10.1 Rankine cycle and its modifications

The Rankine cycle is the backbone of steam power plants, converting thermal energy into mechanical work. It consists of four main components: pump, boiler, turbine, and condenser, each playing a crucial role in the cycle's efficiency and power output.

Modifications to the basic Rankine cycle, such as reheating, regeneration, and superheating, can significantly improve its performance. These enhancements increase thermal efficiency, work output, and steam quality, making power plants more effective and economical in generating electricity.

Rankine cycle components and processes

Basic Rankine cycle components

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• The basic Rankine cycle consists of four main components: pump, boiler, turbine, and condenser
  • Pump raises the pressure of the working fluid (water) to the boiler pressure
  • Boiler converts the high-pressure liquid water into superheated steam by adding heat at constant pressure
  • Turbine extracts work from the superheated steam, reducing its pressure and temperature
  • Condenser converts the low-pressure steam back to liquid water by rejecting heat at constant pressure

Thermodynamic processes in a Rankine cycle

• The working fluid (water) undergoes four processes in a basic Rankine cycle:
  1. Isentropic compression in the pump (liquid water)
  2. Constant-pressure heat addition in the boiler (liquid to superheated steam)
  3. Isentropic expansion in the turbine (superheated steam)
  4. Constant-pressure heat rejection in the condenser (steam to liquid water)
• The phase change of the working fluid from liquid to vapor and back to liquid allows for efficient heat transfer and work extraction
  • Latent heat of vaporization enables high heat transfer rates in the boiler
  • High enthalpy difference between superheated steam and liquid water maximizes work output in the turbine

Thermal efficiency and Carnot efficiency

• Thermal efficiency of a basic Rankine cycle is determined by the ratio of net work output to heat input
  • Net work output is the difference between turbine work and pump work
  • Heat input is the energy transferred to the working fluid in the boiler
• Carnot efficiency sets an upper limit for the thermal efficiency of a Rankine cycle operating between a given set of high and low temperatures
  • Actual Rankine cycles have lower efficiencies due to irreversibilities (friction, heat transfer across finite temperature differences, non-isentropic processes)

Rankine cycle performance analysis

Applying the First and Second Laws of Thermodynamics

• First Law of Thermodynamics (energy balance) is applied to analyze each component in the Rankine cycle
  • Consider heat transfer, work, and changes in enthalpy for each component
  • Steady-flow energy equation: $Q - W = \Delta H$
• Second Law of Thermodynamics is used to determine irreversibilities and losses in the cycle
  • Entropy generation due to friction, heat transfer across finite temperature differences, and non-isentropic processes
  • Isentropic efficiency of turbine and pump: $\eta_{turbine} = \frac{h_{in} - h_{out,actual}}{h_{in} - h_{out,isentropic}}$, $\eta_{pump} = \frac{h_{out,isentropic} - h_{in}}{h_{out,actual} - h_{in}}$

Improving Rankine cycle efficiency

• Increase the average temperature at which heat is added to the working fluid
  • Increase boiler pressure (supercritical Rankine cycles)
  • Increase turbine inlet temperature (superheating)
• Decrease the average temperature at which heat is rejected from the working fluid
  • Lower condenser pressure (vacuum conditions)
  • Use a lower temperature cooling medium (river water, seawater)
• Minimize irreversibilities in the cycle components
  • Reduce friction losses in pipes and components
  • Improve heat exchanger effectiveness (boiler and condenser)

Steam quality and turbine performance

• Steam quality at the turbine outlet is a critical parameter for turbine performance and longevity
  • Excessive moisture (low steam quality) can cause erosion and damage to turbine blades
  • Maintain steam quality above 90% at the turbine outlet for safe and efficient operation
• Reheat cycles can improve steam quality at the turbine outlet
  • Steam is reheated after partial expansion in the turbine, increasing its temperature and quality
  • Reheating also increases the average temperature of heat addition, improving cycle efficiency

Rankine cycle modifications and efficiency

Reheating

• Steam is expanded in stages, with additional heat input between the stages
  • Increases the average temperature of heat addition
  • Improves steam quality at the turbine outlet
  • Increases cycle efficiency and work output
• Reheat pressure is optimized based on the trade-off between efficiency gains and increased complexity/cost

Regeneration (feedwater heating)

• Extract a portion of the steam at intermediate points in the turbine
  • Use extracted steam to preheat the feedwater before it enters the boiler
  • Reduces the amount of heat input required in the boiler
  • Increases cycle efficiency by reducing irreversibilities in the boiler
• Open feedwater heaters (direct contact) and closed feedwater heaters (heat exchangers) are used for regeneration
  • Multiple feedwater heaters at different pressure levels can be employed for better heat recovery

Superheating

• Heat the steam to higher temperatures before it enters the turbine
  • Increases the work output and efficiency of the cycle
  • Improves steam quality at the turbine outlet
  • Limited by the maximum allowable temperature of the boiler and turbine materials (creep and corrosion resistance)

Lowering condenser pressure

• Reduce the condenser pressure (and thus the temperature) to increase cycle efficiency
  • Decreases the average temperature of heat rejection
  • Requires larger and more expensive condensers
  • May be limited by the available cooling medium temperature (ambient air, cooling water)

Combined cycles (e.g., Rankine-Brayton)

• Utilize the waste heat from a gas turbine (Brayton cycle) to generate steam for a Rankine cycle
  • Gas turbine exhaust acts as the heat source for the steam generator
  • Increases the overall efficiency by maximizing the use of high-temperature heat
  • Achieves higher efficiencies than either cycle alone (up to 60% in modern combined cycle power plants)

Energy and mass balances in Rankine cycles

Applying the steady-flow energy equation

• First Law of Thermodynamics for open systems: $Q - W = \Delta H + \Delta KE + \Delta PE$
  • In most cases, changes in kinetic energy (KE) and potential energy (PE) are negligible
  • Steady-flow energy equation simplifies to: $Q - W = \Delta H$
• Apply the steady-flow energy equation to each component of the Rankine cycle
  • Pump: $W_p = \dot{m}(h_{out} - h_{in})$
  • Boiler: $Q_b = \dot{m}(h_{out} - h_{in})$
  • Turbine: $W_t = \dot{m}(h_{in} - h_{out})$
  • Condenser: $Q_c = \dot{m}(h_{in} - h_{out})$

Using thermodynamic property tables and charts

• Steam tables and thermodynamic property charts are used to determine the state properties of the working fluid at various points in the cycle
  • Properties include temperature, pressure, specific volume, enthalpy, and entropy
  • Interpolation may be necessary for intermediate states
• Mollier diagram (enthalpy-entropy chart) is particularly useful for visualizing the Rankine cycle processes
  • Isentropic processes appear as vertical lines on the Mollier diagram
  • Constant-pressure processes (boiler and condenser) appear as nearly horizontal lines

Calculating mass flow rate and cycle efficiency

• Mass flow rate of the working fluid is determined based on the desired power output and the specific work of the turbine and pump
  • $\dot{m} = \frac{P_{net}}{w_{net}} = \frac{P_{net}}{w_t - w_p}$
• Thermal efficiency of the Rankine cycle is the ratio of net work output to heat input
  • $\eta_{th} = \frac{W_{net}}{Q_{in}} = \frac{W_t - W_p}{Q_b}$
  • Can also be expressed in terms of specific enthalpy differences: $\eta_{th} = \frac{(h_3 - h_4) - (h_2 - h_1)}{h_3 - h_2}$

Analyzing the effect of operating parameters

• Boiler pressure: Increasing boiler pressure generally improves cycle efficiency
  • Higher pressure increases the average temperature of heat addition
  • Supercritical Rankine cycles operate above the critical point of water (22.1 MPa)
• Condenser pressure: Lowering condenser pressure improves cycle efficiency
  • Lower pressure reduces the average temperature of heat rejection
  • Limited by the available cooling medium temperature and the size/cost of the condenser
• Turbine inlet temperature: Higher turbine inlet temperature increases cycle efficiency and work output
  • Limited by the maximum allowable temperature of the boiler and turbine materials
  • Superheating is used to achieve higher turbine inlet temperatures while maintaining acceptable steam quality

Solving problems with Rankine cycle modifications

• Reheating: Analyze the reheat cycle as two separate Rankine cycles in series
  • First cycle: from the boiler to the high-pressure turbine and the reheater
  • Second cycle: from the reheater to the low-pressure turbine and the condenser
  • Total work output is the sum of work from both turbine stages
• Regeneration: Account for the extraction of steam at intermediate points in the turbine
  • Extracted steam reduces the mass flow rate through the remaining turbine stages
  • Feedwater heaters increase the enthalpy of the feedwater, reducing the heat input required in the boiler
  • Perform energy balances on each feedwater heater to determine the extraction mass flow rates
• Combined cycles: Analyze the gas turbine (Brayton) cycle and the steam (Rankine) cycle separately
  • Gas turbine exhaust heat is the input for the steam generator in the Rankine cycle
  • Total power output is the sum of the gas turbine and steam turbine outputs
  • Overall efficiency is the ratio of the total power output to the heat input to the gas turbine