Axial Flow Compressors

• Construction: Alternating stator (fixed) and rotor (moving) blades.

• Velocity Diagram: $$\displaystyle V_{w1} \approx V_{w2} $$ for high efficiency; $U$ constant.

• Degree of Reaction (R): $$\displaystyle R = \frac{\Delta h_{rotor}}{\Delta h_{stage}} $$. For axial, $R \approx 0.5$ common.

• Characteristics: High efficiency, large flow rate, low pressure rise per stage (~1.2:1), many stages (10-20).

• Surging vs. Choking:

  | Surging | Choking | | :--- | :--- | | Low flow, high pressure | High flow, low pressure | | Flow reversal, vibration | Mass flow max, $$\displaystyle M=1 $$ at inlet | | System instability | Compressor limit | | Avoid by staying right of surge line | Avoid by staying left of choke line |

E. Comparative Analysis

Centrifugal Pump vs. Reciprocating Pump:

• Centrifugal: Continuous flow, lower pressure, simpler, less maintenance, no pulsation.

• Reciprocating: Pulsating flow, very high pressure, positive displacement, good for metering.

V. POWER TRANSMISSION & FLUID COUPLING DEVICES

A. Fluid Coupling

• Construction: Impeller (pump) on driving shaft, Runner (turbine) on driven shaft, casing filled with oil.

• Working: Oil transmits torque by momentum change. No mechanical contact.

• Slip (s): $$\displaystyle s = \frac{N_1 - N_2}{N_1} $$ (speed difference).

• Torque Transmission: $T \propto s(1-s)$. Max torque at $$\displaystyle s=0.5 $$.

• Efficiency: $$\displaystyle \eta = \frac{N_2}{N_1} = 1 - s $$.

• Characteristics: No-load speed equal to driving speed; torque capacity limited; smooth engagement.

B. Torque Converter

• Construction: Adds Stator (fixed blades) between impeller and runner.

• Working Principle: Stator redirects oil flow from runner against impeller direction, multiplying torque.

• Performance Parameters:

  • Torque Ratio (T.R.): $$\displaystyle \frac{T_{out}}{T_{in}} > 1 $$ (at low speed ratio).

  • Speed Ratio (S.R.): $$\displaystyle \frac{N_{out}}{N_{in}} $$.

  • Efficiency: $$\displaystyle \eta = \text{S.R.} \times \text{T.R.} $$.

• Characteristics: High torque multiplication at start (e.g., 2:1), efficiency peaks at ~0.85 at high S.R.

• Applications: Automotive automatic transmissions, industrial drives (crushers, conveyors).

C. Applications of Power Transmitting Devices

• Overload Protection: Slip prevents damage.

• Vibration Damping: Isolates torsional vibrations.

• Smooth Starting: Gradual torque build-up.

• Flexibility in Alignment: Compensates for misalignment.

• Industrial Use: Conveyors, mills, pumps, fans, compressors, crushers.

VI. AUXILIARY DEVICES & SPECIAL COMPONENTS

A. Centrifugal Blowers (Fans)

• Types:

  • Radial Blade: Simple, robust, high pressure, low efficiency.

  • Forward-Curved: High flow, low pressure, unstable (stall prone).

  • Backward-Curved: High efficiency, stable, self-limiting power.

• Performance: Similar to pumps but for gases. System curve (pressure vs flow) determines operating point.

B. Hydraulic Intensifier

• Working Principle: Uses large-area, low-pressure piston to drive small-area, high-pressure piston.

• Construction: Two cylinders (large & small) with connecting ram.

• Application: Generates high pressure (for hydraulic presses, test rigs) from existing low-pressure hydraulic system.

C. Draft Tube (Recap from III.C)

• Function: Kinetic energy recovery, allows low-pressure outlet.

• Types: Conical (simple), Elbow (practical), Moody (spreading, efficient for high specific speed).

🔥 HIGH-YIELD EXAM FORMULAS & CONCEPTS

1. Euler's Equation: $$\displaystyle \dot{W} = \dot{m}(U_2 V_{w2} - U_1 V_{w1}) $$

2. Optimal Blade Speed Ratio (Impulse): $$\displaystyle \rho_{opt} = \frac{U}{V_1} = \frac{\cos \alpha_1}{2} $$

3. Max Blade Efficiency (Impulse, Sym): $$\displaystyle \eta_{b,max} = \cos^2 \alpha_1 $$

4. Degree of Reaction (Axial): $$\displaystyle R = \frac{1}{2} + \frac{V_{w2} - V_{w1}}{2U} $$

5. Specific Speed (Turbine): $$\displaystyle N_s = \frac{N\sqrt{P}}{H^{5/4}} $$

6. Pelton Power: $$\displaystyle P = \rho Q V_j U (1 + \cos \phi) $$

7. Pump Theoretical Head: $$\displaystyle H_{th} = \frac{U_2 V_{w2}}{g} $$

8. Affinity Laws: $Q \propto N$, $$\displaystyle H \propto N^2 $$, $$\displaystyle P \propto N^3 $$

9. Fluid Coupling Slip: $$\displaystyle \eta = 1 - s $$

10. Cavitation Criterion: $$\displaystyle NPSH_a > NPSH_r $$

[!CAUTION] Common Pitfalls:

• Confusing whirl velocity ($$\displaystyle V_w $$) with flow velocity ($$\displaystyle V_f $$).

• Using wrong sign convention for $$\displaystyle V_{w1} $$ and $$\displaystyle V_{w2} $$ in Euler's equation.

• Forgetting that reaction means pressure drop in rotor.

• Mixing up NPSH available (system) vs. NPSH required (pump characteristic).

• Not distinguishing between surging (system instability) and choking (component flow limit).