Vapor and Combined Power Cycles
Vapor plants convert heat into shaft work by circulating a condensable fluid through pumps, heat exchangers, and turbines. Sound analysis begins with states and component boundaries, then closes mass, energy, entropy, and exergy accounts for the entire loop.
Learning objectives
- Recognize Carnot, Rankine, reheat, regenerative, cogeneration, and combined-cycle architectures.
- Identify water states from absolute pressure, temperature, and phase before selecting authoritative properties.
- Apply steady control-volume balances and distinguish enthalpy carried by flow from internal energy.
- Calculate pump work, turbine work, heat transfer, thermal efficiency, and extraction fractions on a mass basis.
- Separate ideal isentropic components from real efficiencies and verify cycle closure and entropy generation.
- Use second-law and uncertainty evidence to judge modifications without overstating screening calculations.
The Carnot Vapor Cycle
A limit, not a plant layout
The reversible Carnot benchmark has two isothermal and two isentropic processes. For reservoirs at absolute temperatures, \(\eta_{\rm rev}=1-T_L/T_H\). A vapor implementation would require compression of a two-phase mixture and perfectly reversible heat transfer at vanishing temperature differences. Droplets also threaten turbine blades. Rankine hardware avoids wet compression by condensing fully before a liquid pump, accepting irreversibility for feasibility.
Physical meaning
The Carnot result is an upper bound set only by reservoir temperatures, not a promised efficiency. A real boiler has finite temperature differences, pressure loss, material limits, and combustion irreversibility; equality would require unavailable reversible operation.
Rankine Cycle: The Ideal Cycle for Vapor Power
Place a steady control volume around each component. With heat into and shaft work out positive, negligible kinetic and potential energy changes give \(w_p=h_2-h_1\), \(q_{in}=h_3-h_2\), \(w_t=h_3-h_4\), and \(q_{out}=h_4-h_1\). For an incompressible liquid, \(w_p\approx v_1(P_2-P_1)\); this is not \(c_p\Delta T\). All pressures are absolute, temperatures used in ratios are kelvins, and enthalpies come from one consistent steam-table source.
Problem-solving strategy
- Sketch the four component control volumes and number states in flow order.
- At each absolute pressure and temperature, classify compressed liquid, saturated mixture, or superheated vapor before reading properties.
- Use saturated, superheated, or compressed-water tables as appropriate; interpolate only within a table and record both bracketing entries.
- Apply mass and energy balances component by component, using \(h\), not \(u\), for flowing streams.
- Close the cycle independently and check turbine-exit quality, signs, and nonnegative entropy generation.
Recognition strategy
Four states, a liquid pump, boiler, turbine, and condenser identify an ideal Rankine cycle; component enthalpy differences provide the shortest auditable route.
Given information
- Supplied educational steam-table values: \(h_1=191.8\), \(h_3=3330\), \(h_4=2100\ \mathrm{kJ/kg}\), and \(v_1=0.00101\ \mathrm{m^3/kg}\).
- One-kilogram mass basis; absolute pressures; ideal internally reversible pump and turbine; steady cycle.
Model and interpretation
Each device is a steady open control volume, while all four form a cycle. Heat into and work out are positive; water is a real pure substance, not an ideal gas. The supplied properties are exact inputs to an approximate educational model; kinetic and potential energy changes are neglected.
\(=8.07\ \mathrm{kJ/kg}\).
\(=199.87\ \mathrm{kJ/kg}\), retaining flow enthalpy rather than internal energy.
\(=3130.13\ \mathrm{kJ/kg}\) and \(w_t=3330-2100=1230\ \mathrm{kJ/kg}\).
\(=1221.93\ \mathrm{kJ/kg}\) and \(\eta_{th}=1221.93/3130.13\)
\(=0.3904\).
Verification
Units: \(\mathrm{m^3/kg}\) times \(\mathrm{kPa}\) equals \(\mathrm{kJ/kg}\), so every transfer remains on the stated mass basis.
Signs and direction: Pump work is an input and turbine work is an output; positive net output agrees with heat-engine operation.
Plausibility: \(q_{out}=2100-191.8\)
\(=1908.2\ \mathrm{kJ/kg}\), and \(3130.13-1908.2\)
\(=1221.93\ \mathrm{kJ/kg}\), exactly closing energy. Ideal isentropic devices have zero internal entropy generation; the working fluid returns to state 1, so its cycle changes in mass, energy, entropy, and exergy are zero.
Engineering interpretation
Pump work is under one percent of turbine work because liquid specific volume is small. Release work must replace supplied educational values with traceable steam-table software and include pressure losses and measured component efficiencies.
Deviation of Actual Vapor Power Cycles
Ideal states versus actual states
An adiabatic turbine has \(\eta_t=(h_3-h_{4a})/(h_3-h_{4s})\); a pump has \(\eta_p=(h_{2s}-h_1)/(h_{2a}-h_1)\). The subscripts matter: turbine efficiency makes actual outlet enthalpy higher, while pump inefficiency makes it higher than its isentropic counterpart. Pressure drops and heat leakage alter endpoint pressures too. Isentropic means reversible and adiabatic, not merely adiabatic.
Entropy balances must return \(\dot S_{gen}\ge0\). With environment \((T_0,P_0)\), destroyed exergy is \(\dot X_d=T_0\dot S_{gen}\). Negative generation usually signals a state, heat-direction, or property-source error.
How Can We Increase Rankine-Cycle Efficiency?
The Ideal Reheat Rankine Cycle
Reheat divides expansion between high- and low-pressure turbines. Heat input and turbine work become sums across stages; the reheater is not free energy. Intermediate pressure is selected iteratively because work, moisture, pipe losses, and equipment cost compete.
Recognition strategy
Two turbine enthalpy drops and two heat-addition intervals signal reheat; each interval must appear once in the cycle ledger.
Given information
- Supplied steam-table enthalpies: \((h_1,h_2,h_3)=(200,210,3450)\ \mathrm{kJ/kg}\).
- Also supplied: \((h_4,h_5,h_6)=(3000,3520,2400)\ \mathrm{kJ/kg}\).
- Supplied state-6 quality \(x_6=0.91\); preliminary criterion \(x_6\ge0.90\); all pressures absolute and temperatures on an absolute basis where ratios apply.
Model and interpretation
Use steady open control volumes and a one-kilogram water basis. Heat in and shaft work out are positive. Ideal pump and turbines are isentropic; supplied real-water table values are inputs, not invented predictions. Neglect kinetic and potential energy, heat leakage, and pressure loss.
Verification
Units: Every enthalpy difference is \(\mathrm{kJ/kg}\); their ratio is dimensionless, as thermal efficiency must be.
Signs and direction: Both turbine drops are positive work outputs, while the pump difference is positive required input.
Plausibility: Condenser rejection is \(h_6-h_1=2200\ \mathrm{kJ/kg}\), and \(3760-2200=1560\ \mathrm{kJ/kg}\), closing energy. Cycle mass, state, entropy, and exergy changes are zero; ideal isentropic stages generate zero entropy, while real replacements must yield nonnegative generation.
Engineering interpretation
Reheat adds both heat and work, so its value cannot be judged from turbine work alone. The quality threshold is only a screening criterion; blade vendor limits, droplet erosion, and traceable state calculations govern release.
The Ideal Regenerative Rankine Cycle
Extraction changes every downstream mass flow
In an open feedwater heater, extraction steam mixes directly with condensate. On a unit boiler-flow basis, mass conservation gives \(y+(1-y)=1\), and an adiabatic mixing balance gives \(y h_b+(1-y)h_c=h_o\). Therefore \(y=(h_o-h_c)/(h_b-h_c)\). Closed heaters require separate stream balances and drain routing. Never apply turbine work to the full kilogram after a bleed point.
Second-Law Analysis of Vapor Power Cycles
The first law locates energy quantities; the second law locates lost work opportunity. Apply entropy and exergy balances to consistent boundaries. Boiler combustion, finite-temperature heat exchange, throttling, and non-isentropic expansion can dominate. Exergy depends on a declared environment; energy efficiency does not.
Cogeneration
Cogeneration credits process heat only when it is useful at the required temperature, schedule, and reliability. Utilization is not thermal efficiency and does not measure work quality. A defensible study defines the fuel basis—higher or lower heating value—and the boundary around auxiliary loads.
Combined Gas–Vapor Power Cycles
A Brayton topping cycle supplies high-temperature work; its exhaust feeds a heat-recovery steam generator and Rankine bottoming cycle. Pinch temperatures and stack limits couple the cycles, so detailed design is iterative rather than a simple sum of nameplate efficiencies.
Recognition strategy
A topping-cycle output plus a bottoming-cycle output sharing one fuel boundary identifies a combined-cycle energy cascade calculation.
Given information
- Supplied screening inputs: gas net efficiency 0.36, exhaust heat available 44 MW, recovery fraction 0.82, and steam conversion efficiency 0.31.
- Fuel input is 100 MW on the stated supplier basis; reference environment is \(T_0=298\ \mathrm{K}\), \(P_0=101.3\ \mathrm{kPa}\) absolute.
Model and interpretation
The boundary encloses both steady open cycles and the heat-recovery unit. Heat input and net work output are positive; water properties and gas idealization are embedded only in the supplied factors. Values are approximate, mass flows close within each loop, and kinetic and potential energy are neglected.
\(=36.0\ \mathrm{MW}\).
\(=36.08\ \mathrm{MW}\), leaving \(7.92\ \mathrm{MW}\) unrecovered.
\(=11.1848\ \mathrm{MW}\).
\(=47.18\%\).
Verification
Units: Dimensionless supplied factors multiply megawatts, leaving all heat and work rates consistently in megawatts.
Signs and direction: Both cycle outputs are positive, and unrecovered stack energy remains an outward loss rather than false output.
Plausibility: Steam rejection is \(0.69(36.08)=24.8952\ \mathrm{MW}\). Then \(47.1848+24.8952+7.92+20\)
\(=100\ \mathrm{MW}\), closing the stated boundary. Steady cycle mass and stored energy changes vanish. A full entropy/exergy audit at the declared environment must show \(\dot S_{gen}\ge0\) and \(\dot X_d=T_0\dot S_{gen}\ge0\); insufficient state data preclude inventing those values.
Engineering interpretation
Heat recovery adds about 11.2 MW without additional stated fuel, but this estimate is not a performance guarantee. Vendor maps, pinch analysis, ambient corrections, auxiliary loads, emissions constraints, and uncertainty bands are required before release.
Chapter recap
- Define component control volumes, cycle mass bases, signs, and absolute state coordinates before selecting equations.
- Use traceable phase-appropriate properties and distinguish ideal isentropic endpoints from actual efficiencies.
- Evaluate Rankine modifications by complete heat, work, moisture, entropy, exergy, and cost consequences.
- Before moving on, be able to close a cycle ledger, reject negative entropy generation, and explain what further evidence a professional decision requires.