Rankine Cycle Efficiency: Steam Power Plant Analysis Using Tables
Most of India's electricity still comes from thermal power stations where coal heats water into steam that spins a turbine. The ideal thermodynamic model of such a plant is the Rankine cycle. Unlike the gas cycles used for IC engines, the working fluid changes phase between liquid and vapour, so ideal gas formulas do not apply. Instead you read properties from steam tables or a Mollier chart. This article explains the four components, sets out the energy equations and solves a complete cycle using standard steam table values.
Components and Processes
| State change | Component | Process | Energy equation per kg |
|---|---|---|---|
| 1 to 2 | Feed pump | Isentropic compression of liquid | w_p = h2 - h1 = v1 (p2 - p1) |
| 2 to 3 | Boiler and superheater | Constant pressure heat addition | q_in = h3 - h2 |
| 3 to 4 | Turbine | Isentropic expansion | w_t = h3 - h4 |
| 4 to 1 | Condenser | Constant pressure heat rejection | q_out = h4 - h1 |
Net work w_net = w_t - w_p Thermal efficiency eta = w_net / q_in Back work ratio bwr = w_p / w_t Steam rate SSC = 3600 / w_net (kg/kWh, w_net in kJ/kg) Dryness fraction x4 = (s4 - sf) / sfg at condenser pressure Enthalpy at 4 h4 = hf + x4 * hfg
Each component is an open steady-flow device, so these are just the steady flow energy equation with kinetic and potential energy changes neglected.
Worked Example: 3 MPa, 350 Degrees C, 10 kPa
Steam leaves the boiler at 3 MPa and 350 degrees C and expands in the turbine to a condenser pressure of 10 kPa. Find the thermal efficiency, dryness fraction at turbine exit and specific steam consumption.
Step 1: Properties from steam tables
| Property | Value | Source |
|---|---|---|
| Saturation temperature at 10 kPa | 45.81 C | Saturated pressure table |
| hf at 10 kPa | 191.81 kJ/kg | Saturated pressure table |
| hfg at 10 kPa | 2392.1 kJ/kg | Saturated pressure table |
| vf at 10 kPa | 0.00101 m^3/kg | Saturated pressure table |
| sf at 10 kPa | 0.6492 kJ/kg K | Saturated pressure table |
| sfg at 10 kPa | 7.4996 kJ/kg K | Saturated pressure table |
| h at 3 MPa, 350 C | 3116.1 kJ/kg | Superheated steam table |
| s at 3 MPa, 350 C | 6.7450 kJ/kg K | Superheated steam table |
Step 2: Pump
h1 = hf = 191.81 kJ/kg w_p = v1 (p2 - p1) = 0.00101 * (3000 - 10) = 3.02 kJ/kg h2 = 191.81 + 3.02 = 194.83 kJ/kg
Step 3: Turbine
s4 = s3 = 6.7450 kJ/kg K x4 = (6.7450 - 0.6492) / 7.4996 = 0.813 h4 = 191.81 + 0.813 * 2392.1 = 2136.1 kJ/kg w_t = 3116.1 - 2136.1 = 980.0 kJ/kg
Step 4: Boiler, efficiency and steam rate
q_in = 3116.1 - 194.83 = 2921.3 kJ/kg w_net = 980.0 - 3.02 = 977.0 kJ/kg eta = 977.0 / 2921.3 = 33.4 % bwr = 3.02 / 980.0 = 0.3 % SSC = 3600 / 977.0 = 3.68 kg/kWh
The back work ratio is tiny because pumping a liquid takes very little work compared with compressing a gas. This is the great advantage of the Rankine cycle over a gas turbine cycle, where the compressor can absorb a large share of the turbine output.
For comparison, a Carnot engine between the maximum and minimum temperatures of this cycle (623 K and 319 K) would reach about 48.8 percent. Rankine falls short mainly because much of the heat is added to cold feed water at temperatures far below 350 degrees C.
How Power Plants Raise Efficiency
- Lower condenser pressure: reduces the temperature at which heat is rejected. Limited by cooling water temperature and by wetness at the turbine exit.
- Higher boiler pressure: raises the average temperature of heat addition, but increases moisture at turbine exit unless combined with superheat or reheat.
- Higher superheat temperature: increases efficiency and dryness; limited by the creep strength of boiler and turbine materials.
- Reheat: steam is returned to the boiler after partial expansion and reheated, which keeps the final dryness high and allows higher boiler pressure.
- Regenerative feed heating: steam bled from the turbine preheats feed water, raising the average heat addition temperature. All large plants use several feed heaters.
Common Mistakes
- Taking h1 from the boiler pressure instead of the condenser pressure.
- Forgetting that s4 = s3 for isentropic expansion and reading h4 at the wrong state.
- Mixing kPa and MPa in the pump work formula; with v in m^3/kg and p in kPa, work comes out in kJ/kg.
- Using a superheated table value when the state is actually wet, or vice versa. Compare s with sg first.
Summary
The Rankine cycle uses a pump, boiler, turbine and condenser, and each is analysed with the steady flow energy equation and steam table properties. The worked example gives about 33 percent efficiency with a very small pump work. Raising boiler pressure and temperature, lowering condenser pressure, and adding reheat and feed heating are how modern plants push efficiency higher.