Turbomachinery (ME-603 (A)) - Unit 2 Short Notes

UNIT 2: TURBOMACHINERY - EXAM-FOCUSED SHORT NOTES

1.0 FUNDAMENTAL PRINCIPLES & CLASSIFICATION

1.1 Definition & Classification

• Turbomachine: A rotating machine that transfers energy between a fluid and a rotor via dynamic action.

• Classification by Energy Transfer:

  • Turbines: Extract energy from fluid (H → P & ω).

  • Compressors/Pumps: Add energy to fluid (P & ω → H).

• Classification by Flow Path:

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

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

  • Mixed Flow: Combination (e.g., Francis turbine).

1.2 Thermodynamic Analysis (SFEE & Efficiency)

• Steady Flow Energy Equation (SFEE): For a turbomachine, neglecting heat & potential energy changes:

$$ \dot{W}_{\text{shaft}} = \dot{m} \left( h_1 + \frac{V_1^2}{2} - h_2 - \frac{V_2^2}{2} \right) $$

• Isentropic Efficiency:

  • Turbine: $$\displaystyle \eta_{\text{turbine}} = \frac{h_1 - h_2}{h_1 - h_{2s}} = \frac{\text{Actual Work}}{\text{Isentropic Work}} $$

  • Compressor/Pump: $$\displaystyle \eta_{\text{comp}} = \frac{h_{2s} - h_1}{h_2 - h_1} = \frac{\text{Isentropic Work}}{\text{Actual Work}} $$

1.3 Moment of Momentum Theorem & Euler's Equation (VHF)

• Theorem: The torque exerted on a fluid by a rotor equals the rate of change of angular momentum of the fluid.

• Euler's Turbine Equation (Derivation from 1st Principles):

$$ \dot{W} = \dot{m} (U_2 V_{w2} - U_1 V_{w1}) $$

Where:

*   $U$ = Blade speed (tangential)

*   $$\displaystyle V_w $$ = Velocity of whirl (tangential component of absolute velocity)

• Significance: The $$\displaystyle UV_w $$ term represents the work transfer per unit mass. Positive work (turbine) when $$\displaystyle U_2V_{w2} > U_1V_{w1} $$.

[!TIP] Exam Alert: Euler's equation is universal for all turbomachines (turbines, compressors, pumps). Always identify $U$ and $$\displaystyle V_w $$ at inlet (1) and outlet (2) from velocity triangles.

1.4 Velocity Diagram Construction & Analysis (VHF)

• General Construction:

  1. Draw absolute velocity $V$ (magnitude & angle α from tangential direction).

  2. Draw blade speed $U$ (perpendicular to axis, tangential direction).

  3. Relative velocity $$\displaystyle V_r = V - U $$ (vector subtraction). Blade angle β is angle of $$\displaystyle V_r $$ with tangential direction.

• Key Relationships:

  • $$\displaystyle V_w = V \cos \alpha $$

  • $$\displaystyle V_f = V \sin \alpha $$ (Velocity of flow, axial/radial component)

  • $$\displaystyle V_r^2 = (V_w - U)^2 + V_f^2 $$

• Impulse Turbine (Symmetrical Blades): $$\displaystyle |\beta_1| = |\beta_2| $$, $$\displaystyle V_{r1} = V_{r2} $$ (if no friction).

• Reaction Turbine (Parsons, R=0.5): $$\displaystyle \alpha_1 = \beta_2 $$, $$\displaystyle \beta_1 = \alpha_2 $$. $$\displaystyle V_{r1} = V_{r2} $$ for shockless entry.

• Calculation Sequence: Given $$\displaystyle D, N, \dot{m}, \alpha_1, C_1 $$ etc. → Find $$\displaystyle U = \pi D N / 60 $$ → Draw inlet triangle → Apply $$\displaystyle V_{r1} $$ condition → Draw outlet triangle → Compute $$\displaystyle V_{w2} $$ → Use Euler's equation.

2.0 STEAM TURBINES - THEORY & ANALYSIS (DOMINANT TOPIC)

2.1 Impulse Turbines

• Principle: All pressure drop occurs in nozzle. Blades operate at constant pressure. Work done by change in kinetic energy of steam.

• Velocity Diagram (Single-Stage):

  • Nozzle gives high $$\displaystyle C_1 $$ at angle $$\displaystyle \alpha_1 $$.

  • Moving blades change direction of $$\displaystyle V_1 $$ to $$\displaystyle V_2 $$.

  • For symmetrical blades ($$\displaystyle \beta_1 = -\beta_2 $$), $$\displaystyle V_{r2} = V_{r1} $$ (no friction).

  • $$\displaystyle C_2 $$ is large → Leaving loss.

  • DiagramSEARCH: impulse turbine velocity diagram symmetrical blades

• Blade Efficiency ($$\displaystyle \eta_b $$): Ratio of work done on blades to kinetic energy supplied.

$$ \eta_b = \frac{2U(V_{w1} + V_{w2})}{C_1^2} \quad \text{(for frictionless, symmetrical blades)} $$

• Stage Efficiency ($$\displaystyle \eta_s $$): $$\displaystyle \eta_s = \eta_b \times \eta_{\text{nozzle}} $$

• Condition for Maximum Blade Efficiency: $$\displaystyle U = \frac{C_1 \cos \alpha_1}{2} $$ (for frictionless, symmetrical blades). This gives $$\displaystyle \alpha_1 = \beta_2 $$.

• Effect of Blade Friction: Introduced via velocity coefficient $$\displaystyle K = V_{r2}/V_{r1} < 1 $$. Reduces $$\displaystyle V_{w2} $$, increases $$\displaystyle V_{r2} $$, lowers efficiency.

• Losses:

  • Nozzle friction loss: Reduces $$\displaystyle C_1 $$.

  • Blade friction loss: $$\displaystyle V_{r2} < V_{r1} $$.

  • Leaving loss: $$\displaystyle \frac{V_2^2}{2} $$ wasted.

  • Residual velocity loss: Difference between $$\displaystyle \frac{V_2^2}{2} $$ and $$\displaystyle \frac{V_3^2}{2} $$ in next stage (if any).

2.2 Reaction Turbines

• Principle: Pressure drop occurs both in nozzles (fixed blades) and moving blades. Blade rows act as both nozzle and rotor.

• Degree of Reaction (R) (VHF):

$$ R = \frac{\text{Static enthalpy drop in rotor}}{\text{Total static enthalpy drop in stage}} = \frac{\Delta h_{\text{rotor}}}{\Delta h_{\text{stage}}} $$

*   **For Parsons (50% Reaction) Turbine:** $$\displaystyle R = 0.5 $$. Key property: $$\displaystyle \alpha_1 = \beta_2 $$ and $$\displaystyle \beta_1 = \alpha_2 $$. Velocity triangles are symmetrical.

*   
DiagramSEARCH: parsons reaction turbine velocity diagram

• Blade Angle for Shockless Entry: $$\displaystyle \alpha_1 = \beta_1 $$ (steam enters moving blade without shock relative to blade).

• Comparison with Impulse:

  • Reaction: Lower blade speed for same work → more stages needed for high pressure ratio.

  • Impulse: High blade speed, large leaving loss, suitable for high head.

  • Reaction: Better part-load efficiency, smoother operation, used for large power outputs.

2.3 Compounding of Steam Turbines (HF)

• Purpose: Reduce blade speed for high-pressure ratio turbines (avoid excessive centrifugal stress).

• Velocity Compounding (Curtis):

  • Principle: One nozzle, multiple moving blade rows separated by fixed blades (guide vanes). Velocity is compounded (reduced stepwise).

  • Diagram: Multiple velocity triangles in series. $$\displaystyle C_1 $$ high, after each moving blade row $$\displaystyle V_r $$ reduced by friction, $C$ reduced.

  • Advantages: Fewer stages, shorter shaft, less leakage.

  • Disadvantages: Higher leaving loss, lower efficiency than pressure compounding.

  • Application: High-pressure, low-flow rate turbines.

• Pressure Compounding (Rateau):

  • Principle: Multiple nozzle rows and moving blade rows in series. Pressure drop is divided (compounded) across nozzle rows.

  • Diagram: Similar to impulse turbine repeated. Each stage is essentially an impulse stage.

  • Advantages: Higher efficiency (lower leaving loss per stage).

  • Disadvantages: Longer shaft, more components, more leakage.

  • Application: Large steam turbines.

• Comparison:

  | Feature | Velocity Compounding (Curtis) | Pressure Compounding (Rateau) | | :--- | :--- | :--- | | Energy Drop | Mostly in one nozzle | Divided among many nozzles | | Blade Speed | High | Moderate | | Leaving Loss | High (large $$\displaystyle C_2 $$ from last stage) | Low (each stage has moderate $$\displaystyle C_2 $$) | | Efficiency | Lower | Higher | | Size/Length | Shorter | Longer |

2.4 Multi-Stage Turbines & Performance

• Reheat Factor (RF) (HF):

  • Definition: $$\displaystyle \text{RF} = \frac{\text{Isentropic enthalpy drop for whole turbine}}{\text{Sum of isentropic enthalpy drops of individual stages}} $$

  • Physical Meaning: > 1.0 due to reheating effect in later stages (steam gets hotter as pressure drops, increasing average specific heat). This allows more work extraction in later stages.

  • Relation: $$\displaystyle \eta_{\text{overall}} = \eta_{\text{stage}} \times \text{RF} $$

• Stage Efficiency vs. Overall Efficiency: Stage efficiency is for one stage. Overall efficiency accounts for reheat factor and compounding losses.

• Governing (Brief): Speed control by regulating steam flow. Methods: Throttle governing (nozzle control), Nozzle governing (grouped nozzles), Bypass governing.

2.5 Losses in Steam Turbines (HF)

• Nozzle Losses: Friction, shock (off-design), incomplete expansion (discharge coefficient).

• Blade Passage Losses: Surface friction, secondary flows (corner vortices), tip leakage flow.

• Disc Friction & Windage: Friction between rotating disc and steam, air friction (for high-speed rotors in vacuum).

• Leaving Losses: Kinetic energy of steam leaving last stage unused ($$\displaystyle \frac{C_2^2}{2} $$). Minimized by using diffuser or last stage blades with large $$\displaystyle V_w $$ reduction.

• Clearance Losses: Steam leakage over blade tips (rotor-stator clearance) and gland seals.

3.0 HYDRAULIC TURBINES (HF)

3.1 Classification

• Impulse (High Head, Low Flow): Pelton Wheel. Pressure drop in nozzle only.

• Reaction (Medium/Low Head, High Flow): Francis (Medium), Kaplan (Low). Pressure drop in both stator and rotor.

3.2 Pelton Wheel (Impulse)

• Main Parts: Nozzle with spear valve, Runner with double-cup buckets, casing, governing mechanism.

• Velocity Diagram for Bucket:

  • Jet velocity: $$\displaystyle C_j = \sqrt{2gH} $$ (theoretical), actual $$\displaystyle C_1 = C_j \times C_v $$ ($$\displaystyle C_v $$ = velocity coeff.).

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

  • Bucket deflects jet by ~165°–180° (φ = 15°–20° tip angle).

  • Relative velocity $$\displaystyle V_r $$ reverses direction. $$\displaystyle V_{w} = C_1(1 + K\cos\phi) $$.

  • DiagramSEARCH: pelton wheel bucket velocity diagram

• Power & Efficiency:

$$ P = \rho Q C_1 u (1 + K \cos \phi) \quad \text{(Diagram Power)} $$

$$ \eta_h = \frac{2u(1 + K\cos\phi)C_1}{C_1^2} = \frac{2u(1 + K\cos\phi)}{C_1} \quad \text{(Hydraulic Efficiency)} $$

• Design Parameters:

  • Number of Buckets: $Z \approx 15 + 0.5\sqrt{D}$ (to avoid jet impinging on adjacent bucket).

  • Jet Ratio: $$\displaystyle m = D/d_j $$ (runner dia. / jet dia.) ≈ 10–12.

  • Runner Diameter: $$\displaystyle D = \frac{60u}{\pi N} $$.

3.3 Reaction Turbines (Francis & Kaplan)

• Principle: Water fills runner passages. Pressure drop occurs in spiral casing → stay vanes → guide vanes → runner blades → draft tube.

• Draft Tube (VHF):

  • Function:

    1. Converts kinetic energy at runner exit to pressure energy (recovers head).

    2. Creates negative pressure at runner exit → allows setting runner above tailrace.

  • Types:

    • Conical: Simple, limited height.

    • Elbow: For limited space, 90° turn.

    • Moody (Spreading): Diverging tube, best recovery, used for high specific speed.

    • Simple: Straight divergent tube.

  • Recovery Head: $$\displaystyle H_d = \frac{(V_2^2 - V_3^2)}{2g} + (p_2 - p_3)/(\rho g) $$ (Bernoulli from 2 to 3).

  • DiagramSEARCH: draft tube types conical elbow moody

• Manometric Head vs. Total Head:

  • Total Head (H): Gross head available at turbine.

  • Manometric Head (H_m): Head actually converted to mechanical power. $$\displaystyle H_m = H - h_f - \text{losses} $$. Power $$\displaystyle P = \rho g Q H_m \eta_o $$.

3.4 Performance Efficiencies (HF)

• Hydraulic Efficiency ($$\displaystyle \eta_h $$): $$\displaystyle \frac{\text{Power developed on runner shaft}}{\rho g Q H} $$ = Accounts for hydraulic losses (friction, leakage, vortex).

• Mechanical Efficiency ($$\displaystyle \eta_m $$): $$\displaystyle \frac{\text{Shaft Power}}{\text{Power developed on runner shaft}} $$ = Accounts for bearing friction, windage.

• Volumetric Efficiency ($$\displaystyle \eta_v $$): $$\displaystyle \frac{\text{Water used for power}}{\text{Water supplied}} $$ = Accounts for leakage.

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

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

• Definition: Speed at which a geometrically similar turbine would develop 1 kW under 1 m head.

• Expression (Dimensional):

$$ N_s = \frac{N \sqrt{P}}{H^{5/4}} \quad \text{(Units: rpm, kW, m)} $$

• Physical Significance: Indicates shape and type of turbine.

  • Low $$\displaystyle N_s $$ (20–100): Pelton (Impulse, high head).

  • Medium $$\displaystyle N_s $$ (100–400): Francis (Reaction, medium head).

  • High $$\displaystyle N_s $$ (400–1000+): Kaplan (Reaction, low head, high flow).

• Use: Selection of turbine type for given $(H, P, N)$.

4.0 CENTRIFUGAL PUMPS & POSITIVE DISPLACEMENT PUMPS

4.1 Centrifugal Pump - Main Parts

• Impeller: Rotating disc with vanes (Open, Semi-open, Closed).

• Casing: Volute (spiral, converts kinetic to pressure) or Diffuser (vanes, more efficient).

• Suction & Delivery Pipes with foot valve (suction) and check valve (delivery).

4.2 Velocity Triangle & Work Done

• Inlet (Radial Flow): $$\displaystyle \alpha_1 = 90^\circ $$, $$\displaystyle V_{w1} = 0 $$. $$\displaystyle V_{f1} = V_1 $$.

• Outlet: $$\displaystyle \beta_2 $$ depends on blade shape.

  • Backward Curved ($$\displaystyle \beta_2 < 90^\circ $$): High head, high efficiency, self-stable.

  • Radial ($$\displaystyle \beta_2 = 90^\circ $$): Medium.

  • Forward Curved ($$\displaystyle \beta_2 > 90^\circ $$): High flow, low head, unstable.

• Work Done per kg (Euler's Eq.):

$$ gH = u_2 V_{w2} - u_1 V_{w1} \approx u_2 V_{w2} \quad (\text{since } V_{w1} \approx 0) $$

$$ V_{w2} = u_2 - V_{r2} \cos \beta_2 \quad \text{(for backward/radial)} $$

• Effect of Outlet Angle: Larger $$\displaystyle \beta_2 $$ (forward) reduces $$\displaystyle V_{w2} $$ for given $$\displaystyle u_2 $$, hence lower head.

4.3 Performance Characteristics & Cavitation (VHF)

• Manometric Head (H): $$\displaystyle H = \frac{p_d - p_s}{\rho g} + \frac{V_d^2 - V_s^2}{2g} + (z_d - z_s) $$.

• Net Positive Suction Head (NPSH):

  • NPSH Available (NPSHa): $$\displaystyle \text{NPSHa} = \frac{p_s}{\rho g} + \frac{V_s^2}{2g} - \frac{p_v}{\rho g} $$ (absolute).

  • NPSH Required (NPSHr): Minimum NPSHa needed to avoid cavitation (provided by manufacturer).

  • Cavitation Criterion: NPSHa > NPSHr.

• Cavitation (VHF):

  • Cause: Local pressure $$\displaystyle p < p_v $$ (vapor pressure) → vapor bubbles form & collapse.

  • Effects: Noise, vibration, material damage (pitting), efficiency drop, flow interruption.

  • Prevention: Ensure NPSHa > NPSHr, lower suction lift, use cavitation-resistant materials, larger suction pipe.

• Characteristic Curves:

  • H-Q: Head decreases with flow.

  • η-Q: Efficiency peaks at Best Efficiency Point (BEP).

  • P-Q: Power increases with flow.

  • DiagramSEARCH: centrifugal pump characteristic curves H-Q eta-Q

4.4 Positive Displacement Pumps

• Principle: Traps fixed volume & forces it into discharge pipe (positive displacement).

• Types:

  • Reciprocating: Piston, Plunger, Diaphragm. High pressure, pulsating flow, needs priming.

  • Rotary: Gear, Vane, Lobe. Continuous flow, medium pressure.

• Comparison with Centrifugal Pumps:

  | Feature | Centrifugal Pump | Positive Displacement Pump | | :--- | :--- | :--- | | Flow Rate | Variable with head | Nearly constant | | Pressure | Moderate | Very High | | Priming | Self-priming (if flooded) | Needs priming (except some rotary) | | Efficiency | High at BEP | High over wide range | | Pulsation | Smooth | Pulsating (reciprocating) | | Applications | Water supply, circulation | Fuel injection, hydraulic presses |

5.0 AXIAL FLOW COMPRESSORS & CENTRIFUGAL COMPRESSORS

5.1 Axial Flow Compressor

• Construction: Alternating rotor blades (moving) and stator blades (fixed guide vanes). Annulus area may vary.

• Velocity Diagram for a Stage:

  • Inlet/outlet velocities relative to stage axis.

  • $$\displaystyle V_{w} $$ changes sign across rotor (work input).

  • $$\displaystyle V_f $$ roughly constant (mean line design).

  • Degree of Reaction (R): For axial compressor, $$\displaystyle R = \frac{\Delta h_{\text{rotor}}}{\Delta h_{\text{stage}}} $$. For symmetrical velocity diagram ($$\displaystyle \alpha_1 = \beta_2 $$, $$\displaystyle \beta_1 = \alpha_2 $$), $$\displaystyle R = 0.5 $$.

  • DiagramSEARCH: axial flow compressor velocity diagram stage

• Work Done Factor & Stage Efficiency: Accounts for losses (blade friction, secondary flows). $$\displaystyle \eta_{\text{stage}} < \eta_{\text{polytropic}} $$.

• Surging and Choking (VHF):

  • Surging: System instability at low flow/high pressure. Compressor cannot overcome downstream backpressure → flow reversal, vibration, possible damage. Operating limit on left side of characteristic curve.

  • Choking: Flow limit at high flow. Mach number reaches 1 at narrowest passage (throat) → mass flow cannot increase further. Operating limit on right side of characteristic curve.

  • Stable Operating Range: Between surge line and choke line.

  • DiagramSEARCH: compressor map surge choke lines

5.2 Centrifugal Compressor

• Construction: Inlet (axial), impeller (radial/backward curved), diffuser (vaneless or vaned), volute.

• Vector Diagram:

  • Inlet: $$\displaystyle V_1 $$ axial, $$\displaystyle U_1 $$ small.

  • Outlet (impeller): $$\displaystyle V_2 $$ at angle $$\displaystyle \alpha_2 $$ (often radial for backward curved, $$\displaystyle \alpha_2 \approx 90^\circ $$). $$\displaystyle U_2 $$ large.

  • Diffuser: Converts $$\displaystyle V_2 $$ to pressure, $$\displaystyle V_3 $$ small.

  • DiagramSEARCH: centrifugal compressor vector diagram

• Pressure Rise Mechanism: Kinetic energy imparted by impeller ($$\displaystyle u_2^2/2 $$) + pressure recovery in diffuser.

• Slip Factor (σ): $$\displaystyle \sigma = \frac{V_{w2,\text{actual}}}{V_{w2,\text{ideal}}} < 1 $$. Accounts for non-ideal flow turning due to boundary layer & finite blades. Reduces actual $$\displaystyle V_{w2} $$ and pressure ratio.

5.3 Comparison: Axial vs. Centrifugal Compressors (HF)

5.4 Polytropic Efficiency (HF)

• Definition: Efficiency of an infinitesimal stage in a multistage machine. Assumes reversible process with small pressure rise.

$$ \eta_p = \frac{d(\text{Isentropic work})}{d(\text{Actual work})} $$

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

$$ \eta_s = \frac{\text{Total Isentropic Work}}{\text{Total Actual Work}} = \frac{\int dW_s}{\int dW_a} = \frac{\int \eta_p dW_a}{\int dW_a} \approx \eta_p \quad \text{(if } \eta_p \text{ constant)} $$

Actually, $$\displaystyle \eta_s < \eta_p $$ for multistage compression with intercooling? **Correction:** For same overall pressure ratio, multistage with intercooling has higher $$\displaystyle \eta_s $$. Polytropic efficiency is **constant** for a given machine design, while isentropic efficiency varies with pressure ratio.

• Significance: Better for performance comparison of compressors of different pressure ratios and staging.

6.0 CAVITATION, NPSH & PERFORMANCE PARAMETERS

• Cavitation Mechanism: Local static pressure $$\displaystyle p < p_v $$ → vapor bubbles form → carried to high-pressure region → collapse violently.

• Effects: Pitting, noise, vibration, efficiency drop, flow reduction, material damage.

• NPSH (VHF):

  • NPSHa (Available): $$\displaystyle \frac{p_{\text{suction, abs}}}{\rho g} + \frac{V_s^2}{2g} - \frac{p_v}{\rho g} $$

  • NPSHr (Required): Minimum NPSHa to avoid cavitation (from pump testing).

  • Criterion: NPSHa ≥ NPSHr + 0.5–1.0 m (safety margin).

7.0 POWER TRANSMISSION & FLUID POWER DEVICES (HF)

7.1 Fluid Coupling

• Construction: Impeller (driving, mounted on engine), Runner (driven, mounted on load), casing filled with oil.

• Working: Oil momentum transfer. No mechanical contact.

• Torque Transmission: $$\displaystyle T = \frac{\dot{m} u_2 (1 + K)}{g} $$ (simplified). $K$ depends on blade shape.

• Slip (s): $$\displaystyle s = \frac{\omega_1 - \omega_2}{\omega_1} $$. Efficiency $$\displaystyle \eta = 1 - s $$.

• Characteristics: Smooth start, overload protection (slip limits torque), no shock loading.

• Applications: Conveyors, crushers, machine tools.

7.2 Torque Converter

• Construction: Pump (driving), Turbine (driven), Stator (reaction member, fixed to casing via one-way clutch).

• Working Principle: Stator redirects oil flow from turbine to pump, multiplying torque at low speed ratio.

• Torque Multiplication: $$\displaystyle T_{\text{turbine}} = T_{\text{pump}} \times \text{Multiplication Factor} $$. Max at stall ($$\displaystyle \omega_2=0 $$).

• Efficiency: Low at high multiplication (low speed ratio), increases to max at speed ratio ~0.6–0.7, then decreases.

• Applications: Automotive automatic transmissions, industrial drives.

7.3 Hydraulic Intensifier

• Construction: Large diameter ram (low pressure side), small diameter plunger (high pressure side).

• Working (Pascal's Law): $$\displaystyle \frac{F_1}{A_1} = \frac{F_2}{A_2} $$. Small force $$\displaystyle F_1 $$ on large area $$\displaystyle A_1 $$ generates large force $$\displaystyle F_2 $$ on small area $$\displaystyle A_2 $$.

• Pressure Intensification: $$\displaystyle P_2 = P_1 \times (A_1/A_2) $$.

• Applications: Generating high pressure from low-pressure source (e.g., hydraulic presses, test rigs).

8.0 DIMENSIONAL ANALYSIS & SIMILITUDE (HF)

• Buckingham Pi Theorem: For $n$ variables with $r$ fundamental dimensions, we get $(n-r)$ independent dimensionless Pi groups.

• Application to Turbomachinery:

  • Pump/Fan Efficiency: $$\displaystyle \eta = f(\rho, \mu, \omega, D, Q) $$

  • Pi Groups: Reynolds number $$\displaystyle Re = \frac{\rho \omega D^2}{\mu} $$, Flow Coefficient $$\displaystyle \phi = \frac{Q}{\omega D^3} $$, Head Coefficient $$\displaystyle \psi = \frac{gH}{\omega^2 D^2} $$.

  • Result: $$\displaystyle \eta = f(Re, \phi, \psi) $$.

• Similarity Conditions:

  • Geometric: Same shape, $$\displaystyle D_1/D_2 = \text{const} $$.

  • Kinematic: Same $\phi$, $\psi$ → same velocity triangles.

  • Dynamic: Same $Re$, $$\displaystyle C_f $$, $$\displaystyle C_D $$ → same force ratios.

• Key Dimensionless Parameters:

  • Flow Coefficient: $$\displaystyle \phi = \frac{Q}{N D^3} $$

  • Head Coefficient: $$\displaystyle \psi = \frac{gH}{N^2 D^2} $$

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

  • Note: For pumps, $$\displaystyle \psi \propto \phi^2 $$ (affinity laws).

9.0 SPECIAL TOPICS & APPLICATIONS (HF - SHORT NOTES)

9.1 Centrifugal Blower/Fan

• Similar to centrifugal pump but handles gas (air).

• Lower pressure rise (typically < 0.1 bar), higher flow rates.

• Types: Forward-curved (high flow, low pressure, unstable), backward-curved (stable, efficient), radial.

• Applications: HVAC, combustion air supply, dust collection.

9.2 Surging & Efficiency of Axial Compressor

• Surging: See 5.1. Flow reversal, vibration, possible damage. Avoid by operating away from surge line.

• Efficiency: Stage efficiency $$\displaystyle \eta_s = \frac{\text{Isentropic work per stage}}{\text{Actual work per stage}} $$. Polytropic efficiency $$\displaystyle \eta_p $$ is more fundamental for multistage.

9.3 Applications of Power Transmitting Devices

• Fluid Coupling: Smooth start, overload protection in conveyors, mills, crushers.

• Torque Converter: Automotive transmissions, marine propulsion, industrial drives needing variable torque.

• Hydraulic Clutch: Engage/disengage power transmission (e.g., in machine tools).

• Hydraulic Intensifier: Generate high pressure from low-pressure source for presses, jacks.

9.4 Draft Tube Types and Applications (HF)

• Conical: Simple, limited height recovery. Used for small turbines.

• Elbow: 90° turn, saves space. Used where tailrace is at side.

• Moody (Spreading): Diverging tube, best pressure recovery. Used for high specific speed turbines (Kaplan).

• Function: All types recover kinetic energy and create negative pressure at runner exit.

10.0 DESIGN & CALCULATION PROBLEMS (SYNTHESIS)

General Approach for Velocity Diagram Problems:

1. Calculate Blade Speed: $$\displaystyle U = \frac{\pi D N}{60} $$.

2. Draw Inlet Triangle using given $$\displaystyle V_1 $$ (or $$\displaystyle C_1 $$), $$\displaystyle \alpha_1 $$, and $$\displaystyle U_1 \approx 0 $$ (pump) or known $U$.

3. Apply Conditions:

   • Impulse (symm., frictionless): $$\displaystyle V_{r1} = V_{r2} $$, $$\displaystyle \beta_1 = -\beta_2 $$.

   • Impulse (friction): $$\displaystyle V_{r2} = K V_{r1} $$.

   • Parsons (R=0.5): $$\displaystyle \alpha_1 = \beta_2 $$, $$\displaystyle \beta_1 = \alpha_2 $$.

   • Shockless entry: $$\displaystyle \alpha_1 = \beta_1 $$ (relative velocity aligned with blade).

4. Draw Outlet Triangle satisfying condition.

5. Compute:

   • $$\displaystyle V_{w2} $$ from triangle.

   • Tangential Force: $$\displaystyle F_t = \dot{m} (V_{w1} - V_{w2}) $$ (sign convention: turbine: $$\displaystyle F_t = \dot{m}(V_{w1} - V_{w2}) $$ if $$\displaystyle V_{w1} > V_{w2} $$).

   • Diagram Power: $$\displaystyle P = F_t \times U = \dot{m} U (V_{w1} - V_{w2}) $$.

   • Axial Thrust: $$\displaystyle F_a = \dot{m} (V_{f1} \pm V_{f2}) $$ (sign depends on direction).

   • Blade Efficiency: $$\displaystyle \eta_b = \frac{2U(V_{w1} - V_{w2})}{V_1^2} $$ (impulse, frictionless).

6. Always sketch velocity triangles.

10.1 Parsons Reaction Turbine Stage (VHF)

• Given: $$\displaystyle D, N, C_2 $$ (or $$\displaystyle V_2 $$), $$\displaystyle \beta_2 $$, $\dot{m}$.

• Find: $$\displaystyle \alpha_1 $$, $$\displaystyle F_t $$, $P$, $$\displaystyle \eta_b $$.

• Steps:

  1. $$\displaystyle U = \pi D N / 60 $$.

  2. For Parsons: $$\displaystyle \alpha_1 = \beta_2 $$, $$\displaystyle \beta_1 = \alpha_2 $$. Also $$\displaystyle V_{r1} = V_{r2} $$ (shockless).

  3. At outlet: $$\displaystyle V_{w2} = V_2 \cos \beta_2 $$, $$\displaystyle V_{f2} = V_2 \sin \beta_2 $$.

  4. Since $$\displaystyle V_{f1} = V_{f2} $$ (for constant area), $$\displaystyle V_1 \sin \alpha_1 = V_2 \sin \beta_2 $$.

  5. $$\displaystyle V_{w1} = V_1 \cos \alpha_1 $$. Euler: $$\displaystyle P = \dot{m} U (V_{w1} - V_{w2}) $$.

  6. But $$\displaystyle V_{w1} $$ unknown. Use $$\displaystyle V_{r1} = V_{r2} $$:

     $$\displaystyle (V_{w1} - U)^2 + V_{f1}^2 = (U - V_{w2})^2 + V_{f2}^2 $$

     Substitute $$\displaystyle V_{f1}=V_{f2} $$ → Solve for $$\displaystyle V_{w1} $$.

  7. Then find $$\displaystyle V_1 = V_{f1}/\sin \alpha_1 $$, check consistency.

  8. $$\displaystyle \eta_b = \frac{2U(V_{w1} - V_{w2})}{V_1^2} $$.

10.2 Impulse Turbine Blade Optimization (HF)

• Given: $$\displaystyle C_1, \alpha_1, U $$, symmetrical blades, frictionless.

• Find: $$\displaystyle \beta_1, \beta_2 $$, $$\displaystyle F_t $$, $P$, $$\displaystyle \eta_b $$, $$\displaystyle F_a $$.

• Steps:

  1. $$\displaystyle V_{w1} = C_1 \cos \alpha_1 $$, $$\displaystyle V_{f1} = C_1 \sin \alpha_1 $$.

  2. For max efficiency: $$\displaystyle U = \frac{C_1 \cos \alpha_1}{2} $$ (given condition).

  3. Symmetrical & frictionless: $$\displaystyle V_{r1} = V_{r2} $$, $$\displaystyle \beta_1 = -\beta_2 $$.

  4. Inlet triangle: $$\displaystyle V_{r1}^2 = (V_{w1} - U)^2 + V_{f1}^2 $$.

  5. Outlet triangle: $$\displaystyle V_{w2} = U - V_{r1} \cos \beta_2 $$? Better: Since $$\displaystyle V_{r2}=V_{r1} $$ and $$\displaystyle \beta_2 = -\beta_1 $$, use geometry.

     Actually, from inlet: $$\displaystyle \tan \beta_1 = \frac{V_{f1}}{U - V_{w1}} $$. Compute $$\displaystyle \beta_1 $$.

     Then $$\displaystyle \beta_2 = -\beta_1 $$.

  6. $$\displaystyle V_{w2} = V_{r1} \cos \beta_2 + U $$? Careful: $$\displaystyle V_{w2} = U - V_{r2} \cos \beta_2 $$ (if $$\displaystyle \beta_2 $$ measured from opposite direction). Standard: $$\displaystyle V_{w2} = U - V_{r2} \cos \beta_2 $$ for $$\displaystyle \beta_2 $$ on other side. Since $$\displaystyle \beta_2 = -\beta_1 $$, $$\displaystyle \cos \beta_2 = \cos \beta_1 $$.

     So $$\displaystyle V_{w2} = U - V_{r1} \cos \beta_1 $$.

  7. $$\displaystyle F_t = \dot{m}(V_{w1} - V_{w2}) $$, $$\displaystyle P = F_t U $$.

  8. $$\displaystyle \eta_b = \frac{2U(V_{w1} - V_{w2})}{C_1^2} $$.

  9. $$\displaystyle F_a = \dot{m}(V_{f1} + V_{f2}) = 2 \dot{m} V_{f1} $$ (symm.).

10.3 Pelton Wheel Design (VHF)

• Given: $H, N, D, Q, \phi$ (clearance angle).

• Find: $$\displaystyle C_j $$, $$\displaystyle P_{\text{available}} $$, $$\displaystyle \eta_h $$, $$\displaystyle d_j $$, $D$ (if not given).

• Steps:

  1. Jet Velocity: $$\displaystyle C_j = \sqrt{2gH} $$ (theoretical). Actual $$\displaystyle C_1 = C_v C_j $$ ($$\displaystyle C_v \approx 0.98 $$).

  2. Power Available at Nozzle: $$\displaystyle P_{\text{avail}} = \rho Q C_j^2 / 2 $$ (or $\rho g Q H$).

  3. Bucket Speed: $$\displaystyle u = \frac{\pi D N}{60} $$. Given $$\displaystyle u = K \sqrt{2gH} $$, find $K$.

  4. Hydraulic Efficiency: $$\displaystyle \eta_h = \frac{2u(1 + \cos \phi)}{C_j} $$ (for ideal, frictionless).

  5. Jet Diameter: $$\displaystyle d_j = \sqrt{\frac{4Q}{\pi C_1}} $$.

  6. Runner Diameter: From $$\displaystyle u = \pi D N / 60 \rightarrow D = \frac{60u}{\pi N} $$.

  7. Number of Jets: Usually 1–4 depending on $Q$.

10.4 Hydraulic Intensifier

• Given: $$\displaystyle A_1, A_2, P_1, F_2 $$ (or vice versa).

• Find: Other force/pressure.

• Principle (Pascal): $$\displaystyle \frac{F_1}{A_1} = \frac{F_2}{A_2} = P $$ (intensified pressure).

• Formulas:

  • $$\displaystyle P_2 = P_1 \times (A_1/A_2) $$

  • $$\displaystyle F_2 = F_1 \times (A_2/A_1) $$

• Note: $$\displaystyle F_1 $$ is applied on large ram area $$\displaystyle A_1 $$, $$\displaystyle F_2 $$ is output on small plunger area $$\displaystyle A_2 $$. $$\displaystyle A_1 > A_2 $$, so $$\displaystyle P_2 > P_1 $$, $$\displaystyle F_2 < F_1 $$? Correction: For intensifier, small force on large area generates high pressure, which acts on small area to give moderate force? Actually typical: Low pressure $$\displaystyle P_1 $$ on large piston area $$\displaystyle A_1 $$ gives force $$\displaystyle F_1 = P_1 A_1 $$. This force acts on small plunger area $$\displaystyle A_2 $$ → $$\displaystyle P_2 = F_1/A_2 = P_1 A_1/A_2 $$. Then output force on load: $$\displaystyle F_{\text{load}} = P_2 A_{\text{load}} $$. So pressure is intensified, force on load depends on its area.

[!TIP] Final Exam Strategy:

1. For derivation questions (Euler's Eq, Degree of Reaction, Specific Speed): Write step-by-step from first principles (momentum, energy, similarity).

2. For numerical problems: Always draw the velocity/vector diagram first. Label all knowns ($$\displaystyle U, V, V_w, V_f, \alpha, \beta $$). Use geometry/trigonometry.

3. For definition questions: State clear definition, formula, physical significance, and typical values/applications.

4. For comparison questions: Use tabular form (as shown) for full marks.

5. For "Explain" questions: Start with basic principle, then construction, then working, then applications/advantages/disadvantages. Use diagrams where possible.

6. Remember key VHF topics: Euler's equation, Velocity diagrams (Parsons/Impulse), Degree of Reaction, Pelton calculations, Draft tube, Surging, Cavitation, Polytropic Efficiency, Fluid Coupling/Torque Converter.