PE Exam Exam > PE Exam Notes > Mechanical Engineering for PE > Cheatsheet: Power Cycles (Rankine, Brayton)
Cheatsheet: Power Cycles (Rankine, Brayton)
1. Rankine Cycle Fundamentals
1.1 Basic Components and Processes
| Process | Description |
|---|---|
| 1→2: Isentropic Compression | Pump increases pressure of saturated/compressed liquid; work input required |
| 2→3: Constant Pressure Heat Addition | Boiler adds heat to convert liquid to superheated vapor |
| 3→4: Isentropic Expansion | Turbine extracts work as vapor expands to lower pressure |
| 4→1: Constant Pressure Heat Rejection | Condenser rejects heat to convert vapor to saturated liquid |
1.2 Key Assumptions
- Steady-state, steady-flow operation
- Negligible kinetic and potential energy changes
- No pressure drops in boiler and condenser
- Isentropic pump and turbine (ideal cycle)
- Saturated liquid exits condenser
1.3 Performance Equations
| Parameter | Formula |
|---|---|
| Pump Work (ideal) | wp = v1(P2 - P1) ≈ h2 - h1 |
| Turbine Work (ideal) | wt = h3 - h4 |
| Heat Added | qin = h3 - h2 |
| Heat Rejected | qout = h4 - h1 |
| Net Work | wnet = wt - wp = qin - qout |
| Thermal Efficiency | ηth = wnet/qin = 1 - qout/qin |
| Back Work Ratio | bwr = wp/wt |
2. Rankine Cycle Variations and Improvements
2.1 Reheat Rankine Cycle
| Feature | Description |
|---|---|
| Configuration | Steam expanded in high-pressure turbine, reheated in boiler, then expanded in low-pressure turbine |
| Benefit | Increases turbine exit quality, reduces moisture damage; improves efficiency |
| Turbine Work | wt = (h3 - h4) + (h5 - h6) |
| Heat Added | qin = (h3 - h2) + (h5 - h4) |
2.2 Regenerative Rankine Cycle
| Type | Description |
|---|---|
| Open Feedwater Heater | Extracted steam mixes directly with feedwater; requires pump between each heater |
| Closed Feedwater Heater | Heat exchange without mixing; extracted steam condensed and can be pumped forward or backward |
| Benefit | Reduces exergy destruction by preheating feedwater; increases thermal efficiency |
| Extraction Fraction | y = ṁextracted/ṁtotal; determined by energy balance on feedwater heater |
2.3 Supercritical Rankine Cycle
- Operating pressure above critical point of water (22.09 MPa, 374.14°C)
- No distinct phase change; continuous transition from liquid-like to vapor-like
- Higher efficiency due to higher average temperature of heat addition
- Modern power plants operate at 25-35 MPa
3. Actual Rankine Cycle Performance
3.1 Component Isentropic Efficiencies
| Component | Definition |
|---|---|
| Pump Efficiency | ηp = (h2s - h1)/(h2a - h1) = ws/wa |
| Turbine Efficiency | ηt = (h3 - h4a)/(h3 - h4s) = wa/ws |
| Actual Pump Work | wp,a = wp,s/ηp |
| Actual Turbine Work | wt,a = ηt × wt,s |
3.2 Additional Real-World Effects
- Pressure drops in boiler, condenser, and piping reduce efficiency
- Heat losses from components to surroundings
- Mechanical losses in turbine and pump bearings
- Generator electrical losses (efficiency 95-98%)
- Auxiliary equipment power consumption (3-5% of gross output)
3.3 Typical Performance Values
| Parameter | Typical Range |
|---|---|
| Pump Isentropic Efficiency | 75-85% |
| Turbine Isentropic Efficiency | 85-90% |
| Overall Thermal Efficiency (simple) | 30-40% |
| Overall Thermal Efficiency (reheat/regen) | 40-45% |
| Back Work Ratio | 1-3% |
4. Brayton Cycle Fundamentals
4.1 Basic Components and Processes
| Process | Description |
|---|---|
| 1→2: Isentropic Compression | Compressor increases air pressure and temperature; work input required |
| 2→3: Constant Pressure Heat Addition | Combustor adds heat (fuel combustion) at constant pressure |
| 3→4: Isentropic Expansion | Turbine extracts work as gas expands to lower pressure |
| 4→1: Constant Pressure Heat Rejection | Heat rejected to atmosphere (open cycle) or heat exchanger (closed cycle) |
4.2 Key Assumptions
- Steady-state, steady-flow operation
- Air-standard analysis (air as working fluid, constant specific heats)
- Negligible kinetic and potential energy changes
- No pressure drops in combustor and heat exchanger
- Isentropic compressor and turbine (ideal cycle)
4.3 Performance Equations (Constant Specific Heats)
| Parameter | Formula |
|---|---|
| Compressor Work | wc = h2 - h1 = cp(T2 - T1) |
| Turbine Work | wt = h3 - h4 = cp(T3 - T4) |
| Heat Added | qin = h3 - h2 = cp(T3 - T2) |
| Heat Rejected | qout = h4 - h1 = cp(T4 - T1) |
| Net Work | wnet = wt - wc = cp[(T3 - T4) - (T2 - T1)] |
| Thermal Efficiency | ηth = wnet/qin = 1 - qout/qin = 1 - T1/T2 |
| Back Work Ratio | bwr = wc/wt |
5. Brayton Cycle Pressure Ratio Relations
5.1 Isentropic Relations
| Relation | Formula |
|---|---|
| Pressure Ratio | rp = P2/P1 = P3/P4 |
| Temperature Ratio (Compressor) | T2/T1 = (P2/P1)(k-1)/k = rp(k-1)/k |
| Temperature Ratio (Turbine) | T3/T4 = (P3/P4)(k-1)/k = rp(k-1)/k |
| Efficiency (k = 1.4 for air) | ηth = 1 - rp-(k-1)/k = 1 - rp-0.286 |
5.2 Effect of Design Parameters
- Increasing pressure ratio increases thermal efficiency
- Increasing turbine inlet temperature (T3) increases net work and efficiency
- Decreasing compressor inlet temperature (T1) increases net work and efficiency
- Back work ratio for gas turbines: 40-80% (much higher than steam cycles)
- Optimal pressure ratio for maximum work output: rp,opt = (T3/T1)k/[2(k-1)]
6. Brayton Cycle Variations and Improvements
6.1 Regenerative Brayton Cycle
| Feature | Description |
|---|---|
| Configuration | Heat exchanger (regenerator) transfers heat from turbine exhaust to compressor exit |
| Condition for Benefit | T4 > T2 (turbine exhaust hotter than compressor exit) |
| Regenerator Effectiveness | ε = (T5 - T2)/(T4 - T2) = (h5 - h2)/(h4 - h2) |
| Heat Added | qin = h3 - h5 = cp(T3 - T5) |
| Thermal Efficiency | ηth = 1 - (T1/T3)rp(k-1)/k |
6.2 Reheat Brayton Cycle
- Gas expanded in high-pressure turbine, reheated in second combustor, expanded in low-pressure turbine
- Increases turbine work output
- Optimal intermediate pressure: Pint = √(P3P4)
- Total turbine work: wt = cp[(T3 - T4) + (T5 - T6)]
6.3 Intercooling Brayton Cycle
- Compression in two stages with cooling between stages
- Reduces compressor work input
- Optimal intermediate pressure: Pint = √(P1P2)
- Total compressor work: wc = cp[(T2 - T1) + (T4 - T3)]
6.4 Combined Reheat, Intercooling, and Regeneration
- Maximum efficiency and work output achieved by combining all improvements
- Modern gas turbine power plants use multiple stages
- Complexity and cost increase with additional components
7. Actual Brayton Cycle Performance
7.1 Component Isentropic Efficiencies
| Component | Definition |
|---|---|
| Compressor Efficiency | ηc = (h2s - h1)/(h2a - h1) = ws/wa |
| Turbine Efficiency | ηt = (h3 - h4a)/(h3 - h4s) = wa/ws |
| Actual Compressor Work | wc,a = wc,s/ηc |
| Actual Turbine Work | wt,a = ηt × wt,s |
7.2 Additional Real-World Effects
- Pressure drops in combustor, heat exchangers, and ducts (2-5% per component)
- Combustion inefficiency (combustion efficiency 95-99%)
- Heat losses from turbine and combustor casings
- Mechanical losses in bearings and gearbox
- Generator electrical losses
7.3 Typical Performance Values
| Parameter | Typical Range |
|---|---|
| Compressor Isentropic Efficiency | 80-88% |
| Turbine Isentropic Efficiency | 85-92% |
| Pressure Ratio | 10-40 |
| Turbine Inlet Temperature | 1200-1700°C |
| Overall Thermal Efficiency (simple) | 30-40% |
| Overall Thermal Efficiency (regenerative) | 35-42% |
| Back Work Ratio | 40-80% |
8. Combined Cycle Power Plants
8.1 Configuration and Components
| Component | Description |
|---|---|
| Topping Cycle | Gas turbine (Brayton cycle) operates at high temperature |
| Bottoming Cycle | Steam turbine (Rankine cycle) uses waste heat from gas turbine exhaust |
| Heat Recovery Steam Generator (HRSG) | Boiler that uses gas turbine exhaust as heat source |
8.2 Performance Equations
| Parameter | Formula |
|---|---|
| Total Heat Input | qin,total = qin,gas |
| Total Work Output | wnet,total = wnet,gas + wnet,steam |
| Combined Cycle Efficiency | ηCC = wnet,total/qin,total |
| Alternative Expression | ηCC = ηgas + ηsteam(1 - ηgas)ηHRSG |
8.3 Typical Performance
- Combined cycle thermal efficiency: 50-60%
- Gas turbine efficiency: 35-40%
- Steam turbine efficiency: 30-35%
- HRSG effectiveness: 85-95%
- Multiple pressure levels in HRSG improve performance
- Supplementary firing can increase steam cycle output
9. Cycle Comparison and Selection Criteria
9.1 Rankine vs. Brayton Characteristics
| Characteristic | Rankine Cycle |
|---|---|
| Working Fluid | Water/steam undergoes phase change |
| Back Work Ratio | 1-3% (low) |
| Typical Efficiency | 30-45% |
| Startup Time | Several hours (cold start) |
| Capital Cost | Moderate to high |
| Applications | Base load power plants, nuclear, geothermal |
| Characteristic | Brayton Cycle |
|---|---|
| Working Fluid | Gas (air) remains in gas phase |
| Back Work Ratio | 40-80% (high) |
| Typical Efficiency | 30-42% |
| Startup Time | Minutes to 1 hour |
| Capital Cost | Low to moderate |
| Applications | Peaking power, aircraft propulsion, industrial |
9.2 Key Selection Factors
- Load profile: Rankine for base load, Brayton for peaking/intermediate
- Fuel availability: Brayton more flexible with fuel types
- Water availability: Rankine requires significant cooling water
- Space constraints: Brayton more compact per unit output
- Environmental regulations: Combined cycle offers lowest emissions per kWh
- Economic considerations: capital cost, fuel cost, maintenance cost, efficiency
10. Important Thermodynamic Properties and Constants
10.1 Water/Steam Properties
| Property | Value |
|---|---|
| Critical Pressure | 22.09 MPa |
| Critical Temperature | 374.14°C (647.3 K) |
| Triple Point | 0.01°C, 0.6117 kPa |
| Normal Boiling Point | 100°C at 101.325 kPa |
10.2 Air Properties (at 300 K)
| Property | Value |
|---|---|
| Specific Heat (cp) | 1.005 kJ/(kg·K) |
| Specific Heat (cv) | 0.718 kJ/(kg·K) |
| Specific Heat Ratio (k) | 1.4 |
| Gas Constant (R) | 0.287 kJ/(kg·K) |
10.3 Common Unit Conversions
- 1 bar = 100 kPa = 0.1 MPa
- 1 atm = 101.325 kPa = 14.696 psia
- 1 kW·h = 3600 kJ
- 1 Btu = 1.055 kJ
- °C = K - 273.15
- 1 hp = 0.7457 kW
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Apr 20, 2026 Last updated
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