Turbomachinery (ME-603 (A)) - Unit 4 Short Notes

UNIT 4: PUMPS, COMPRESSORS & POWER TRANSMISSION DEVICES

IV. PUMPS

Centrifugal Pumps

• Main Components:

  • Impeller: Rotating disc with curved vanes.

  • Casing: Volute (spiral) or diffuser type, converts kinetic energy to pressure.

  • Suction & Delivery Pipes: With non-return valves.

• Working Principle: Centrifugal force throws fluid outward from impeller eye, creating low pressure at suction. Fluid gains kinetic energy in impeller, which is converted to pressure energy in the casing.

• Velocity Triangles:

  • Inlet (Suction): Ideally radial flow ($$\displaystyle \alpha_1 = 90^\circ $$), $$\displaystyle u_1 = \frac{\pi D_1 N}{60} $$.

  • Outlet (Discharge: $$\displaystyle u_2 = \frac{\pi D_2 N}{60} $$, $$\displaystyle V_{r2} $$ makes angle $$\displaystyle \beta_2 $$ with tangent.

  [!TIP] Exam Focus: Often asked to draw velocity triangles for centrifugal pump. Key: $$\displaystyle V_w $$ (whirl velocity) at outlet is crucial for Euler's equation.

• Performance Characteristics:

  • Head-Flow Curve: Head decreases with increase in flow rate.

  • Efficiency-Flow Curve: Efficiency peaks at design flow rate (BEP - Best Efficiency Point).

  • Power-Flow Curve: Power increases with flow rate.

• Advantages over Reciprocating Pumps:

  • Smooth, continuous discharge (no pulsations).

  • Higher capacity for same size.

  • Lower maintenance, simpler construction.

  • Can handle dirty/suspended fluids.

  • Direct coupling to motor possible.

Positive Displacement Pumps

• Types & Principle:

  • Reciprocating: Piston/plunger moves in cylinder, creating suction/discharge via valves. Delivers pulsating flow at high pressure.

  • Rotary (Gear, Vane, Lobe): Rotating element traps fluid and forces it from suction to discharge. Smoother than reciprocating.

• Characteristics:

  • Flow rate nearly independent of head (positive displacement).

  • High pressure generation capability.

  • Self-priming generally.

  • Pulsations require air chambers/smoothers.

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

• Definition: The speed of a geometrically similar pump that would deliver unit flow rate (1 m³/s) against unit head (1 m) when operating at its maximum efficiency.

• Derivation (from similarity laws):

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

Where, $N$ = RPM, $Q$ = Discharge (m³/s), $H$ = Head (m).

• Significance:

  • Low $$\displaystyle N_s $$ (< 500): Radial flow pumps (Centrifugal).

  • Medium $$\displaystyle N_s $$ (500-5000): Mixed flow pumps.

  • High $$\displaystyle N_s $$ (> 5000): Axial flow pumps.

  \boxed{N_s = \frac{N \sqrt{Q}}{H^{3/4}} \quad \text{(SI Units)}}

V. COMPRESSORS

Centrifugal Compressors

• Construction: High-speed rotating impeller, diffuser vanes (or volute), multi-stage possible.

• Velocity Diagram:

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

  • Outlet: High radial component, $$\displaystyle u_2 $$ is high.

  • Key Parameter: Slip Factor ($\sigma$) accounts for deviation of $$\displaystyle V_{w2} $$ from ideal due to boundary layer and curvature. $$\displaystyle \sigma = \frac{V_{w2 \text{ actual}}}{V_{w2 \text{ ideal}}} $$.

• Work Done (per kg, for one stage):

$$\Delta h_0 = \sigma u_2 V_{w2} \quad \text{(or } u_2 V_{w2} \text{ ideal)}$$

• Pressure Rise: $$\displaystyle \Delta P = \rho \Delta h_0 $$ (isentropic assumption).

• Performance: High flow rate, moderate pressure rise per stage. Efficiency drops at off-design.

Axial Flow Compressors

• Construction: Alternating rows of rotating (rotor) and stationary (stator) blades. Many stages for high pressure ratio.

• Velocity Diagram: Similar triangles for each stage. Degree of Reaction ($R$) is key.

• Degree of Reaction:

$$R = \frac{\text{Static enthalpy rise in rotor}}{\text{Total static enthalpy rise in stage}}$$

For 50% reaction (Parsons turbine/compressor), $$\displaystyle \alpha_1 = \beta_2 $$, $$\displaystyle \beta_1 = \alpha_2 $$, and blade angles symmetrical.

• Multi-stage: Overall pressure ratio = product of individual stage pressure ratios. Reheat factor improves efficiency.

• Comparison with Centrifugal:

  | Feature | Centrifugal Compressor | Axial Flow Compressor | | :--- | :--- | :--- | | Flow Path | Radial | Axial | | Pressure Ratio/Stage | High (3:1 to 5:1) | Low (1.1:1 to 1.4:1) | | Flow Capacity | Lower | Higher | | Efficiency | Moderate | High (at design) | | Size/Weight | Compact for low flow | Larger, slender for high flow | | Application | Gas turbines, small plants | Large gas turbines, aircraft engines |

Compressor Efficiencies

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

$$\eta_s = \frac{\text{Ideal isentropic work input}}{\text{Actual work input}} = \frac{h_{02s} - h_{01}}{h_{02} - h_{01}}$$

Most common, but depends on pressure ratio.

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

  • Definition: Efficiency of an infinitesimal stage. Constant for a given compressor over its operating range.

  • Advantage: True measure of aerodynamic quality, independent of pressure ratio.

  • Calculation:

$$\eta_p = \frac{n-1}{n} \cdot \frac{\gamma-1}{\gamma} \quad \text{for ideal gas}$$

Where $n$ = polytropic index, $\gamma$ = specific heat ratio.

> \boxed{\eta_p = \frac{\text{Incremental isentropic work}}{\text{Incremental actual work}}}

Operating Limits and Instabilities

• Surging:

  • Definition: Complete breakdown of steady flow through the compressor, causing large-amplitude flow reversal and vibration. Occurs at low flow rates.

  • Cause: Flow separation on stator/rotor blades at high incidence angles, leading to stall and flow reversal.

  • Characteristics: Sharp drop in pressure, flow reversal, violent oscillations. Surge line on compressor map (left boundary).

  • Control: Avoid operation left of surge line, use variable stators, bleed valves.

• Choking:

  • Definition: Sonic velocity reached at the smallest flow area (throat). Mass flow rate becomes maximum and constant for given speed, regardless of downstream pressure.

  • Effect: Limits maximum mass flow at high speeds. Right boundary on compressor map.

• Stall vs Surge:

  • Stall: Localized flow separation on a blade row (rotating stall). Can lead to surge.

  • Surge: System instability involving the entire compressor and ducting.

VI. POWER TRANSMISSION DEVICES

Fluid Coupling

• Construction: Impeller (pump) and Runner (turbine) in a casing, filled with oil. No mechanical link.

• Working Principle: Impeller driven by motor imparts kinetic energy to oil. Oil strikes runner blades, transferring momentum and torque. Slip exists ($$\displaystyle \omega_t < \omega_p $$).

• Slip & Efficiency:

$$\text{Slip} = \frac{\omega_p - \omega_t}{\omega_p}$$

$$\eta = \frac{\omega_t}{\omega_p} \quad \text{(for equal diameters)}$$

> \boxed{\eta = \frac{2}{1 + \frac{u_t}{u_p}} \quad \text{(if } u_t/u_p \text{ is slip ratio)}}

• Applications: Soft start, overload protection, vibration damping in conveyors, crushers, pumps.

Torque Converter

• Construction: Impeller (pump), Turbine, Stator (redirects flow). Three elements.

• Working Principle: Oil circulates in a loop. Stator redirects oil to turbine, providing torque multiplication at low turbine speed (high slip).

• Performance:

  • Torque Ratio ($$\displaystyle T_{t}/T_{p} $$): >1 at low speed ratio.

  • Efficiency: Low at high multiplication, peaks at speed ratio ~0.6-0.7.

• Applications: Automatic transmissions, marine propulsion, heavy machinery.

Hydraulic Intensifier

• Principle: Pressure Multiplication using two pistons of different diameters in a single cylinder.

$$P_1 A_1 = P_2 A_2 \quad \Rightarrow \quad P_2 = P_1 \frac{A_2}{A_1}$$

Small piston area $$\displaystyle A_1 $$ receives low pressure $$\displaystyle P_1 $$, large piston area $$\displaystyle A_2 $$ delivers high pressure $$\displaystyle P_2 $$.

• Applications: Hydraulic presses, test rigs, systems requiring high pressure from a low-pressure source.

Centrifugal Blowers

• Working: Similar to centrifugal compressor but for low pressure rise (typically < 0.5 bar). Often with forward-curved blades.

• Applications: Ventilation, dust collection, drying, combustion air supply.

Hydraulic Press

• Principle: Pascal's Law - Pressure transmitted undiminished in confined fluid.

$$F_1/A_1 = F_2/A_2 \quad \Rightarrow \quad F_2 = F_1 \frac{A_2}{A_1}$$

• Force Multiplication: Small force $$\displaystyle F_1 $$ on small plunger area $$\displaystyle A_1 $$ generates large force $$\displaystyle F_2 $$ on large ram area $$\displaystyle A_2 $$.

Draft Tube (Reaction Turbines - Francis, Kaplan)

• Function: Converts kinetic energy at turbine runner exit to pressure energy, increasing net head. Allows setting turbine above tailrace (regains static head).

• Types:

  • Conical: Simple, efficient.

  • Elbow: For horizontal shaft turbines, space-saving.

  • Simple Tapered: Straight divergent tube.

• Design: Gradually expanding area to decelerate flow with minimal losses. Pressure recovery coefficient is key.

VIII. ADDITIONAL TOPICS (Relevant to UNIT 4)

Polytropic Efficiency of Compressors (Detailed)

• Why Polytropic? Isentropic efficiency varies with pressure ratio. Polytropic efficiency is constant for a given compressor design, representing true aerodynamic blade efficiency.

• Relation to Isentropic:

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

Where $$\displaystyle r_p $$ = pressure ratio, $n$ = polytropic index.

• For Multi-stage: Overall $$\displaystyle \eta_s $$ can be calculated from stage $$\displaystyle \eta_p $$:

$$\eta_s = \frac{r_p^{(\gamma-1)/\gamma} - 1}{r_p^{(n-1)/n} - 1} \quad \text{with } \frac{n-1}{n} = \frac{\gamma-1}{\gamma} \ln \eta_p$$

Vector Diagram of Centrifugal Compressor

• Inlet: $$\displaystyle V_1 \approx u_1 $$ (radial), $$\displaystyle \alpha_1 \approx 90^\circ $$.

• Outlet: $$\displaystyle u_2 $$ (tangential), $$\displaystyle V_{r2} $$ at angle $$\displaystyle \beta_2 $$ (vane angle), $$\displaystyle V_2 = u_2 - V_{w2} $$ (vector subtraction).

• Slip: $$\displaystyle V_{w2 \text{ actual}} = \sigma u_2 - V_{r2} \cos \beta_2 $$.

  DiagramSEARCH: centrifugal compressor velocity diagram

Vector Diagram of Axial Flow Compressor

• Constant Mean Diameter: $$\displaystyle u_1 = u_2 = u $$.

• Symmetrical 50% Reaction Stage: $$\displaystyle \alpha_1 = \beta_2 $$, $$\displaystyle \beta_1 = \alpha_2 $$.

  • Rotor: Changes $$\displaystyle \alpha_1 $$ to $$\displaystyle \beta_1 $$ (adds energy).

  • Stator: Changes $$\displaystyle \beta_2 $$ to $$\displaystyle \alpha_2 $$ (redirects flow, no work).

• Work Input: $$\displaystyle \Delta h_0 = u (V_{w2} - V_{w1}) $$.

  DiagramSEARCH: axial flow compressor velocity diagram stage

Surging and Efficiency of Axial Flow Compressor

• Surge occurs at low mass flow, high incidence → separation → efficiency drops sharply.

• Choking occurs at high mass flow, sonic at throat → flow max, efficiency may still be moderate.

• Efficiency Map: Peak efficiency near design point. Efficiency falls rapidly as surge line is approached from right.

Short Notes

Centrifugal Blower:

A low-pressure, high-volume flow device. Uses centrifugal impeller (often forward-curved blades) to move air against moderate resistance. Simple, compact, used in HVAC, drying, combustion air.

Hydraulic Intensifier:

Pressure multiplication device. Two pistons of different diameters in common cylinder. $$\displaystyle P_2 = P_1 (A_2/A_1) $$. Converts large volume/low pressure to small volume/high pressure. Used in hydraulic presses where system pressure is low.

Draft Tube:

Diffuser at exit of reaction turbine runner. Converts kinetic energy to pressure, recovers static head. Allows turbine placement above tailrace. Types: Conical (most efficient), Elbow (for horizontal shafts), Simple Tapered. Design goal: minimize kinetic energy loss at exit.

Applications of Power Transmitting Devices:

• Fluid Coupling: Soft starters for conveyors, mills, pumps; vibration isolation.

• Torque Converter: Automatic transmissions (cars, earthmovers), marine propulsion.

• Hydraulic Intensifier: High-pressure testing, hydraulic presses with low-pressure source.

• Centrifugal Blowers: Industrial ventilation, dust collection, drying ovens, boiler forced draft.

• Hydraulic Press: Metal forming, molding, assembly (press fits), scrap processing.

KEY FORMULAS FOR UNIT 4

[!TIP] Exam Strategy: For numerical problems (Pelton, centrifugal pump, compressor stages), always start with a clear velocity diagram. Label $u$, $V$, $$\displaystyle V_w $$, $$\displaystyle V_r $$, $\alpha$, $\beta$. Apply Euler's equation $$\displaystyle \Delta h_0 = u V_w $$ (often $$\displaystyle u_1 V_{w1} \approx 0 $$). Calculate forces ($$\displaystyle F = \dot{m} \Delta V_w $$), power ($$\displaystyle P = \dot{m} \Delta h_0 $$), and efficiency ($$\displaystyle \eta = \frac{\text{Useful output}}{\text{Input}} $$). For specific speed, ensure units are consistent (SI: $N$ in RPM, $Q$ in m³/s, $H$ in m).