Rankine Cycle in Steam Power Plants: Diagrams & Formulas

Posted on Dec 24, 2024 in Technology

A steam power plant converts heat energy into mechanical work by utilizing the Rankine cycle, which involves processes such as compression, heat addition, expansion, and heat rejection. Below, we will explain the P-V, T-S, and H-S diagrams of a steam power plant, the working of each cycle process, and the relevant formulas.

1. P-V Diagram (Pressure-Volume Diagram) for Steam Power Plants

The P-V diagram shows the relationship between pressure and volume during the Rankine cycle in a steam power plant. This cycle includes four key processes: isentropic compression, isobaric heat addition, isentropic expansion, and isobaric heat rejection.

Steps of the Rankine Cycle:

  • Process 1-2 (Isentropic Compression – Pump):
    • Water (liquid) is pumped from low pressure to high pressure.
    • The volume is very small, and the pressure increases significantly while the temperature remains almost constant.
    • The pump requires work input, and the water is compressed into a high-pressure liquid state.
    • Equation:
      \(W_{\text{pump}} = \frac{V}{\eta_{\text{pump}}} (P_2 – P_1)\)
      where \(V\) is the specific volume and \(P_1, P_2\) are the initial and final pressures.
  • Process 2-3 (Isobaric Heat Addition – Boiler):
    • The high-pressure water is heated at constant pressure in the boiler, turning it into steam.
    • The specific volume increases significantly as the liquid water turns into saturated or superheated steam.
    • Heat input to the system:
      \(Q_{\text{in}} = m \cdot (h_3 – h_2)\)
      where \(m\) is the mass flow rate, and \(h_3, h_2\) are the enthalpies at points 3 and 2, respectively.
  • Process 3-4 (Isentropic Expansion – Turbine):
    • The steam expands through the turbine, performing work. The steam loses energy, and both pressure and temperature decrease.
    • The volume increases significantly during expansion.
    • Work output from the turbine:
      \(W_{\text{turbine}} = m \cdot (h_3 – h_4)\)
  • Process 4-1 (Isobaric Heat Rejection – Condenser):
    • The low-pressure steam is condensed back into water at constant pressure by rejecting heat to the surroundings.
    • The specific volume decreases significantly, and the steam returns to the liquid phase.
    • Heat rejected:
      \(Q_{\text{out}} = m \cdot (h_4 – h_1)\)

P-V Diagram:

P (Pressure)
|
|          3
|         / |
|        /  |
|       /   |
|      1    / |
|      \   / |
|       \ /  |
|        \/   4
| --------------> V (Volume)

In the P-V diagram:

  • The area under the curve represents the work done by the steam in the turbine and the pump.
  • The area between processes 1-2 and 3-4 represents the heat added and rejected.

2. T-S Diagram (Temperature-Entropy Diagram) for Steam Power Plants

The T-S diagram visually represents the entropy (S) and temperature (T) changes during the Rankine cycle.

Steps of the Rankine Cycle on T-S Diagram:

  • Process 1-2 (Isentropic Compression – Pump): The liquid water is compressed in the pump, which is represented by a nearly vertical line (since entropy change is negligible).
  • Process 2-3 (Isobaric Heat Addition – Boiler): The water is heated at constant pressure, which is represented by a horizontal line as entropy increases.
  • Process 3-4 (Isentropic Expansion – Turbine): The steam expands through the turbine, reducing its pressure and temperature while increasing entropy. The expansion is represented by a downward curve in the T-S diagram.
  • Process 4-1 (Isobaric Heat Rejection – Condenser): The steam is cooled and condensed at constant pressure. The entropy decreases as heat is rejected to the surroundings.

T-S Diagram:

T (Temperature)
^
|          3
|         /|
|        / |
|       /  |
|      1   / | 4
| ----+---->
|         S (Entropy)
  • The area under the curve between points 2-3 and 3-4 represents the heat input, while the area between points 4-1 represents the heat rejection.

3. H-S Diagram (Enthalpy-Entropy Diagram) for Steam Power Plants

The H-S diagram, also known as Mollier diagram, shows the relationship between enthalpy (H) and entropy (S). It is useful for calculating the work done and heat added or rejected during each process.

Steps of the Rankine Cycle on H-S Diagram:

  • Process 1-2 (Isentropic Compression – Pump): The pump requires energy to compress the liquid water. The enthalpy increases, but the entropy remains nearly constant.
  • Process 2-3 (Isobaric Heat Addition – Boiler): The water absorbs heat at constant pressure, and the enthalpy increases while entropy also increases.
  • Process 3-4 (Isentropic Expansion – Turbine): The steam expands through the turbine, and its enthalpy decreases while the entropy increases.
  • Process 4-1 (Isobaric Heat Rejection – Condenser): The steam is condensed back to water. The enthalpy decreases, and the entropy decreases as heat is rejected to the surroundings.

H-S Diagram:

H (Enthalpy)
^
|          3
|         / |
|        /  |
|       /   |
|      1    / | 4
| ----+---->
|         S (Entropy)
  • The area under the curve represents the work output and heat input, while the vertical distances between processes indicate changes in enthalpy and entropy.

4. Efficiency of Steam Power Plants (Rankine Cycle)

The thermal efficiency of a steam power plant using the Rankine cycle can be calculated using the formula:

\(\eta = \frac{\text{Net Work Output}}{\text{Total Heat Input}} = 1 – \frac{Q_{\text{out}}}{Q_{\text{in}}}\)

Where:

  • \(Q_{\text{in}}\) is the heat supplied to the system in the boiler.
  • \(Q_{\text{out}}\) is the heat rejected in the condenser.

The net work output is the difference between the work output from the turbine and the work input to the pump:

\(W_{\text{net}} = W_{\text{turbine}} – W_{\text{pump}}\)

Where:

  • \(W_{\text{turbine}} = m \cdot (h_3 – h_4)\)
  • \(W_{\text{pump}} = m \cdot (h_2 – h_1)\)

The efficiency improves with:

  • A higher temperature and pressure in the boiler (leading to higher \(Q_{\text{in}}\)).
  • A lower temperature and pressure in the condenser (leading to lower \(Q_{\text{out}}\)).

5. Key Formulas and Relations:

  1. Pump Work:
    \(W_{\text{pump}} = m \cdot (h_2 – h_1)\)
    where \(h_2\) and \(h_1\) are the enthalpies at points 2 and 1, respectively.
  2. Turbine Work:
    \(W_{\text{turbine}} = m \cdot (h_3 – h_4)\)
    where \(h_3\) and \(h_4\) are the enthalpies at points 3 and 4, respectively.
  3. Heat Input (Boiler):
    \(Q_{\text{in}} = m \cdot (h_3 – h_2)\)
  4. Heat Output (Condenser):
    \(Q_{\text{out}} = m \cdot (h_4 – h_1)\)
  5. Thermal Efficiency:
    \(\eta = 1 – \frac{Q_{\text{out}}}{Q_{\text{in}}}\)

Conclusion

The Rankine cycle is the backbone of steam power plants, and understanding the processes using P-V, T-S, and H-S diagrams helps engineers design more efficient systems. The efficiency of the system depends on the quality of steam extraction, temperature, and pressure conditions. The key formulas allow for calculating the work done by the turbine, the heat added in the boiler, and the thermal efficiency of the system.