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Rankine cycle
Rankine cycle
Rankine cycle
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Physical layout of the Rankine cycle
1. Pump, 2. Boiler, 3. Turbine, 4. Condenser
Thermodynamics
The classical Carnot heat engine
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Note: Conjugate variables in italics
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The Rankine cycle is an idealized thermodynamic cycle describing the process by which certain heat engines, such as steam turbines or reciprocating steam engines, allow mechanical work to be extracted from a fluid as it moves between a heat source and heat sink. The Rankine cycle is named after William John Macquorn Rankine, a Scottish polymath professor at Glasgow University.

Heat energy is supplied to the system via a boiler where the working fluid (typically water) is converted to a high-pressure gaseous state (steam) in order to turn a turbine. After passing over the turbine the fluid is allowed to condense back into a liquid state as waste heat energy is rejected before being returned to boiler, completing the cycle. Friction losses throughout the system are often neglected for the purpose of simplifying calculations as such losses are usually much less significant than thermodynamic losses, especially in larger systems.

Description

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The Rankine cycle closely describes the process by which steam engines commonly found in thermal power generation plants harness the thermal energy of a fuel or other heat source to generate electricity. Possible heat sources include combustion of fossil fuels such as coal, natural gas, and oil, use of mined resources for nuclear fission, renewable fuels like biomass and ethanol, and energy capture of natural sources such as concentrated solar power and geothermal energy. Common heat sinks include ambient air above or around a facility and bodies of water such as rivers, ponds, and oceans.

The ability of a Rankine engine to harness energy depends on the relative temperature difference between the heat source and heat sink. The greater the differential, the more mechanical power can be efficiently extracted out of heat energy, as per Carnot's theorem.

The efficiency of the Rankine cycle is limited by the high heat of vaporization of the working fluid. Unless the pressure and temperature reach supercritical levels in the boiler, the temperature range over which the cycle can operate is quite small. As of 2022, most supercritical power plants adopt a steam inlet pressure of 24.1 MPa and inlet temperature between 538°C and 566°C, which results in plant efficiency of 40%. However, if pressure is further increased to 31 MPa the power plant is referred to as ultra-supercritical, and one can increase the steam inlet temperature to 600°C, thus achieving a thermal efficiency of 42%.[1] This low steam turbine entry temperature (compared to a gas turbine) is why the Rankine (steam) cycle is often used as a bottoming[clarification needed] cycle to recover otherwise rejected heat in combined-cycle gas turbine power stations. The idea is that very hot combustion products are first expanded in a gas turbine, and then the exhaust gases, which are still relatively hot, are used as a heat source for the Rankine cycle, thus reducing the temperature difference between the heat source and the working fluid and therefore reducing the amount of entropy generated by irreversibility.

Rankine engines generally operate in a closed loop in which the working fluid is reused. The water vapor with condensed droplets often seen billowing from power stations is created by the cooling systems (not directly from the closed-loop Rankine power cycle). This "exhaust" heat is represented by the "Qout" flowing out of the lower side of the cycle shown in the T–s diagram below. Cooling towers operate as large heat exchangers by absorbing the latent heat of vaporization of the working fluid and simultaneously evaporating cooling water to the atmosphere.

While many substances can be used as the working fluid, water is usually chosen for its simple chemistry, relative abundance, low cost, and thermodynamic properties. By condensing the working steam vapor to a liquid, the pressure at the turbine outlet is lowered, and the energy required by the feed pump consumes only 1% to 3% of the turbine output power. These factors contribute to a higher efficiency for the cycle. The benefit of this is offset by the low temperatures of steam admitted to the turbine(s). Gas turbines, for instance, have turbine entry temperatures approaching 1500 °C. However, the thermal efficiencies of actual large steam power stations and large modern gas turbine stations are similar.

The four processes in the Rankine cycle

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T–s diagram of a typical Rankine cycle operating between pressures of 0.06 bar and 50 bar. Left from the bell-shaped curve is liquid, right from it is gas, and under it is saturated liquid–vapour equilibrium.

There are four processes in the Rankine cycle. The states are identified by numbers (in brown) in the T–s diagram.

Successive processes of the Rankine cycle
Name Summary Explanation
Process 1–2 Isentropic compression The working fluid is pumped from low to high pressure. As the fluid is a liquid at this stage, the pump requires little input energy.
Process 2–3 Constant pressure heat addition in boiler The high-pressure liquid enters a boiler, where it is heated at constant pressure by an external heat source to become a dry saturated vapour. The input energy required can be easily calculated graphically, using an enthalpy–entropy chart (h–s chart, or Mollier diagram), or numerically, using steam tables or software.
Process 3–4 Isentropic expansion The dry saturated vapour expands through a turbine, generating power. This decreases the temperature and pressure of the vapour, and some condensation may occur. The output in this process can be easily calculated using the chart or tables noted above.
Process 4–1 Constant pressure heat rejection in condenser The wet vapour then enters a condenser, where it is condensed at a constant pressure to become a saturated liquid.

In an ideal Rankine cycle the pump and turbine would be isentropic: i.e., the pump and turbine would generate no entropy and would hence maximize the net work output. Processes 1–2 and 3–4 would be represented by vertical lines on the T–s diagram and more closely resemble that of the Carnot cycle. The Rankine cycle shown here prevents the state of the working fluid from ending up in the superheated vapor region after the expansion in the turbine, [1] which reduces the energy removed by the condensers.

The actual vapor power cycle differs from the ideal Rankine cycle because of irreversibilities in the inherent components caused by fluid friction and heat loss to the surroundings; fluid friction causes pressure drops in the boiler, the condenser, and the piping between the components, and as a result the steam leaves the boiler at a lower pressure; heat loss reduces the net work output, thus heat addition to the steam in the boiler is required to maintain the same level of net work output.

Variables

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Heat flow rate to or from the system (energy per unit time)
Mass flow rate (mass per unit time)
Mechanical power consumed by or provided to the system (energy per unit time)
Thermodynamic efficiency of the process (net power output per heat input, dimensionless)
Isentropic efficiency of the compression (feed pump) and expansion (turbine) processes, dimensionless
The "specific enthalpies" at indicated points on the T–s diagram
The final "specific enthalpy" of the fluid if the turbine were isentropic
The pressures before and after the compression process

Equations

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defines the thermodynamic efficiency of the cycle as the ratio of net power output to heat input. As the work required by the pump is often around 1% of the turbine work output, it can be simplified:

Each of the next four equations[1] is derived from the energy and mass balance for a control volume.

When dealing with the efficiencies of the turbines and pumps, an adjustment to the work terms must be made:

Real Rankine cycle (non-ideal)

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Rankine cycle with superheat

In a real power-plant cycle (the name "Rankine" cycle is used only for the ideal cycle), the compression by the pump and the expansion in the turbine are not isentropic. In other words, these processes are non-reversible, and entropy is increased during the two processes. This somewhat increases the power required by the pump and decreases the power generated by the turbine.[2]

In particular, the efficiency of the steam turbine will be limited by water-droplet formation. As the water condenses, water droplets hit the turbine blades at high speed, causing pitting and erosion, gradually decreasing the life of turbine blades and efficiency of the turbine. The easiest way to overcome this problem is by superheating the steam. On the T–s diagram above, state 3 is at a border of the two-phase region of steam and water, so after expansion the steam will be very wet. By superheating, state 3 will move to the right (and up) in the diagram and hence produce a drier steam after expansion.

Variations of the basic Rankine cycle

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The overall thermodynamic efficiency can be increased by raising the average heat input temperature

of that cycle. Increasing the temperature of the steam into the superheat region is a simple way of doing this. There are also variations of the basic Rankine cycle designed to raise the thermal efficiency of the cycle in this way; two of these are described below.

Rankine cycle with reheat

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Rankine cycle with reheat

The purpose of a reheating cycle is to remove the moisture carried by the steam at the final stages of the expansion process. In this variation, two turbines work in series. The first accepts vapor from the boiler at high pressure. After the vapor has passed through the first turbine, it re-enters the boiler and is reheated before passing through a second, lower-pressure, turbine. The reheat temperatures are very close or equal to the inlet temperatures, whereas the optimal reheat pressure needed is only one fourth of the original boiler pressure. Among other advantages, this prevents the vapor from condensing during its expansion and thereby reducing the damage in the turbine blades, and improves the efficiency of the cycle, because more of the heat flow into the cycle occurs at higher temperature. The reheat cycle was first introduced in the 1920s, but was not operational for long due to technical difficulties. In the 1940s, it was reintroduced with the increasing manufacture of high-pressure boilers, and eventually double reheating was introduced in the 1950s. The idea behind double reheating is to increase the average temperature. It was observed that more than two stages of reheating are generally unnecessary, since the next stage increases the cycle efficiency only half as much as the preceding stage. Today, double reheating is commonly used in power plants that operate under supercritical pressure.

Regenerative Rankine cycle

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Regenerative Rankine cycle

The regenerative Rankine cycle is so named because after emerging from the condenser (possibly as a subcooled liquid) the working fluid is heated by steam tapped from the hot portion of the cycle. On the diagram shown, the fluid at 2 is mixed with the fluid at 4 (both at the same pressure) to end up with the saturated liquid at 7. This is called "direct-contact heating". The Regenerative Rankine cycle (with minor variants) is commonly used in real power stations.

Another variation sends bleed steam from between turbine stages to feedwater heaters to preheat the water on its way from the condenser to the boiler. These heaters do not mix the input steam and condensate, function as an ordinary tubular heat exchanger, and are named "closed feedwater heaters".

Regeneration increases the cycle heat input temperature by eliminating the addition of heat from the boiler/fuel source at the relatively low feedwater temperatures that would exist without regenerative feedwater heating. This improves the efficiency of the cycle, as more of the heat flow into the cycle occurs at higher temperature.

Organic Rankine cycle

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Main article: Organic Rankine cycle

The organic Rankine cycle (ORC) uses an organic fluid such as n-pentane[3] or toluene[4] in place of water and steam. This allows use of lower-temperature heat sources, such as solar ponds, which typically operate at around 70 –90 °C.[5] The efficiency of the cycle is much lower as a result of the lower temperature range, but this can be worthwhile because of the lower cost involved in gathering heat at this lower temperature. Alternatively, fluids can be used that have boiling points above water, and this may have thermodynamic benefits (See, for example, mercury vapour turbine). The properties of the actual working fluid have great influence on the quality of steam (vapour) after the expansion step, influencing the design of the whole cycle.

The Rankine cycle does not restrict the working fluid in its definition, so the name "organic cycle" is simply a marketing concept and the cycle should not be regarded as a separate thermodynamic cycle.

Supercritical Rankine cycle

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The Rankine cycle applied using a supercritical fluid[6] combines the concepts of heat regeneration and supercritical Rankine cycle into a unified process called the regenerative supercritical cycle (RGSC). It is optimised for temperature sources 125–450 °C.

See also

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References

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Revisions and contributorsEdit on WikipediaRead on Wikipedia

Rankine cycle

View on Grokipedia
from Grokipedia
The Rankine cycle is a thermodynamic cycle that serves as the fundamental model for vapor power systems, converting heat energy into mechanical work in a closed-loop process typically using water as the working fluid. It comprises four primary reversible processes: isentropic compression of liquid in a pump to increase pressure, isobaric heat addition in a boiler to produce high-pressure vapor, isentropic expansion of the vapor in a turbine to generate work, and isobaric heat rejection in a condenser to return the fluid to its liquid state. Developed by Scottish engineer and physicist William John Macquorn Rankine (1820–1872) in the mid-19th century as an idealized representation of steam engine operation, the cycle provides a practical framework for analyzing the efficiency of heat engines operating between two temperature reservoirs, though its thermal efficiency is inherently lower than that of the Carnot cycle due to the lower average temperature of heat addition. This cycle underpins the majority of conventional steam power plants, including those fueled by , , , and concentrated , where it enables large-scale by driving turbines connected to electrical generators. Key performance metrics, such as (typically 30–45% in practical implementations), depend on factors like pressure, maximum temperature, and condenser temperature, with higher values achieved through modifications including the vapor to avoid wet in the , reheating after partial expansion to boost work output, and regeneration via feedwater heaters to preheat the pumped liquid using extracted . These enhancements address real-world irreversibilities like and losses, making the Rankine cycle a cornerstone of modern energy production while variants like the adapt it for lower-temperature sources using organic fluids for applications in recovery and .

Fundamentals

Overview

The Rankine cycle is an idealized used in engines to convert into mechanical work, serving as the fundamental model for vapor power systems such as steam turbines. It operates by circulating a , typically , through a closed loop where is absorbed to generate high-pressure vapor, which then drives mechanical components to produce power. This cycle forms the basis for efficient energy conversion in large-scale systems, enabling the practical harnessing of from various sources. The Rankine cycle finds primary application in steam power plants for , where it powers the majority of global power infrastructure. sources include fossil fuels such as and in conventional plants, nuclear reactors in atomic power stations, and renewable options like concentrating solar systems and geothermal flash plants. In these settings, the cycle's adaptability to high-temperature heat inputs makes it essential for reliable, scalable power production across diverse energy portfolios. Compared to the , which offers the highest theoretical efficiency between two temperature limits but proves impractical for vapor-based systems due to the difficulties in managing isothermal processes amid phase changes, the Rankine cycle provides a more feasible alternative through its use of constant pressure addition and rejection. This design avoids the challenges of compressing two-phase mixtures and enables straightforward and , aligning better with real-world constraints. The basic components include a for vaporizing the via input, a for work extraction during expansion, a condenser for cooling and liquefying the exhaust vapor, and a for returning the to . Overall , measured as the net work output divided by input, generally achieves 30-40% in operational power plants, reflecting practical limitations like irreversibilities and heat losses.

Historical Development

The Rankine cycle originated from advancements in steam engine technology during the 18th century, particularly through the improvements made by , who enhanced the efficiency of Newcomen engines by introducing a separate condenser and other mechanisms that reduced energy loss, laying the groundwork for more practical heat engines. Building on these foundations, Scottish engineer William John Macquorn Rankine formalized the cycle in 1859 through his seminal work Manual of the Steam Engine and Other Prime Movers, providing the first systematic thermodynamic analysis of steam power processes and establishing it as a theoretical model for heat-to-work conversion. During the from the late 1700s to the mid-1800s, the principles underlying the Rankine cycle were adopted in reciprocating engines, which powered factories, mines, and transportation, driving widespread industrialization despite their relatively low compared to modern standards. The late 19th century marked a pivotal transition from reciprocating engines to , with Swedish engineer Carl Gustaf Patrick de Laval developing the first impulse in 1883 and British engineer inventing the multi-stage reaction turbine in , enabling higher speeds and power outputs. Key milestones included Parsons' demonstration of the first commercial power plant in , which generated 7.5 kW and showcased the potential for electrical integration. By the early , turbines were routinely coupled with electrical generators, revolutionizing power as seen in General Electric's 500-kW Curtis turbine in 1901 and widespread adoption in utility plants. In the mid-20th century, the Rankine cycle evolved toward high-pressure and supercritical designs to boost in plants, with the first commercial supercritical unit operational at the Philo Power Plant in in 1957, operating above water's critical point of 22.1 MPa and 374°C for improved performance. As of 2025, the cycle remains central to plants, where it converts fission to , and renewable systems like concentrated solar facilities, with ongoing incremental gains through advanced materials and controls rather than fundamental shifts.

Cycle Processes

The Four Processes

The Rankine cycle operates as a in a , typically using as the , and involves four sequential processes that convert into mechanical work under steady-flow conditions. These processes occur in components such as the , , , and condenser, assuming reversible behavior in the ideal case and familiarity with phase transitions between and vapor states. In the first process (1-2), isentropic compression takes place in the , where subcooled liquid exiting the condenser is pressurized to the boiler's . The mechanism involves adiabatic work input to the incompressible liquid, resulting in a small increase in and with negligible . The purpose is to elevate the fluid's efficiently, enabling subsequent addition at elevated levels while minimizing expenditure due to the liquid's low . The second process (2-3) is isobaric heat addition in the , where the pressurized absorbs at constant from an external source. This involves three stages: preheating the subcooled to the saturation temperature, evaporative to produce saturated vapor, and optional to raise the vapor above its saturation point, increasing its . The mechanism relies on steady to drive the phase change and thermal excitation, producing high-energy ready for expansion. The purpose is to maximize the availability of for conversion into work in the subsequent process. During the third process (3-4), isentropic expansion occurs in the , where the high-pressure, high-temperature flows through and expands adiabatically. The mechanism entails the 's driving turbine blades, leading to a decrease in , , and while generating mechanical work output, often resulting in a wet vapor mixture at the exit. The purpose is to extract useful work from the fluid's stored , powering generators or other machinery. The fourth process (4-1) involves isobaric heat rejection in the condenser, where the low-pressure exhaust steam from the turbine is cooled at constant pressure. The mechanism includes latent heat removal to condense the vapor back into saturated or subcooled liquid, with heat transferred to a cooling medium such as water or air. The purpose is to restore the working fluid to its initial liquid state, facilitating efficient recirculation and completing the cycle while rejecting waste heat to the environment.

Thermodynamic Representation

The thermodynamic representation of the Rankine cycle is typically illustrated using temperature-entropy (T-s) and pressure-volume (P-v) diagrams, which provide visual insights into the state changes of the working fluid, usually water, during the cycle's processes. On the T-s diagram, the cycle is plotted with entropy (s) on the horizontal axis and temperature (T) on the vertical axis, featuring the saturation dome that separates the liquid, vapor, and two-phase regions. State 1 represents the saturated liquid at the condenser pressure, located on the liquid saturation line under the dome. From state 1 to 2, isentropic compression in the pump occurs nearly vertically upward along a constant entropy line, moving the fluid to a compressed liquid state at the boiler pressure, with minimal entropy change due to the low specific volume of the liquid. The process from 2 to 3 involves constant-pressure heat addition in the boiler, following the constant-pressure line that rises from the compressed liquid state 2 to the saturated liquid line, proceeds horizontally through the two-phase region under the dome, and then rises into the superheated vapor region to state 3 as superheated vapor. Isentropic expansion from 3 to 4 in the turbine follows another vertical line downward, ending in the wet vapor region under the dome at state 4. Finally, constant-pressure heat rejection from 4 to 1 is a horizontal line leftward through the two-phase region back to the saturated liquid at state 1. The area under the curve from 2 to 3 represents the heat input (q_in), while the area under the curve from 4 to 1 represents the heat rejected (q_out), allowing for visual estimation of thermal efficiency as the ratio of the difference in these areas to q_in. Constant-pressure lines are horizontal in the two-phase region due to constant saturation temperature but curve upward in the single-phase regions. The P-v diagram, with pressure (P) on the vertical axis and specific volume (v) on the horizontal axis, highlights the phase boundaries and volume changes more prominently than the T-s diagram. State 1 is again the saturated liquid at low pressure, near the left side of the dome. The pump compression from 1 to 2 is a nearly vertical line with a small increase in v, reflecting the incompressible nature of the liquid and resulting in state 2 as compressed liquid at high pressure. Heat addition from 2 to 3 at constant high pressure appears as a horizontal line extending far to the right into the superheated vapor region, with state 3 having a large specific volume. Isentropic expansion from 3 to 4 slopes downward to the right initially then leftward as it enters the two-phase dome, reaching state 4 as wet vapor at low pressure with reduced v. Heat rejection from 4 to 1 at constant low pressure is a horizontal line leftward to the saturated liquid dome boundary. Unlike gas power cycles such as the Otto or Diesel, which operate entirely in the vapor phase with closed loops avoiding phase changes, the Rankine cycle's P-v diagram crosses the saturation dome, illustrating the benefits of latent heat utilization for higher work output. The enclosed area within the cycle loop directly represents the net work output (w_net), as it quantifies the difference between expansion and compression work. These diagrams facilitate the interpretation of cycle performance by visually delineating the state points—1 (saturated liquid), 2 (compressed liquid), 3 (superheated vapor), and 4 (wet vapor)—and revealing key thermodynamic relationships without numerical computation. They are particularly advantageous for identifying irreversibilities, such as non-isentropic processes that cause deviations from vertical lines on the T-s diagram or slanted lines on the P-v diagram, and for spotting opportunities for improvements like increasing superheat or reducing condenser pressure to enlarge the work area.

Mathematical Model

Key Variables

The key variables in the Rankine cycle analysis encompass thermodynamic state properties at the four cycle points and overarching parameters that define operating conditions, enabling the evaluation of energy transfers and cycle performance. These properties are primarily for water as the working fluid and are determined using thermodynamic tables or charts due to the fluid's behavior across phase boundaries. At state point 1 (condenser exit and pump inlet), the fluid is saturated liquid under condenser pressure P1=PlowP_1 = P_\text{low}, with temperature T1T_1 equal to the saturation temperature at PlowP_\text{low}, specific volume v1v_1 as the saturated liquid volume vfv_f, enthalpy h1h_1 as the saturated liquid enthalpy hfh_f, and entropy s1s_1 as the saturated liquid entropy sfs_f. State point 2 (pump exit and boiler inlet) features compressed liquid at boiler pressure P2=PhighP_2 = P_\text{high}, where entropy s2≈s1s_2 \approx s_1 under ideal isentropic compression, temperature T2T_2 slightly exceeds T1T_1, specific volume v2≈v1v_2 \approx v_1, and enthalpy h2h_2 accounts for the small work input during compression. At point 3 (boiler exit and turbine inlet), the fluid is superheated vapor at P3=PhighP_3 = P_\text{high} and maximum cycle temperature T3=TmaxT_3 = T_\text{max}, with enthalpy h3h_3 and entropy s3s_3 obtained from superheated vapor tables. Point 4 (turbine exit and condenser inlet) is typically a two-phase mixture at P4=PlowP_4 = P_\text{low}, defined by vapor quality x4x_4 (the mass fraction of vapor), where h4=hf+x4(hg−hf)h_4 = h_f + x_4 (h_g - h_f), s4=sf+x4(sg−sf)s_4 = s_f + x_4 (s_g - s_f), v4=vf+x4(vg−vf)v_4 = v_f + x_4 (v_g - v_f), and T4=Tsat(Plow)T_4 = T_\text{sat}(P_\text{low}). Cycle parameters include PhighP_\text{high} (typically 10–100 bar in conventional steam power plants), condenser PlowP_\text{low} (often around 0.1 bar under saturated conditions), maximum TmaxT_\text{max} (ranging from 500–600°C), and exit quality x4x_4 (ideally near 0.9–1.0 to minimize ). The m˙\dot{m} (in kg/s) scales the cycle for capacity, while rate Q˙in\dot{Q}_\text{in} to the and rejection rate Q˙out\dot{Q}_\text{out} from the condenser (both in kW), along with pump work rate W˙pump\dot{W}_\text{pump} and work rate W˙turbine\dot{W}_\text{turbine} (in kW), quantify energy interactions; these are often analyzed on a per-unit-mass basis before applying m˙\dot{m}. Standard units include in bar or kPa, in °C or , in m³/kg, enthalpy in kJ/kg, and entropy in kJ/(kg·). These variables facilitate analysis by encapsulating phase transitions—from subcooled liquid at point 1, to compressed liquid at point 2, superheated vapor at point 3, and wet vapor at point 4—allowing precise property retrieval from steam tables to model non-ideal behaviors like latent heat absorption without relying on ideal gas approximations.

Governing Equations

The governing equations for the ideal Rankine cycle are derived by applying the first law of thermodynamics to each component as a steady-flow open system, assuming steady-state operation, negligible changes in kinetic and potential energies, and no heat or work losses other than those specified for each process. The first law for a steady-flow process per unit mass simplifies to q−w=h2−h1q - w = h_2 - h_1, where qq is heat transfer, ww is shaft work (positive when done by the system), and hh is specific enthalpy. For the ideal cycle, processes 1-2 (pump) and 3-4 (turbine) are isentropic, so s2=s1s_2 = s_1 and s4=s3s_4 = s_3, enabling determination of outlet states from inlet conditions using thermodynamic property relations. For the pump (process 1-2), the process is adiabatic (q=0q = 0) and reversible. From the first law, w=h1−h2<0w = h_1 - h_2 < 0 (negative, indicating work input to the system). Approximating the working fluid as an incompressible liquid, the enthalpy change is h2−h1=v1(P2−P1)h_2 - h_1 = v_1 (P_2 - P_1), so the magnitude of the pump work input per unit mass is wpump=h2−h1≈v1(P2−P1)>0w_\text{pump} = h_2 - h_1 \approx v_1 (P_2 - P_1) > 0. For mass flow rate m˙\dot{m}, the total pump work input is W˙pump=m˙v1(P2−P1)\dot{W}_\text{pump} = \dot{m} v_1 (P_2 - P_1). For the (process 2-3), there is no shaft work (w=0w = 0), so qin=h3−h2q_\text{in} = h_3 - h_2. The total input rate is therefore Q˙in=m˙(h3−h2)\dot{Q}_\text{in} = \dot{m} (h_3 - h_2). For the (process 3-4), the process is adiabatic (q=0q = 0) and reversible, yielding wturbine=h3−h4w_\text{turbine} = h_3 - h_4 (work output). The total work rate is W˙turbine=m˙(h3−h4)\dot{W}_\text{turbine} = \dot{m} (h_3 - h_4). For the condenser (process 4-1), there is no shaft work (w=0w = 0), so q=h1−h4<0q = h_1 - h_4 < 0 (heat leaving the system). The magnitude of the heat rejected per unit mass is thus qout=h4−h1>0q_\text{out} = h_4 - h_1 > 0. The total heat rejection rate is Q˙out=m˙(h4−h1)\dot{Q}_\text{out} = \dot{m} (h_4 - h_1). The net work output per unit mass is wnet=wturbine−wpump=(h3−h4)−v1(P2−P1)w_\text{net} = w_\text{turbine} - w_\text{pump} = (h_3 - h_4) - v_1 (P_2 - P_1), and the total net work rate is W˙net=W˙turbine−W˙pump\dot{W}_\text{net} = \dot{W}_\text{turbine} - \dot{W}_\text{pump}. The is η=wnetqin=1−qoutqin=1−h4−h1h3−h2\eta = \frac{w_\text{net}}{q_\text{in}} = 1 - \frac{q_\text{out}}{q_\text{in}} = 1 - \frac{h_4 - h_1}{h_3 - h_2}.

Ideal and Real Cycles

Ideal Rankine Cycle

The of the ideal Rankine cycle is calculated using the formula η=1−h4−h1h3−h2,\eta = 1 - \frac{h_4 - h_1}{h_3 - h_2}, where h1h_1, h2h_2, h3h_3, and h4h_4 are the specific enthalpies at the inlet, outlet, inlet, and outlet, respectively. These enthalpies are determined from steam tables based on the cycle conditions, assuming isentropic processes in the and . For typical operating conditions of a pressure of 100 bar and temperature of 500°C, with a condenser pressure of 0.1 bar, the enthalpies are approximately h1=192h_1 = 192 kJ/kg, h2=202h_2 = 202 kJ/kg, h3=3375h_3 = 3375 kJ/kg, and h4=2090h_4 = 2090 kJ/kg, yielding an efficiency of about 40%. The of the ideal Rankine cycle improves with higher or greater superheat , as these parameters raise the average at which is supplied to the , thereby increasing the net work output relative to input. Lowering the condenser also enhances by reducing the average of rejection, which widens the overall span of the cycle. These effects are evident in performance analyses using steam property data, where, for instance, increasing superheat from saturation to 500°C at a fixed can boost by 5-10 percentage points. Compared to the operating between the same maximum and minimum s, the ideal Rankine cycle achieves lower because addition occurs over a range of s from the saturation point to the superheat , rather than isothermally at the maximum . Similarly, rejection is isothermal at the condenser TLT_L, but the effective mean for addition TH,meanT_{H,\text{mean}} is lower than the peak THT_H, resulting in η=1−TL/TH,mean<1−TL/TH=ηCarnot\eta = 1 - T_L / T_{H,\text{mean}} < 1 - T_L / T_H = \eta_{\text{Carnot}}. For example, under conditions yielding 40% Rankine , the corresponding Carnot might exceed 60%. The Rankine cycle assumes perfect isentropic compression and expansion, with no irreversibilities, and includes the work in calculations, though this work is small—typically 1-3% of the work—and sometimes approximated as negligible in preliminary estimates. These assumptions establish theoretical benchmarks but highlight limits, as real cycles incorporate additional losses.

Deviations in Real Cycles

In real Rankine cycles, deviations from the ideal model arise primarily due to irreversibilities in the components, leading to reduced compared to the theoretical predictions of 40-50% for ideal cases. These non-idealities include , losses, drops, and property effects, which increase generation and alter the work and heat transfers. Pump inefficiencies stem from non-isentropic compression caused by mechanical , , and hydraulic losses, resulting in higher actual work input than the ideal isentropic value. The actual pump work is given by Wpump, real=Wpump, idealηpumpW_{\text{pump, real}} = \frac{W_{\text{pump, ideal}}}{\eta_{\text{pump}}}, where the isentropic efficiency ηpump\eta_{\text{pump}} typically ranges from 70% to 90% in practical steam power plants. This inefficiency elevates the at the pump outlet, thereby decreasing the net cycle work output. Boiler losses occur due to drops across the complex circuits, including fittings, headers, and tubing, as well as incomplete and imperfect from the combustion gases to the . These effects reduce the effective heat input QinQ_{\text{in}} by 2-5% or more, depending on the type and burner design, as incomplete combustion leads to unburned and excess air requirements that lower the overall energy transfer . drops can amount to 5-10% of the boiler inlet , further diminishing the cycle's performance. Turbine inefficiencies arise from non-isentropic expansion due to aerodynamic , steam leakage past blade tips, finite stage expansions, and partial admission losses, causing the actual exhaust to be higher than the isentropic value. The isentropic is defined as ηt=h3−h4,realh3−h4,s\eta_t = \frac{h_3 - h_{4,\text{real}}}{h_3 - h_{4,s}}, where h3h_3 is the inlet , h4,realh_{4,\text{real}} is the actual outlet , and h4,sh_{4,s} is the isentropic outlet ; typical values for turbines range from 80% to 90%. In the low-pressure stages, in the expanding exacerbates losses through , reduced aerodynamic , and wet , potentially dropping stage efficiencies by an additional 5-10%. Condenser deviations include pressure drops in the and of the condensate below the saturation , which increases the actual rejection QoutQ_{\text{out}} beyond the ideal saturated state. , often by 5-10°C to ensure proper drainage and prevent in the feedwater system, is constrained by the cooling water inlet , typically limiting the minimum condenser pressure to 5-10 kPa and raising QoutQ_{\text{out}} by 1-3%. These factors, combined with air inleakage and , contribute to higher turbine backpressure and reduced net work. Overall, these deviations result in real Rankine cycle thermal of 30-40% in conventional steam power plants, significantly lower than ideal benchmarks due to the cumulative effects of component losses and generation. Blade losses and in low-pressure stages alone can account for 5-15% of the work reduction. To account for these in , steam tables are used to determine real enthalpies by incorporating irreversibilities, such as through isentropic corrections and balances, enabling more accurate performance predictions.

Advanced Variations

Reheat Rankine Cycle

The reheat Rankine cycle enhances the basic Rankine cycle by introducing a reheating to improve performance in high-pressure steam power plants. In this configuration, from the at and maximum expands isentropically in a to an intermediate . The partially expanded steam is then returned to the or a separate reheater, where it is heated at constant back to the original maximum before entering a low-pressure for further isentropic expansion to the condenser . This setup effectively divides the turbine work into two while elevating the temperature profile during heat addition. The primary processes in the reheat cycle modify the expansion step of the ideal Rankine cycle: the isentropic expansion from state 3 (boiler exit) to state 4 is split into expansion from state 3 to 4' in the high-pressure turbine, followed by constant-pressure reheating from 4' to 5, and then expansion from 5 to 6 in the low-pressure turbine. The reheat heat input is calculated as Qreheat=m˙(h5−h4′)Q_{\text{reheat}} = \dot{m} (h_5 - h_{4'}), where m˙\dot{m} is the of and hh denotes specific . The work and other processes ( and compression) remain similar to the basic cycle. The is given by ηreheat=(h3−h4′)+(h5−h6)−Wpump(h3−h2)+(h5−h4′),\eta_{\text{reheat}} = \frac{ (h_3 - h_{4'}) + (h_5 - h_6) - W_{\text{pump}} }{ (h_3 - h_2) + (h_5 - h_{4'}) }, where WpumpW_{\text{pump}} is the pump work input, typically small compared to turbine work. This equation reflects the increased net work output relative to total heat input, with the reheat term contributing to higher average heat addition temperature. Key benefits of the reheat cycle include an efficiency improvement of 4-5% over the simple Rankine cycle, achieved by raising the average temperature at which heat is added, which approaches the Carnot limit more closely. Additionally, reheating reduces moisture content in the low-pressure turbine exhaust steam—typically limiting it to under 10% wetness fraction—thereby minimizing erosion of turbine blades caused by liquid droplets and extending equipment life. These advantages are particularly valuable in cycles operating at high boiler pressures, where excessive moisture would otherwise occur during deep expansion. Despite these gains, the reheat cycle involves drawbacks such as increased from additional reheater components, , and control systems, which add to the plant design. It is thus economically justified mainly in large-scale applications, such as coal-fired or plants with capacities exceeding 500 MW, where the efficiency benefits offset the higher upfront . The reheat Rankine cycle was introduced in the mid-1920s as pressures rose in power generation, enabling reliable operation at elevated conditions.

Regenerative Rankine Cycle

The regenerative Rankine cycle modifies the basic cycle by extracting steam from the turbine at intermediate pressure stages to preheat the feedwater before it enters the boiler, thereby reducing the heat input required from the external source and improving overall efficiency. This preheating is accomplished using feedwater heaters, which transfer heat from the extracted steam to the subcooled liquid, minimizing the temperature difference during heat addition in the boiler and approaching the ideal of reversible heat transfer. In the configuration, steam is bled from the after partial expansion and directed to one or more feedwater heaters arranged in series or parallel along the feedwater line. The preheated feedwater is then returned to the at a higher , while the condensed steam from the heaters is either mixed back into the cycle or pumped separately. Large power plants typically employ 4 to 8 stages of regeneration to optimize the profile, with extraction points selected at pressures that match the saturation conditions suitable for the heaters. Feedwater heaters are classified into open and closed types. Open feedwater heaters operate on direct mixing, where extracted steam bubbles into the feedwater stream, achieving excellent due to intimate contact, but requiring the streams to enter at the same ; the outlet is a single saturated liquid stream at the heater . In contrast, closed feedwater heaters use a shell where condenses on one side without mixing, and the feedwater is heated on the other side before being pumped to the next stage; this allows operation at different pressures but introduces additional pumping requirements for the condensate drain. Open heaters are simpler and common for lower stages, while closed heaters predominate in higher-pressure sections to maintain cycle integrity. The processes in a regenerative cycle involve multiple extraction points during expansion, typically between the high-pressure and low-pressure stages (corresponding to points 3 and 4 in the basic cycle T-s diagram). For an open , the balance ensures : (1 - y) h_6 + y h_4 = h_7, where y is the fraction of steam extracted, h_4 is the at the extraction point, h_6 is the incoming feedwater , and h_7 is the preheated outlet (saturated at heater ). This yields y = (h_7 - h_6) / (h_4 - h_6). For closed heaters, the balance is across the : \dot{m}{\text{extracted}} (h{\text{extracted}} - h_{\text{condensate}}) = \dot{m}{\text{feed}} (h{\text{preheated}} - h_{\text{incoming}}), with no mass mixing. These balances determine the extraction fractions needed to achieve desired preheating temperatures. The primary benefits include raising the average temperature of heat addition, which enhances the Carnot efficiency factor and reduces exergy losses from irreversible mixing in the boiler; this typically improves thermal efficiency by 5-10% over the basic cycle, depending on the number of heaters and operating conditions. Additionally, preheating reduces thermal stresses on boiler tubes by minimizing the temperature gradient between feedwater and combustion gases, extending equipment life. The modified efficiency calculation accounts for reduced turbine work output due to extractions but offsets it with lower boiler heat input: ηregen=Wturb,net−WpumpQin,reduced\eta_{\text{regen}} = \frac{W_{\text{turb,net}} - W_{\text{pump}}}{Q_{\text{in,reduced}}}, often approximated as ηbasic+ΔTavg/Thigh×ηCarnot\eta_{\text{basic}} + \Delta T_{\text{avg}} / T_{\text{high}} \times \eta_{\text{Carnot}}, where ΔTavg\Delta T_{\text{avg}} reflects the temperature rise from regeneration. Limitations arise from the increased system complexity, including additional , valves, and pumps, which elevate capital and costs; optimal extraction pressures are determined through economic analyses balancing gains against these expenses. In practice, regeneration is most viable in large-scale where the improvements justify the added hardware.

Organic Rankine Cycle

The (ORC) is a variation of the Rankine cycle designed for low-temperature recovery, employing organic working fluids with high molecular weights instead of to enable efficient power generation from sources typically below 400°C. Unlike steam-based cycles, which struggle with at low temperatures due to the need for operation, the ORC maintains positive pressures throughout, facilitating compact system design and operation with sources as low as 80°C. This adaptation preserves the core four processes—pumping, , expansion, and —but shifts to subcritical conditions with reduced pressures (often below 20 bar) and temperatures, making it ideal for low-grade where traditional steam cycles yield efficiencies under 5%. Fluid selection in ORC systems prioritizes organic compounds with low boiling points and favorable thermodynamic properties to match low-temperature sources, such as refrigerants like R134a or hydrocarbons like n-pentane, which exhibit dry vapor expansion curves that avoid liquid droplet formation during expansion and subsequent erosion. These fluids, often with molecular weights exceeding 100 g/mol, enable at temperatures 50–100°C lower than , enhancing and cycle performance while minimizing isentropic losses. Selection criteria also consider critical temperature alignment with the heat source to maximize vapor density and expander efficiency. ORC applications focus on waste heat recovery from industrial processes, such as exhaust gases at 100–300°C, alongside geothermal and solar thermal systems, where net efficiencies range from 10% to 20%—significantly lower than the 30–40% of high-temperature steam cycles but valuable for otherwise unused energy. These systems convert low-grade heat into electricity at scales from kilowatts to megawatts, with modular designs supporting distributed generation in biomass plants or automotive exhaust recovery. Advantages include compact turbines suited to organic vapors' higher densities and the absence of vacuum condensers, reducing complexity and enabling partial-load operation above 50% efficiency. Post-2000 advancements, driven by regulatory incentives for renewables, have scaled modular ORC units up to 10 MW for biomass and industrial cogeneration, enhancing reliability through non-corrosive fluids. Despite these benefits, systems face challenges including of organic fluids above 300–350°C, which limits maximum operating temperatures and requires careful fluid matching, alongside higher specific costs (often 2–3 times that of cycles per kW due to specialized components). Environmental concerns arise from some fluids' (e.g., older HFCs) or high , prompting shifts to low-ODP alternatives like hydrofluoroolefins. Performance optimization centers on the approximate given by the Carnot-like formula: ηORC=1−TlowThigh, mean\eta_{\text{ORC}} = 1 - \frac{T_{\text{low}}}{T_{\text{high, mean}}} where TlowT_{\text{low}} is the condenser temperature and Thigh, meanT_{\text{high, mean}} is the mean evaporator temperature (both in Kelvin), further refined by expander isentropic efficiencies of 70–85%, which directly impact net power output.

Supercritical Rankine Cycle

The supercritical Rankine cycle operates above the critical point of water (374°C and 221 bar), where there is no distinct boiling phase transition; instead, the working fluid transitions continuously from a liquid-like state to a dense, gas-like supercritical fluid during heat addition, enabling a single-phase process that eliminates the two-phase evaporation stage typical of subcritical cycles. This configuration allows for more efficient heat transfer in the boiler, as the fluid's properties change gradually without the formation of bubbles or steam drums. In practice, supercritical cycles typically employ pressures exceeding 250 bar and maximum temperatures above 550°C, while ultra-supercritical variants push these limits further to around 300 bar and 600–700°C, often incorporating double reheat stages to manage thermal stresses and enhance performance. These plants utilize once-through boilers, which feedwater directly through tubes without recirculation drums, simplifying the but requiring precise control to achieve uniform heating. Components must withstand extreme conditions, necessitating specialized materials such as nickel-based alloys (e.g., ) to resist , oxidation, and creep at high temperatures. Efficiency in supercritical cycles typically achieves 38–42%, while ultra-supercritical variants reach 42–48%, compared to 33–38% in subcritical plants, primarily due to a closer approximation to the Carnot limit through higher average addition temperatures and reduced irreversibilities in the single-phase process. This improvement translates to lower specific fuel consumption and reduced CO₂ emissions per in coal-fired applications, potentially cutting emissions by 10–15% relative to subcritical units. The first commercial supercritical plant, the 125 MW Philo Unit 6 in , , entered operation in 1957, marking the transition from experimental to practical deployment. By 2025, supercritical and ultra-supercritical technologies dominate new coal-fired capacity in , particularly in and , where they support "clean coal" initiatives by maximizing from domestic resources; many recent installations utilize ultra-supercritical conditions exceeding 600°C. Ongoing research focuses on and cycles targeting 700°C+ for efficiencies above 50%, driven by international collaborations including the Department of Energy's Advanced Ultra-Supercritical (A-USC) program. However, drawbacks include significantly higher —up to 20–30% more than subcritical plants—due to exotic materials and complex controls, as well as heightened sensitivity to feedwater impurities, which can cause rapid or tube failures. In the temperature-entropy (T-s) , the heat addition process features a characteristic "slide region" near the pseudo-critical line, where peaks, mimicking pseudo-boiling but introducing challenges in flow stability and distribution.

References

  1. https://civil.colorado.edu/~silverst/aren2110/Power-RefrigerationCyclesChapter.pdf
  2. https://eaglepubs.erau.edu/introductiontoaerospaceflightvehicles/chapter/thermodynamic-foundations/
  3. https://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/node65.html
  4. https://www.sciencedirect.com/topics/engineering/rankine-cycle
  5. https://users.wpi.edu/~sullivan/ES3001/Lectures/ch08-jms.pdf
  6. https://www.sciencedirect.com/topics/engineering/organic-rankine-cycle
  7. https://www.thermopedia.com/content/1072/
  8. https://www.sciencedirect.com/topics/engineering/steam-rankine-cycle
  9. https://wires.onlinelibrary.wiley.com/doi/10.1002/wene.420
  10. https://www.sfu.ca/~mbahrami/ENSC%20461/Notes/Vapor%20Power%20Cycles.pdf
  11. https://engineering.purdue.edu/~wassgren/teaching/ME20000/NotesAndReading/Lec33_Reading_Wassgren.pdf
  12. https://www.britannica.com/technology/Watt-steam-engine
  13. https://www.britannica.com/science/Rankine-cycle
  14. https://www.britannica.com/biography/William-John-Macquorn-Rankine
  15. https://www.thoughtco.com/steam-in-the-industrial-revolution-1221643
  16. https://www.sciencedirect.com/topics/engineering/steam-engines
  17. https://encyclopedia.che.engin.umich.edu/turbines/
  18. https://www.infinityturbine.com/search/orc-09/the-historical-evolution-turbomachinery-006.html
  19. https://collection.sciencemuseumgroup.org.uk/objects/co51109/parsons-steam-turbine-generator-1884
  20. https://www.powermag.com/history-of-power-the-evolution-of-the-electric-generation-industry/
  21. https://asmedigitalcollection.asme.org/memagazineselect/article/122/09/68/368339/Power-to-the-PeopleA-Century-of-Mechanical
  22. https://www.asme.org/about-asme/engineering-history/landmarks/228-philo-6-steam-electric-generating-unit
  23. https://www.sciencedirect.com/topics/engineering/supercritical-cycle
  24. https://www.sciencedirect.com/org/science/article/pii/S0199859525000375
  25. https://medium.com/%40ashwinpalo/from-theory-to-practice-exploring-real-world-applications-of-the-rankine-cycle-9eeaea30e44c
  26. https://world-nuclear.org/news-and-media/press-statements/world-nuclear-performance-report-2025-nuclear-delivers-record-breaking-year-in-electricity-generation
  27. https://courses.ems.psu.edu/eme812/node/708
  28. https://ocw.mit.edu/courses/res-2-008-thermodynamics-and-climate-change-summer-2020/mitres-2-008su22_ch5.pdf
  29. https://www3.nd.edu/~powers/ame.50531/notes.pdf
  30. https://www.qrg.northwestern.edu/thermo/design-library/rankine/rankine.html
  31. https://www.nuclear-power.com/nuclear-engineering/thermodynamics/thermodynamic-cycles/rankine-cycle-steam-turbine-cycle/theory-of-rankine-cycle-equations-and-calculation/
  32. https://www.eeer.org/journal/view.php?doi=10.4491/eer.2020.683
  33. https://www.power-eng.com/environmental-emissions/maintaining-maximum-efficiency-in-power-generation-units/
  34. https://ntrs.nasa.gov/api/citations/19840025233/downloads/19840025233.pdf
  35. https://staff.emu.edu.tr/devrimaydin/documents/meng246/chapter_10_lecture.pdf
  36. https://www.sciencedirect.com/topics/physics-and-astronomy/rankine-cycle
  37. https://www.mdpi.com/2076-3417/14/16/6999
  38. https://indico.ictp.it/event/a10196/session/12/contribution/9/material/0/1.pdf
  39. https://ocw.mit.edu/courses/2-60j-fundamentals-of-advanced-energy-conversion-spring-2020/1437d5925093b270c871160ec861fd12_MIT2_60s20_lec16.pdf
  40. https://www.qrg.northwestern.edu/thermo/design-library/regen/regen.html
  41. https://web1.eng.famu.fsu.edu/~alvi/eml4304/webpage_old/documents/rankine/Rankine_Lecture1_2.pdf
  42. https://documents.uow.edu.au/~buyung/ThermodynamicLecture.pdf
  43. https://www.electrical4u.com/rankine-cycle-for-closed-feed-water-heaters-and-cogeneration/
  44. https://www.sciencedirect.com/science/article/abs/pii/S0360544206001812
  45. https://asmedigitalcollection.asme.org/gasturbinespower/article/137/10/100801/373802/Organic-Rankine-Cycle-Power-Systems-From-the
  46. https://pubs.aip.org/aip/acp/article-pdf/doi/10.1063/5.0138444/18085983/020015_1_5.0138444.pdf
  47. https://www.researchgate.net/publication/347847086_Working_fluid_selection_of_Organic_Rankine_Cycles
  48. http://www1.eere.energy.gov/manufacturing/intensiveprocesses/pdfs/waste_heat_recovery.pdf
  49. https://www.turboden.com/products/2463/orc-system
  50. https://inpressco.com/wp-content/uploads/2016/03/Paper420-26.pdf
  51. https://www.researchgate.net/publication/277673991_Experimental_Investigation_of_the_Performance_of_a_Hermetic_Screw-Expander_Organic_Rankine_Cycle
  52. https://docs.lib.purdue.edu/cgi/viewcontent.cgi?article=1207&context=coolingpubs
  53. https://www.sciencedirect.com/science/article/pii/S294991782300024X
  54. https://www.chemtreat.com/solutions/water-essentials-handbook-fundamentals-of-industrial-boilers-and-steam-generation-systems/
  55. https://nickelinstitute.org/media/2746/nickelmagapril_fb_final.pdf
  56. https://journals.sagepub.com/doi/10.1177/16878132231221065
  57. https://www.researchgate.net/publication/335727384_Thermodynamic_Analysis_of_Supercritical_and_Subcritical_Rankine_Cycles
  58. https://www.ourenergypolicy.org/wp-content/uploads/2014/01/R43343.pdf
  59. https://www.sciencedirect.com/topics/engineering/ultrasupercritical-plant
  60. https://www.gktoday.in/advanced-ultra-supercritical-ausc-thermal-plant/
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