10.1 Rankine cycle and its modifications

Rankine Cycle Components and Processes

The Rankine cycle is the foundation of nearly all steam power plants. It converts thermal energy into mechanical work using water as the working fluid, cycling it between liquid and vapor phases. Understanding this cycle and its modifications is essential for analyzing real power plant performance.

Basic Rankine Cycle Components

Four components make up the basic Rankine cycle, and each one corresponds to a specific thermodynamic process:

• Pump: Raises the pressure of liquid water to boiler pressure. This requires a relatively small work input because compressing a liquid takes far less energy than compressing a gas.
• Boiler: Adds heat at constant pressure, converting high-pressure liquid water into superheated steam. The phase change from liquid to vapor absorbs a large amount of energy (latent heat of vaporization), which makes the boiler the primary site of heat input.
• Turbine: Extracts work from the superheated steam as it expands, reducing its pressure and temperature. The large enthalpy difference between superheated steam entering and the lower-energy fluid exiting is what drives the power output.
• Condenser: Rejects heat at constant pressure, converting low-pressure steam back into liquid water so the cycle can repeat.

Thermodynamic Processes in the Rankine Cycle

The working fluid undergoes four processes in sequence:

1. Process 1→2: Isentropic compression in the pump (liquid water)
2. Process 2→3: Constant-pressure heat addition in the boiler (liquid → superheated steam)
3. Process 3→4: Isentropic expansion in the turbine (superheated steam)
4. Process 4→1: Constant-pressure heat rejection in the condenser (steam → liquid water)

The key advantage of using a fluid that changes phase is efficiency: the latent heat of vaporization allows the boiler to absorb large amounts of energy at a nearly constant temperature, and the high enthalpy of superheated steam maximizes the work extracted by the turbine.

Thermal Efficiency and Carnot Efficiency

Thermal efficiency of the Rankine cycle is the ratio of net work output to heat input:

$\eta_{th} = \frac{W_{net}}{Q_{in}} = \frac{W_t - W_p}{Q_b}$

• Net work output = turbine work minus pump work
• Heat input = energy added to the working fluid in the boiler

The Carnot efficiency sets the theoretical upper limit for any heat engine operating between a high temperature $T_H$ and a low temperature $T_L$:

$\eta_{Carnot} = 1 - \frac{T_L}{T_H}$

Actual Rankine cycles always fall below this limit because of irreversibilities: friction in piping and components, heat transfer across finite temperature differences, and non-isentropic behavior in the turbine and pump.

Rankine Cycle Performance Analysis

Applying the First and Second Laws

The First Law (energy balance) is applied to each component individually using the steady-flow energy equation. For most Rankine cycle analyses, changes in kinetic and potential energy are negligible, so it simplifies to:

$Q - W = \dot{m}(h_{out} - h_{in})$

The Second Law identifies where losses occur. Entropy generation arises from friction, heat transfer across finite temperature differences, and deviations from ideal (isentropic) processes. To quantify these deviations, you use isentropic efficiencies:

• Turbine: $\eta_{turbine} = \frac{h_{in} - h_{out,actual}}{h_{in} - h_{out,isentropic}}$
• Pump: $\eta_{pump} = \frac{h_{out,isentropic} - h_{in}}{h_{out,actual} - h_{in}}$

Notice the structure is different for each: the turbine efficiency compares actual work produced to ideal work produced, while the pump efficiency compares ideal work consumed to actual work consumed. Both ratios are less than 1 for real devices.

Strategies for Improving Rankine Cycle Efficiency

There are three fundamental strategies, each tied directly to thermodynamic principles:

Increase the average temperature of heat addition:

• Raise boiler pressure (supercritical cycles operate above 22.1 MPa, the critical point of water)
• Increase turbine inlet temperature through superheating

Decrease the average temperature of heat rejection:

• Lower condenser pressure (operate under vacuum conditions)
• Use a colder cooling medium (river water, seawater)

Minimize irreversibilities:

• Reduce friction losses in pipes and components
• Improve heat exchanger effectiveness in the boiler and condenser

Steam Quality and Turbine Performance

Steam quality (the mass fraction of vapor in a liquid-vapor mixture) at the turbine outlet directly affects turbine longevity. Excessive moisture causes erosion and pitting on turbine blades. A common design rule: maintain steam quality above 90% at the turbine exit.

Reheat cycles address this problem. By partially expanding steam in a high-pressure turbine, then sending it back to the boiler for reheating before further expansion, you raise the steam quality at the final turbine outlet. Reheating also increases the average temperature of heat addition, which improves cycle efficiency.

Rankine Cycle Modifications

Reheating

In a reheat cycle, steam expands in stages with additional heat input between them:

1. Steam expands partially in the high-pressure (HP) turbine.
2. The partially expanded steam returns to the boiler and is reheated to a high temperature.
3. The reheated steam expands through the low-pressure (LP) turbine to the condenser pressure.

This modification increases the average temperature of heat addition, improves steam quality at the turbine outlet, and increases both cycle efficiency and net work output. The reheat pressure is chosen to balance efficiency gains against added complexity and cost.

Regeneration (Feedwater Heating)

Regeneration extracts a portion of steam from the turbine at intermediate pressures and uses it to preheat the feedwater before it enters the boiler. This reduces the heat input the boiler must supply and raises the average temperature at which heat is added to the cycle.

Two types of feedwater heaters are used:

• Open feedwater heaters (direct contact): Extracted steam mixes directly with the feedwater. These also serve as deaerators, removing dissolved gases.
• Closed feedwater heaters (shell-and-tube heat exchangers): Extracted steam and feedwater remain in separate streams, allowing operation at different pressures.

Real power plants often use multiple feedwater heaters at different pressure levels for better heat recovery. You'll need to perform an energy balance on each heater to find the extraction mass flow rates.

Superheating

Superheating raises the steam temperature well above the saturation temperature before it enters the turbine. This increases work output, improves cycle efficiency, and keeps steam quality high at the turbine exit. The practical limit is set by material constraints: boiler tubes and turbine blades must resist creep and high-temperature corrosion, which currently caps turbine inlet temperatures around 600°C for conventional materials.

Lowering Condenser Pressure

Reducing condenser pressure lowers the temperature at which heat is rejected, directly improving cycle efficiency. The trade-off: lower pressures require larger condensers (to handle the increased specific volume of steam) and are limited by the temperature of the available cooling medium. You can't condense steam at a temperature below that of your cooling water.

Combined Cycles (Rankine-Brayton)

A combined cycle pairs a gas turbine (Brayton cycle) with a steam turbine (Rankine cycle). The hot exhaust from the gas turbine, which would otherwise be wasted, serves as the heat source for the steam generator in the Rankine cycle.

This arrangement captures energy from both high-temperature combustion gases and the steam cycle, achieving overall efficiencies up to about 60% in modern plants. That's significantly higher than either cycle could achieve alone.

Energy and Mass Balances in Rankine Cycles

Applying the Steady-Flow Energy Equation

Starting from the First Law for open systems:

$Q - W = \dot{m}\left(\Delta h + \Delta KE + \Delta PE\right)$

Since changes in kinetic and potential energy are typically negligible in power plant components, the equation for each component becomes:

• Pump: $W_p = \dot{m}(h_2 - h_1)$
• Boiler: $Q_b = \dot{m}(h_3 - h_2)$
• Turbine: $W_t = \dot{m}(h_3 - h_4)$
• Condenser: $Q_c = \dot{m}(h_4 - h_1)$

Note the sign conventions: turbine work and condenser heat rejection are written as positive quantities here (energy leaving the system), while pump work and boiler heat are energy inputs.

Using Steam Tables and the Mollier Diagram

To evaluate these equations, you need enthalpy and entropy values at each state point. Steam tables provide properties (temperature, pressure, specific volume, enthalpy, entropy) for saturated and superheated states. Interpolation is often necessary for states that fall between tabulated values.

The Mollier diagram (enthalpy-entropy or $h$-$s$ chart) is especially useful for visualizing the Rankine cycle:

• Isentropic processes (ideal pump and turbine) appear as vertical lines
• Constant-pressure processes (boiler and condenser) appear as curves moving left to right

Plotting the cycle on a Mollier diagram makes it easy to see the effect of modifications like reheating or superheating on work output and steam quality.

Calculating Mass Flow Rate and Cycle Efficiency

To find the required mass flow rate for a given net power output:

$\dot{m} = \frac{\dot{W}_{net}}{w_{net}} = \frac{\dot{W}_{net}}{(h_3 - h_4) - (h_2 - h_1)}$

Thermal efficiency in terms of specific enthalpies at each state point:

$\eta_{th} = \frac{(h_3 - h_4) - (h_2 - h_1)}{h_3 - h_2}$

The numerator is the net specific work (turbine work minus pump work), and the denominator is the specific heat input in the boiler.

Effect of Operating Parameters

• Boiler pressure: Increasing boiler pressure generally improves efficiency by raising the average temperature of heat addition. Supercritical cycles operate above 22.1 MPa, where there is no distinct phase change from liquid to vapor.
• Condenser pressure: Lowering condenser pressure improves efficiency by reducing the average temperature of heat rejection. Practical limits depend on cooling water temperature and condenser size/cost.
• Turbine inlet temperature: Higher inlet temperatures increase both efficiency and work output, but are constrained by material limits. Superheating is the standard way to raise this temperature while keeping acceptable steam quality at the turbine exit.

Solving Problems with Cycle Modifications

Reheat cycle analysis:

1. Treat the HP turbine and LP turbine as separate expansion stages.

2. Calculate work output for each stage independently.

3. Add the reheat heat input ($Q_{reheat} = \dot{m}(h_5 - h_4)$ where states 4 and 5 are the reheater inlet and outlet).

4. Total work = HP turbine work + LP turbine work. Total heat input = boiler heat + reheat heat.

Regenerative cycle analysis:

1. Identify the extraction point(s) and the type of feedwater heater (open or closed).
2. Define a mass fraction $y$ for the extracted steam (fraction of total flow).
3. Perform an energy balance on each feedwater heater to solve for $y$.
4. Account for the reduced mass flow rate through the remaining turbine stages when calculating turbine work: $W_t = \dot{m}[(h_3 - h_6) + (1 - y)(h_6 - h_4)]$ for a single extraction at state 6.

Combined cycle analysis:

1. Analyze the gas turbine (Brayton) cycle first to find its power output and exhaust temperature.
2. Use the gas turbine exhaust as the heat source for the Rankine cycle's steam generator.
3. Total power = gas turbine power + steam turbine power.
4. Overall efficiency = total power output ÷ heat input to the gas turbine combustor.