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Steam power cycles form the backbone of many thermal power plants worldwide, converting heat energy into mechanical work and eventually into electrical energy. Among these, the Rankine cycle is the fundamental thermodynamic cycle used in steam power plants. Understanding this cycle requires a solid grasp of the thermodynamic laws and the properties of steam, which govern how energy is transferred and transformed.
At its core, the Rankine cycle describes how water is converted into steam, expanded through a turbine to produce work, condensed back into water, and then pumped back to the boiler to repeat the process. This continuous loop efficiently harnesses thermal energy from fuel combustion or other heat sources.
Before diving into the cycle itself, it is essential to understand the behavior of steam under various pressures and temperatures, which is captured in steam tables and graphical tools like the Mollier diagram. These resources allow engineers to determine key properties such as enthalpy, entropy, and specific volume at different points in the cycle, enabling precise analysis and optimization.
The ideal Rankine cycle consists of four main thermodynamic processes that occur sequentially:
These four processes form a closed loop, continuously converting heat energy into useful work.
Steam, or water vapor, exists in different phases depending on temperature and pressure. To analyze the Rankine cycle accurately, we need to know properties such as pressure (P), temperature (T), specific volume (v), enthalpy (h), and entropy (s) at various points.
Steam tables provide these properties for saturated and superheated steam at different pressures and temperatures. They are essential tools for engineers to find the thermodynamic state of steam without complex calculations.
Steam tables typically include:
Using these tables, you can find enthalpy and entropy values at each state point of the Rankine cycle, which are crucial for calculating work and heat transfer.
| Property | Saturated Liquid (Water) | Saturated Vapor (Steam) | Superheated Steam (500°C) |
|---|---|---|---|
| Pressure (MPa) | 10 | 10 | 10 |
| Temperature (°C) | 311.0 | 311.0 | 500.0 |
| Enthalpy, \(h\) (kJ/kg) | 762.81 | 2796.4 | 3375.1 |
| Entropy, \(s\) (kJ/kg·K) | 2.138 | 6.507 | 6.592 |
| Specific Volume, \(v\) (m³/kg) | 0.001127 | 0.0199 | 0.0281 |
The performance of a Rankine cycle is measured by its thermal efficiency, which is the ratio of the net work output to the heat input supplied in the boiler.
Mathematically, thermal efficiency \(\eta\) is expressed as:
The work done by the turbine and the pump can be calculated using enthalpy differences at the respective state points:
The heat added in the boiler and heat rejected in the condenser are:
Steam enters the turbine at 15 MPa and 600°C and expands isentropically to a condenser pressure of 10 kPa. The condenser pressure is maintained at 10 kPa. Using steam tables, calculate the thermal efficiency of the ideal Rankine cycle.
Step 1: Identify state points and their properties.
State 1 (Turbine inlet): P₁ = 15 MPa, T₁ = 600°C
From steam tables: \(h_1 = 3583.1\) kJ/kg, \(s_1 = 6.6\) kJ/kg·K
State 2 (Turbine outlet): P₂ = 10 kPa, isentropic expansion -> \(s_2 = s_1 = 6.6\)
At 10 kPa, find \(h_2\) corresponding to \(s_2 = 6.6\) (superheated steam).
From steam tables: \(h_2 = 2300.0\) kJ/kg (approximate)
State 3 (Condenser outlet): Saturated liquid at 10 kPa
From steam tables: \(h_3 = 191.8\) kJ/kg
State 4 (Pump outlet): P₄ = 15 MPa, liquid water
Calculate pump work:
Specific volume of liquid water at 10 kPa, \(v = 0.001\) m³/kg
\(W_{pump} = v (P_4 - P_3) = 0.001 \times (15 \times 10^6 - 10 \times 10^3) = 14,990 \text{ Pa·m}^3/\text{kg} = 14.99 \text{ kJ/kg}\)
Enthalpy at pump outlet:
\(h_4 = h_3 + W_{pump} = 191.8 + 14.99 = 206.79 \text{ kJ/kg}\)
Step 2: Calculate turbine work and heat added.
\(W_{turbine} = h_1 - h_2 = 3583.1 - 2300.0 = 1283.1 \text{ kJ/kg}\)
\(Q_{in} = h_1 - h_4 = 3583.1 - 206.79 = 3376.31 \text{ kJ/kg}\)
Step 3: Calculate net work output and thermal efficiency.
\(W_{net} = W_{turbine} - W_{pump} = 1283.1 - 14.99 = 1268.11 \text{ kJ/kg}\)
\(\eta = \frac{W_{net}}{Q_{in}} = \frac{1268.11}{3376.31} = 0.3755 = 37.55\%\)
Answer: The thermal efficiency of the ideal Rankine cycle is approximately 37.55%.
In an ideal Rankine cycle, steam enters the turbine at 8 MPa and 480°C and expands isentropically to 0.008 MPa. Calculate the turbine work output and pump work input per kg of steam. Use steam tables for enthalpy values.
Step 1: Find enthalpy and entropy at turbine inlet (state 1).
At 8 MPa and 480°C:
\(h_1 = 3316.5\) kJ/kg, \(s_1 = 6.7\) kJ/kg·K
Step 2: Find enthalpy at turbine outlet (state 2) assuming isentropic expansion.
At 0.008 MPa, \(s_2 = s_1 = 6.7\)
From steam tables or interpolation, \(h_2 \approx 2300.0\) kJ/kg
Step 3: Calculate turbine work output:
\(W_{turbine} = h_1 - h_2 = 3316.5 - 2300.0 = 1016.5\) kJ/kg
Step 4: Calculate pump work input.
At condenser pressure 0.008 MPa, saturated liquid water specific volume \(v = 0.001\) m³/kg
Pressure rise in pump: \(P_2 - P_1 = 8 \times 10^6 - 0.008 \times 10^6 = 7.992 \times 10^6\) Pa
\(W_{pump} = v (P_2 - P_1) = 0.001 \times 7.992 \times 10^6 = 7.992\) kJ/kg
Answer: Turbine work output is 1016.5 kJ/kg and pump work input is 7.992 kJ/kg.
A Rankine cycle operates with steam entering the turbine at 15 MPa and 600°C. After expanding to 3 MPa, steam is reheated to 500°C and then expanded to 10 kPa. Calculate the thermal efficiency improvement due to reheat compared to the simple Rankine cycle.
Step 1: Identify state points for reheat cycle:
Step 2: Use steam tables to find enthalpies and entropies:
\(s_2 = s_1 = 6.6\), \(h_2 \approx 3120.0\) kJ/kg
\(h_3 = 3410.0\) kJ/kg, \(s_3 = 7.0\) kJ/kg·K
\(s_4 = s_3 = 7.0\), \(h_4 \approx 2400.0\) kJ/kg
\(h_5 = 191.8\) kJ/kg
\(W_{pump} = v (P_6 - P_5) = 0.001 \times (15 \times 10^6 - 10 \times 10^3) = 14.99\) kJ/kg
\(h_6 = h_5 + W_{pump} = 191.8 + 14.99 = 206.79\) kJ/kg
Step 3: Calculate turbine work output:
First stage turbine work: \(W_{t1} = h_1 - h_2 = 3583.1 - 3120.0 = 463.1\) kJ/kg
Second stage turbine work: \(W_{t2} = h_3 - h_4 = 3410.0 - 2400.0 = 1010.0\) kJ/kg
Total turbine work: \(W_{turbine} = 463.1 + 1010.0 = 1473.1\) kJ/kg
Step 4: Calculate heat added:
Heat added in boiler: \(Q_{in,boiler} = h_1 - h_6 = 3583.1 - 206.79 = 3376.31\) kJ/kg
Heat added in reheater: \(Q_{in,reheat} = h_3 - h_2 = 3410.0 - 3120.0 = 290.0\) kJ/kg
Total heat added: \(Q_{in} = 3376.31 + 290.0 = 3666.31\) kJ/kg
Step 5: Calculate net work and efficiency:
Net work: \(W_{net} = W_{turbine} - W_{pump} = 1473.1 - 14.99 = 1458.11\) kJ/kg
Thermal efficiency: \(\eta = \frac{1458.11}{3666.31} = 0.3976 = 39.76\%\)
Step 6: Compare with simple Rankine cycle efficiency (from Example 1, 37.55%)
Efficiency improvement due to reheat: \(39.76\% - 37.55\% = 2.21\%\)
Answer: Reheating improves the thermal efficiency by approximately 2.21%.
Using the Mollier diagram (h-s chart), find the enthalpy and entropy of steam at 5 MPa and 400°C for Rankine cycle analysis.
Step 1: Locate the pressure line corresponding to 5 MPa on the Mollier diagram.
Step 2: Move along the 5 MPa pressure line to the temperature curve of 400°C.
Step 3: Read the enthalpy (h) and entropy (s) values from the chart at this intersection.
From the Mollier diagram, approximate values are:
Answer: At 5 MPa and 400°C, steam has approximately enthalpy 3200 kJ/kg and entropy 6.8 kJ/kg·K.
Calculate the work done by the pump to raise the pressure of water from 10 kPa to 8 MPa. Use the specific volume of liquid water as 0.001 m³/kg.
Step 1: Write down the given data:
Step 2: Calculate pump work using:
\[ W_{pump} = v (P_2 - P_1) \]
\[ W_{pump} = 0.001 \times (8 \times 10^6 - 10 \times 10^3) = 0.001 \times 7,990,000 = 7,990 \text{ Pa·m}^3/\text{kg} \]
Since 1 Pa·m³ = 1 J,
\(W_{pump} = 7,990\) J/kg = 7.99 kJ/kg
Answer: The pump work required is approximately 7.99 kJ/kg.
When to use: When analyzing turbine inlet conditions to ensure realistic efficiency calculations.
When to use: During pump work calculations where high precision is not required.
When to use: For quick efficiency approximations in entrance exam problems.
When to use: During exams or practical situations where rapid property lookup is needed.
When to use: At the start of any Rankine cycle problem solving.
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