Thermal Engineering and Gas Dynamics (ME-601) - Unit 2 Short Notes

UNIT 2: Thermal Engineering and Gas Dynamics

0. Fundamentals of Steam Properties

• Steam tables: Tabulate thermodynamic properties (pressure, temperature, specific volume \(v\), enthalpy \(h\), entropy \(s\)) for saturated and superheated steam.

• Mollier chart: \(h\text{-}s\) diagram for steam, useful for visualizing processes.

• Dryness fraction \(x\): Mass of vapor divided by total mass in a saturated mixture.

• Properties of saturated mixture:

  \[ v = v_f + x v_{fg}, \quad h = h_f + x h_{fg}, \quad s = s_f + x s_{fg} \]

• State determination: Given any two independent properties (e.g., \(P\) and \(x\), \(T\) and \(v\)), use steam tables to find others.

I. Steam Generation Systems

A. Boiler Classifications and Types

• High-pressure boilers:

  • Velox boiler: Forced circulation, high pressure (up to 170 bar), compact. Water circulated by centrifugal pump through tubes heated by flue gases. Steam generated rapidly.

    DiagramSEARCH: Velox boiler diagram

  • Lamont boiler: Forced circulation with separate steam drum. Water circulated by pump through evaporator tubes. High pressure (up to 100 bar).

    DiagramSEARCH: Lamont boiler diagram

• Other types:

  • Fire-tube: Flue gases inside tubes, water outside (e.g., Lancashire). Low–medium pressure.

  • Water-tube: Water inside tubes, flue gases outside (e.g., Babcock & Wilcox). High pressure, high capacity.

B. Draught Systems

• Natural draught: Uses chimney height \(h\) to create pressure difference.

  • Theoretical draught: \(D = h (\rho_a - \rho_g)\) (in mm water if \(h\) in m, \(\rho\) in kg/m³).

  • Where \(\rho_a = \frac{P_{atm} M_a}{R T_a}\), \(\rho_g = \frac{P_g M_g}{R T_g}\) (often \(P_g \approx P_{atm}\)).

• Artificial draught:

  • Induced draught: Fan after boiler, creates negative pressure in boiler.

  • Forced draught: Fan before boiler, creates positive pressure.

  • Advantages: Independent of weather, higher pressure, compact chimney.

  • Disadvantages: Power consumption, maintenance.

• Numerical on chimney draught: Given \(h\), \(T_g\), \(T_a\), \(P_{atm}\), \(D\), find air used per kg fuel.

  • Step 1: Calculate \(\rho_a\) and \(\rho_g\) from ideal gas law.

  • Step 2: From \(D = h(\rho_a - \rho_g)\), find \(\rho_g\) if not given.

  • Step 3: Air supplied \(m_a\) (kg air/kg fuel) often derived from gas flow relations, e.g., \(m_a = \frac{\rho_g}{\rho_a} \cdot \frac{T_a}{T_g} \cdot \frac{D}{0.0125\,h}\) (empirical form). Verify with standard formula in textbooks.

C. Boiler Performance

• Boiler efficiency:

  • Direct: \(\eta = \frac{m_s (h - h_{f1})}{m_f \cdot CV}\)

    • \(m_s\): steam generated, \(h\): steam enthalpy, \(h_{f1}\): feed water enthalpy, \(m_f\): fuel, \(CV\): calorific value.

  • Indirect: \(\eta = 100 - \sum \text{losses}\) (dry flue gas, moisture, incomplete combustion, radiation, etc.).

• Heat balance sheet: Tabular account of energy input (fuel) and output (steam, losses).

• Equivalent evaporation: \(m_e = m_s \cdot \frac{h - h_{f1}}{2257}\) (kg steam from and at 100°C per kg fuel).

D. Boiler Mountings and Accessories

• Mountings (safety): Safety valve, pressure gauge, water level indicator, stop valve, fusible plug.

• Accessories (efficiency): Economizer (preheats feed water), superheater (increases steam temperature), air preheater (heats combustion air), feed pump.

II. Vapor Power Cycles (Steam)

A. Basic Rankine Cycle

• Processes:

  1–2: Isentropic compression in pump (saturated liquid → compressed liquid).

  2–3: Constant-pressure heat addition in boiler (to superheated steam).

  3–4: Isentropic expansion in turbine.

  4–1: Constant-pressure heat rejection in condenser.

• T-s diagram: Shows constant pressure lines, saturation curve.

  DiagramSEARCH: Rankine cycle T-s diagram

• p-v diagram: Closed loop.

• Thermal efficiency:

  \[ \eta = \frac{h_3 - h_4}{h_3 - h_2} \approx \frac{h_3 - h_4}{h_3 - h_{f2}} \]

  where \(h_2 \approx h_{f1} + v_{f1}(P_{boiler} - P_{cond})\).

B. Effect of Cycle Parameters

• Boiler pressure: ↑ Pressure → ↑ efficiency (higher average \(T\) of heat addition) but ↓ quality at turbine exit (more moisture). Requires better materials.

• Condenser pressure: ↓ Pressure → ↑ efficiency (lower \(T\) of heat rejection) but ↑ moisture and vacuum issues.

• Superheat: ↑ Superheat → ↑ efficiency, avoids moisture, allows higher boiler pressure.

C. Steam Turbines

• Impulse turbine: Pressure drop only in nozzles; blades change direction only. Velocity diagram: absolute velocity \(V\), relative velocity \(V_r\), blade speed \(U\). Work per kg: \(W = U(V_{w1} + V_{w2})\).

• Reaction turbine: Pressure drop in both fixed and moving blades. Velocity diagram shows change in \(V_r\) magnitude.

• Compounding:

  • Need: Reduce blade speed and size for high pressure ratio.

  • Velocity compounding: Multiple stages of nozzles and blades (e.g., Curtis turbine).

  • Pressure compounding: Multiple pressure drops with intervening reheat (e.g., Rateau turbine).

D. Cycle Improvements

• Reheat cycle:

  • Steam expands in HP turbine, reheated to high \(T\), then expands in LP turbine.

  • T-s diagram: Two expansion lines with constant-pressure reheat.

    DiagramSEARCH: reheat Rankine cycle T-s diagram

  • Advantages: ↑ Efficiency, ↓ moisture at final stages.

  • Numerical: Given pressures, temperatures, find quality, efficiency, steam rate.

• Regenerative cycle:

  • Uses feed water heaters (FWH) to preheat feed water using extracted steam.

  • Open FWH (de-aerator): Direct contact, removes dissolved gases.

  • Closed FWH: Shell-and-tube, no mixing.

  • T-s diagram: Shows extraction points.

    DiagramSEARCH: regenerative Rankine cycle T-s diagram

  • Advantages: ↑ Efficiency by reducing heat input.

E. Steam Engine Cycles

• Modified Rankine cycle: Includes cut-off, release, back pressure, constant-volume release.

• Efficiency: \(\eta = \frac{\text{work output}}{\text{heat input}}\) from indicator diagram area.

• Numerical: Given pressures, cut-off ratio, back pressure, find efficiency.

III. Gas Turbine Cycles

A. Basic Brayton (Joule) Cycle

• Processes:

  1–2: Isentropic compression.

  2–3: Constant-pressure heat addition.

  3–4: Isentropic expansion.

  4–1: Constant-pressure heat rejection.

• T-s and p-v diagrams: Vertical lines for isentropic processes.

  DiagramSEARCH: Brayton cycle T-s diagram

• Thermal efficiency:

  \[ \eta = 1 - \frac{1}{r_p^{(\gamma-1)/\gamma}} \]

  where \(r_p = P_2/P_1\) (pressure ratio), \(\gamma = C_p/C_v\).

B. Regeneration in Gas Turbines

• Regenerator: Recovers heat from exhaust to preheat compressed air.

  • Open (recuperator): Separate streams.

  • Closed (regenerator): Same stream cycles.

• Effectiveness: \(\varepsilon = \frac{T_5 - T_2}{T_3 - T_2}\), where \(T_5\) is hot-side exit temperature.

C. Reheating in Gas Turbines

• Steam expands in HP turbine, reheated, then LP turbine.

• Advantages: ↑ Work output, may ↑ efficiency with optimal pressure ratio.

D. Conditions for Maximum Output

• With reheater and regenerator, derive optimal pressure ratios for maximum net work.

• For two-stage cycle: Maximum output when pressure ratios equal and reheat temperature equals maximum cycle temperature.

IV. Advanced Vapor Cycles

A. Binary Vapor Cycle

• Working principle: Two vapors (e.g., mercury and water). High-temperature vapor (mercury) expands in turbine, then heats water in boiler.

• Advantages: Higher average temperature of heat addition, improved efficiency.

• Example: Mercury-water cycle.

B. Vapor Carnot Cycle

• T-s diagram: Two isothermal and two adiabatic processes.

  DiagramSEARCH: Carnot cycle T-s diagram

• Processes: Isothermal heat addition, adiabatic expansion, isothermal heat rejection, adiabatic compression.

• Limitations:

  • Isothermal heat addition requires large heat exchanger and slow process.

  • Wet compression in pump requires large volume.

  • Practical only for vapors with high latent heat and low specific volume.

V. Steam Power Plant Components

A. Feed Water Heaters (FWH)

• Open-type (de-aerator): Feed water mixed with steam; removes dissolved gases; operates at saturation temperature.

• Closed-type (shell and tube): Steam on shell side, feed water in tubes; no mixing.

• Differences: Open type removes gases; closed type avoids contamination.

B. Condensers

• Jet condensers: Mixing of steam and cooling water.

  • Low-level, high-level, counter-flow types.

  • Simple but water loss.

• Surface condensers: Steam and water separate (shell and tube). No water loss, better vacuum.

• Air leakage sources: Leaks in condenser, packing, etc.

  • Effects: ↑ Pumping work, ↓ vacuum, ↑ compressor work for air removal.

• Numerical: Given condensate temperature, volume, find mass, etc.

VI. Fundamentals of Compressible Flow

A. Mach Number

• Definition: \(M = V/a\), where \(a = \sqrt{\gamma R T}\) is speed of sound.

• Significance:

  • \(M < 1\): Subsonic

  • \(M = 1\): Sonic

  • \(M > 1\): Supersonic

  • \(M > 5\): Hypersonic

• Mach cone: Angle \(\mu = \sin^{-1}(1/M)\).

• Zone of action: Region affected by disturbances; zone of silence: Region unaffected.

B. One-Dimensional Isentropic Flow

• Flow through variable area ducts:

  • Converging: Accelerates to sonic at throat if back pressure low enough.

  • Diverging: Decelerates subsonic, accelerates supersonic.

  • Convergent–divergent (CD): Achieves supersonic flow if throat choked and divergent section.

• Stagnation properties: \(T_0\), \(P_0\), \(\rho_0\) constant in isentropic flow.

• Area–velocity relationship:

  \[ \frac{dA}{A} = (M^2 - 1) \frac{dV}{V} \]

• Choking: When \(M=1\) at throat, mass flow rate maximum, independent of downstream pressure.

C. Diffusers

• Function: Decelerate flow, increase pressure (pressure recovery).

• Compressible flow: Subsonic diffuser is divergent; supersonic requires CD to avoid shocks.

VII. Compressors

A. Classification

• Reciprocating: Single-stage, multi-stage, single/double acting.

• Rotary: Centrifugal, axial, rotary screw, vane.

B. Reciprocating Compressors

• Working: Piston compresses air; P-V diagram shows intake, compression, discharge, expansion.

• Work input without clearance (polytropic index \(n\)):

  \[ W = \frac{n}{n-1} P_1 V_1 \left[ \left( \frac{P_2}{P_1} \right)^{(n-1)/n} - 1 \right] \]

• With clearance (clearance ratio \(C = V_c / V_s\)):

  \[ \eta_v = 1 + C - C \left( \frac{P_2}{P_1} \right)^{1/n} \]

• Mean effective pressure (MEP):

  \[ \text{MEP} = \frac{\text{Work per cycle}}{\text{Stroke volume}} \]

• Numerical: Find MEP, power, delivery temperature, \(\eta_v\).

C. Multistage Compression

• Advantages: ↓ Work (with intercooling), ↓ temperature, ↑ efficiency.

• Perfect intercooling: Intercooler cools to initial temperature \(T_1\).

• Minimum work: For given pressure ratio, work minimized when pressure ratio equal in each stage:

  \[ r_{p1} = r_{p2} = \sqrt{P_2/P_1} \quad \text{(two stages)} \]

• Heat rejected in intercooler (per kg):

  \[ Q_{\text{intercooler}} = C_p (T_2 - T_1) \]

  where \(T_2\) is exit temperature of first stage.

D. Rotary Compressors

• Centrifugal compressor:

  • Air enters axially, accelerated by impeller, diffuser converts velocity to pressure.

  • Velocity diagrams show absolute and relative velocities.

    DiagramSEARCH: centrifugal compressor velocity diagram

• Comparison with reciprocating:

  • Continuous flow, higher capacity, lower pressure ratio per stage, used for large volumes.

E. Compressor Efficiencies

• Isentropic efficiency: \(\eta_s = \dfrac{\text{Isentropic work}}{\text{Actual work}}\)

• Isothermal efficiency: \(\eta_i = \dfrac{\text{Isothermal work}}{\text{Actual work}}\)

• Mechanical efficiency: \(\eta_m = \dfrac{\text{Indicated work}}{\text{Shaft work}}\)

VIII. Nozzles

A. Nozzle Flow Theory

• Steady flow energy equation: \(h_0 = h + \frac{V^2}{2}\).

• Critical pressure ratio (for convergent nozzle):

  \[ \left( \frac{P_{\text{exit}}}{P_0} \right)_{\text{crit}} = \left( \frac{2}{\gamma+1} \right)^{\gamma/(\gamma-1)} \]

  For \(\gamma = 1.4\), \(P_{\text{exit}}/P_0 \approx 0.528\).

• Derivation for maximum discharge: From continuity and energy, mass flow \(\dot{m} = \frac{A P_0}{\sqrt{T_0}} \sqrt{\frac{\gamma}{R}} M (1 + \frac{\gamma-1}{2} M^2)^{-(\gamma+1)/(2(\gamma-1))}\). Maximum at \(M=1\).

• CD nozzle: Can achieve supersonic if throat choked and back pressure below design.

B. Supersaturated Flow

• Occurrence: In steam nozzles with rapid expansion, nucleation delayed → metastable vapor beyond saturation line.

• Effects:

  • Discharge: Mass flow rate ↑ (specific volume larger than equilibrium wet flow).

  • Heat drop: ↓ (exit enthalpy higher than wet state, so smaller \(\Delta h\)).

• Wilson line: On T-s diagram, limit of supersaturation; beyond which condensation occurs.

• Conditions: High initial dryness, large pressure drop, smooth nozzle surface.

C. Nozzle Efficiency and Friction

• Nozzle efficiency:

  \[ \eta_n = \frac{V_{\text{actual}}^2}{V_{\text{ideal}}^2} = \frac{h_0 - h_{\text{actual}}}{h_0 - h_{\text{ideal}}} \]

• Effect of friction: ↓ Exit velocity, ↑ Exit pressure, ↓ discharge.

• Numerical: Given initial state, exit pressure, friction loss, find velocity, % reduction.

IX. Plant Performance and Economic Analysis

A. Overall Plant Efficiency

• Expression:

  \[ \eta_{\text{overall}} = \eta_{\text{boiler}} \times \eta_{\text{turbine}} \times \eta_{\text{generator}} \times \eta_{\text{cycle}}? \]

  Actually, \(\eta_{\text{overall}} = \frac{\text{Net work output}}{\text{Heat input}} = \frac{m_s (h_3 - h_4) \cdot \eta_{\text{generator}}}{m_f \cdot CV}\).

• Numerical: Given steam rate, generator efficiency, condensate conditions, find \(\eta_{\text{overall}}\).

B. Steam Rate and Specific Steam Consumption

• Steam rate: \(S = \frac{m_s}{\text{Power output}}\) (kg/kWh).

• Relation to efficiency:

  \[ \eta_{\text{overall}} = \frac{3600}{S \cdot (h - h_{f1}) \cdot \eta_{\text{generator}}} \quad \text{(if turbine cycle efficiency included)} \]

• Calculation: From plant data (steam generated, power output).

C. Heat Balance and Energy Accounting

• Heat balance sheet for steam power plant:

  | Input | Output | |-------|--------| | Heat from fuel | Useful work (net) | | | Heat in steam (to process) | | | Losses: stack, radiation, unaccounted |

• Loss quantification: From flue gas analysis (temperature, composition), radiation measurements.

[!TIP] Exam Focus:

• Derive Rankine and Brayton efficiencies.

• Explain impulse vs reaction turbines with velocity diagrams.

• Solve numericals on boiler efficiency, equivalent evaporation, compressor work (with/without clearance), multistage compression with intercooling, nozzle flow (critical pressure ratio, supersaturation), and overall plant efficiency.

• Draw and label T-s diagrams for Rankine (basic, reheat, regenerative), Brayton, and Carnot cycles.

• Differentiate open vs closed feed water heaters, jet vs surface condensers.

• Understand Mach number significance and isentropic flow relations (choking, area-velocity).