There is a thermodynamic process that fulfills almost half of the world’s power demands. This is called the Rankine cycle. In 2022, coal power plants (which generate nearly 40% of the world’s electricity) and nuclear power plants (which satisfy about 10%) employ the Rankine cycle in power generation. The cycle was first theorized in 1859 by William. J.M. Rankine, a Scottish engineer.
Fig. 1. Diagram of heat engine, where heat flows from a hot to a cold reservoir, Wikimedia Commons CC BY-SA 4.0
The Rankine cycle is a type of heat engine. In thermodynamics, a heat engine is a system that converts the flow of thermal energy (heat) into mechanical work. Heat is the transfer of thermal energy from higher to lower temperatures. In heat engines, this is achieved by having the heat flow from a hot reservoir to a cold reservoir. A mechanism, such as a turbine, in between the two reservoirs acts to physically convert some of the thermal energy into useful mechanical work.
Rankine Cycle Process
A Rankine cycle system requires four main components, a boiler, a turbine, a condenser, and a water pump. The most common and relevant example of the Rankine cycle is the steam turbine system, which uses water in its liquid and gaseous phases as the working fluid.
Many different types of liquid could hypothetically be used as the working liquid in the Rankine cycle. However, we use water for several reasons. It is very cheap and abundant, is chemically unreactive, non-toxic, and has helpful thermodynamic properties. For instance, water has one of the highest specific heat capacities of any common liquid, meaning it requires more energy than most other liquids to increase its temperature. This is useful as the water can have relatively high thermal energy, which can be converted to useful mechanical work before it begins to boil into steam.
Fig. 2. Diagram of the Rankine Cycle in a Steam Turbine System, Wikimedia Commons CC BY-SA 3.0
- The water pump is necessary to force water from the condenser back into the boiler at high pressures. This step requires work to be added to the system. Fortunately, the pump only requires approximately 2% of the turbine’s output power to work.
- The high-pressure water is heated from a fuel source within the boiler, which turns the water into steam. The fuel source might be the burning of coal or the controlled nuclear fission of uranium.
- The steam is then used to spin a turbine as the vapor expands. This part of the cycle converts thermal energy into useful mechanical work. The spinning turbine then drives an electric generator which produces electricity for our use.
- Some steam is immediately released into the surrounding environment as waste heat, which can be observed rising from a power plant’s cooling towers. This allows the remaining water to return to a liquid state inside the condenser after driving the turbine as it is cooled.
The condensation step is important for two reasons. Firstly, the larger the difference in temperature between the hot and cold reservoirs (boiler and condenser), the faster the heat will flow between them. This means the steam will travel faster and therefore drive the turbine faster, generating more electricity. Secondly, by condensing the steam back into water, we can reuse that water for the boiler.
Rankine Cycle Diagram
Fig. 3. Pressure-Volume diagram of each stage of the Rankine Cycle, Wikimedia Commons CC-BY-SA-2.0-DE
The Rankine cycle can be represented by a pressure-volume diagram. The four stages seen on the PV diagram above, correspond directly to the steps described in the Rankine cycle process section. The bell curve represents the vapor curve of water, but you do not need to worry about this at a high school level.
Rankine Cycle Equations
The term enthalpy is a thermodynamic quantity used to describe the total heat content within a system. It is defined as the sum of the system’s internal energy and the product of pressure and volume.
The work and heat for each stage of the Rankine cycle can be expressed using equations. It is helpful to understand the proofs of these equations to understand the topic.
$$H=E+PV$$
Where \(H\) is the Enthalpy of the working fluid, \(E\) is the internal energy of the system, \(P\) is pressure, and \(V\) is the volume. This is an important concept when considering the Rankine cycle. As enthalpy considers both the heat added and removed to the system and the work done on or by the system.
The first law of thermodynamics states the change of internal energy of a system is equal to the heat added to the system minus the work done by the system.
\[\Delta E=Q-W\]
Furthermore, the product of pressure and the change in volume is equal to the work done by the system to the environment. An increase in volume will mean that work is done by the system on the environment, while a decrease in volume would mean the environment performs work on the system.
\[W=P\Delta V\]
Rankine Cycle in Thermodynamics
All of the equations below assume an ideal Rankine cycle, where there are no inefficiencies. These inefficiencies can be caused by friction between different layers of fluid, heat loss to the surrounding environment, or mechanical losses due to friction in the turbine or pump. Ideal Rankine cycles cannot exist in reality but are helpful when discussing the fundamentals.
Change in Enthalpy
For each stage of the Rankine cycle, we must know the change of enthalpy to understand how heat and work are added and removed from the system and the environment. \(H_f\) is the final enthalpy and \(H_i\) is the initial enthalpy. We can use the enthalpy equation from earlier as a basis.
\[H_f-H_i=\Delta H\]
\[\Delta H=\Delta E+\Delta PV\]
Using calculus it is possible to derive the change in enthalpy equation below, which is more helpful when considering the Rankine Cycle. At each of the four stages of the Rankine Cycle, heat can either be added or removed from the system. Alternatively, work can be done on the system by the environment on the system, or work can be done on the environment by the system. The \(Q\) term in the equation below accounts for the heat, while the \(V\Delta P\) term describes the work done.
\[\Delta H=Q+V\Delta P\]
This deep dive will help you understand the derivation of the change of enthalpy equation used in the Rankine Cycle. Firstly, the change in the product of pressure and volume (\(\Delta PV\)) can be expanded using the product rule of differentiation from calculus.
\[\Delta H=\Delta E+P\Delta V+V\Delta P\]
Furthermore, the change in internal energy (\(\Delta E\)) can be substituted for the first law of thermodynamics discussed in the previous section.
\[\Delta H=(Q-W)+P\Delta V+V\Delta P\]
The work term is equal to the product of pressure and change in volume, as discussed earlier, this can be substituted into the change of enthalpy equation.
\[\Delta H=Q-P\Delta V+P\Delta V+V\Delta P\]
Lastly, the two \(P\Delta V\) terms cancel each other out to derive the final equation for the change of enthalpy in the Rankine Cycle.
\[\Delta H=Q+V\Delta P\]
First Stage: Compression of working fluid by the water pump
Work is done by the pump to compress the working fluid to high pressure. This is an adiabatic process, which means that there is no heat exchange between the system and the environment. Moreover, this stage is also considered an isochoric process, meaning the volume remains constant throughout.
However, something called adiabatic heating occurs, where the internal energy (temperature) of the working fluid increases due to the work done by the surroundings on the system. Remember that temperature refers to molecular kinetic energy, and heat is the transfer of thermal energy, so adding work to the system can force an increase in the molecular kinetic energy of the fluid particles.
To determine the work done by the pump on the system, we must calculate the change in enthalpy of the working fluid before and after adiabatic compression.
$$H_2-H_1=\Delta H$$
$$\Delta H=Q+V\Delta P$$
In an adiabatic process \(Q=0\) as there is no heat exchange, so that term can be ignored.
\[\Delta H=V\Delta P\]
The change of enthalpy during compression of the working fluid is equal to work done by the environment (pump) on the working fluid.
\[W_{in}=V\Delta P\]
This definition of work done \(W=V\Delta P\) can only be used in an open system. An open thermodynamic system allows both matter and energy to be exchanged between the system and the environment.
Most of the ideal thermodynamic systems you will learn about will be closed systems, where the thermodynamic system only exchanges energy with the environment and not matter.
In this stage of the Rankine cycle, as the volume remains constant, more matter (water) must be pumped into the system to increase the pressure of the working liquid. This requires work done by the pump.
Second Stage: Heating of the working fluid by the boiler
In the next stage, the boiler heats the working liquid. This is an isobaric process, meaning the pressure of the system remains constant throughout the entire process. As the working liquid is heated, it expands (increasing volume). To determine the heat added to the system we must calculate the change of enthalpy of the working fluid before and after boiling.
\[H_3-H_2=\Delta H\]
\[\Delta H=Q+V\Delta P\]
As there is no change in pressure during an isobaric process, meaning \(V\Delta P=0\), the term can be ignored.
\[\Delta H=Q_{in}\]
Third Stage: Expansion of working fluid to spin the turbine
Now, the vapor (steam) from the boiler expands adiabatically to spin the turbine and generate useful work. To determine the useful work done by the system to the environment we must calculate the change of enthalpy before and after expansion.
\[H_4-H_3=\Delta H\]
\[\Delta H=Q+V\Delta P\]
In an adiabatic process \(Q=0\) as there is no heat exchange, so that term can be ignored.
\[\Delta H=V\Delta P\]
The change of enthalpy during compression of the working fluid is equal to the useful work done by the working fluid on the environment (turbine).
\[W_{out}=V\Delta P\]
Fourth Stage: Condensation of working fluid
Finally, the vapor cools by transferring the heat content to the surrounding environment in an isobaric process. The cooled liquid is then used again by the water pump, completing the cycle. To determine the heat that is rejected by the working fluid we must calculate the change of enthalpy before and after the condensation.
\[H_1-H_4=\Delta H\]
\[\Delta H=Q+V\Delta P\]
As there is no change in pressure during an isobaric process \(V\Delta P=0\), then the term can be ignored.
\[\Delta H=Q_{out}\]
Thermal Efficiency of Rankine Cycle
The thermal efficiency of an ideal Rankine cycle is given by the ratio of the net work done by the system on the environment and the total heat produced in the cycle. Which represents the fraction of heat added that is converted to work done. Remember, this is for an ideal cycle where there is no friction, unintended heat loss, or mechanical losses.
\[\text{Thermal Efficiency}=\dfrac{\text{Word done to Turbine}-\text{Work done by Pump}}{\text{Heat added}}\]
\[\nu=\dfrac{W_{out}-W_{in}}{Q_{in}}=\dfrac{W_{net}}{Q_{in}}\]
Rankine Cycle - Key takeaways
- The Rankine cycle is an example of a heat engine, where the flow of heat is converted into useful work done.
- The larger the difference in temperature between the hot and cold reservoir, the faster the heat flow between them.
- The four main components of the Rankine cycle are the water pump, boiler, turbine, and condenser.
- There are four main stages to the Rankine cycle. Compression, heating, expansion, and condensation of the working fluid.
- The Rankine cycle can be represented by a pressure-volume diagram for each stage.
- Enthalpy is a thermodynamic quantity used to describe the total heat content within a system, it is the sum of the system’s internal energy and the product of pressure and volume. \(H=E+PV\).
- We can use enthalpy to help describe the changes in the heat and work done by the system and the environment during the Rankine cycle.
- The thermal efficiency of an ideal Rankine cycle is given by the ratio of the net work done by the system on the environment and the total heat produced in the cycle.
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