# Thermal Efficiency of Steam Turbine

Typically, most nuclear power plants operate multi-stage condensing steam turbines. In modern nuclear power plants, the overall thermal efficiency is about one-third (33%), so 3000 MWth of thermal power from the fission reaction is needed to generate 1000 MWe of electrical power.

Supercritical fossil fuel power plants operated at supercritical pressure (i.e., greater than 22.1 MPa) have efficiencies of around 43%.

The thermal efficiency of the simple Rankine cycle and in terms of specific enthalpies would be:

It is a very simple equation, and for the determination of the thermal efficiency, you can use data from steam tables.

In general, the thermal efficiency, ηth, of any heat engine is defined as the ratio of the work it does, W, to the heat input at the high temperature, QH.

The thermal efficiency, ηth, represents the fraction of heat, QH, that is converted to work. Since energy is conserved according to the first law of thermodynamics and energy cannot be converted to work completely, the heat input, QH, must equal the work done, W, plus the heat that must be dissipated as waste heat QC into the environment. Therefore we can rewrite the formula for thermal efficiency as:

This is a very useful formula, but here we express the thermal efficiency using the first law in terms of enthalpy.

Typically most nuclear power plants operate multi-stage condensing steam turbines. In these turbines, the high-pressure stage receives steam (this steam is nearly saturated steam – x = 0.995 – point C at the figure; 6 MPa; 275.6°C) from a steam generator and exhausts it to moisture separator-reheater (point D). The steam must be reheated to avoid damages caused to the steam turbine blades by low-quality steam. The reheater heats the steam (point D), and then the steam is directed to the low-pressure stage of the steam turbine, where it expands (point E to F). The exhausted steam then condenses in the condenser. It is at a pressure well below atmospheric (absolute pressure of 0.008 MPa) and is in a partially condensed state (point F), typically of a quality near 90%.

In this case, steam generators, steam turbine, condensers, and feedwater pumps constitute a heat engine that is subject to the efficiency limitations imposed by the second law of thermodynamics. In an ideal case (no friction, reversible processes, perfect design), this heat engine would have a Carnot efficiency of

= 1 – Tcold/Thot = 1 – 315/549 = 42.6%

where the temperature of the hot reservoir is 275.6°C (548.7K), the temperature of the cold reservoir is 41.5°C (314.7K). But the nuclear power plant is the real heat engine, in which thermodynamic processes are somehow irreversible. They are not done infinitely slowly. In real devices (such as turbines, pumps, and compressors), mechanical friction and heat losses cause further efficiency losses.

To calculate the thermal efficiency of the simplest Rankine cycle (without reheating), engineers use the first law of thermodynamics in terms of enthalpy rather than in terms of internal energy.

The first law in terms of enthalpy is:

dH = dQ + Vdp

In this equation, the term Vdp is a flow process work. This work,  Vdp, is used for open flow systems like a turbine or a pump in which there is a “dp”, i.e., change in pressure. There are no changes in the control volume. As can be seen, this form of the law simplifies the description of energy transfer. At constant pressure, the enthalpy change equals the energy transferred from the environment through heating:

Isobaric process (Vdp = 0):

dH = dQ     →     Q = H2 – H1

At constant entropy, i.e., in isentropic process, the enthalpy change equals the flow process work done on or by the system:

Isentropic process (dQ = 0):

dH = Vdp     →     W = H2 – H1

Obviously, it will be very useful in the analysis of both thermodynamic cycles used in power engineering, i.e., in the Brayton and Rankine cycles.

The enthalpy can be made into an intensive or specific variable by dividing by the mass. Engineers use the specific enthalpy in thermodynamic analysis more than the enthalpy itself, and it is tabulated in the steam tables along with specific volume and specific internal energy. The thermal efficiency of such a simple Rankine cycle and in terms of specific enthalpies would be:

It is a very simple equation, and for the determination of the thermal efficiency, you can use data from steam tables.

Thermal Efficiency of Steam Turbine

In modern nuclear power plants, the overall thermal efficiency is about one-third (33%), so 3000 MWth of thermal power from the fission reaction is needed to generate 1000 MWe of electrical power. The reason lies in relatively low steam temperature (6 MPa; 275.6°C). Higher efficiencies can be attained by increasing the temperature of the steam. But this requires an increase in pressures inside boilers or steam generators. However, metallurgical considerations place upper limits on such pressures. Compared to other energy sources, the thermal efficiency of 33% is not much. But it must be noted that nuclear power plants are much more complex than fossil fuel power plants, and it is much easier to burn fossil fuel than to generate energy from nuclear fuel. Sub-critical fossil fuel power plants operated under critical pressure (i.e., lower than 22.1 MPa) can achieve 36–40% efficiency.

Efficiency of Engines in Power Engineering
• Ocean thermal energy conversion (OTEC). OTEC is a sophisticated heat engine that uses the temperature difference between cooler deep and warmer surface seawaters to run a low-pressure turbine. Since the temperature difference is low, about 20°C, its thermal efficiency is also very low, about 3%.
• In modern nuclear power plants, the overall thermal efficiency is about one-third (33%), so 3000 MWth of thermal power from the fission reaction is needed to generate 1000 MWe of electrical power. Higher efficiencies can be attained by increasing the temperature of the steam. But this requires an increase in pressures inside boilers or steam generators. However, metallurgical considerations place upper limits on such pressures. Compared to other energy sources, the thermal efficiency of 33% is not much. But it must be noted that nuclear power plants are much more complex than fossil fuel power plants, and it is much easier to burn fossil fuel than to generate energy from nuclear fuel.
• Sub-critical fossil fuel power plants operated under critical pressure (i.e., lower than 22.1 MPa) can achieve 36–40% efficiency.
• Supercritical water reactors are considered a promising advancement for nuclear power plants because of their high thermal efficiency (~45 % vs. ~33 % for current LWRs).
• Supercritical fossil fuel power plants operated at supercritical pressure (i.e., greater than 22.1 MPa) have efficiencies of around 43%. Most efficient and complex coal-fired power plants operate at “ultra critical” pressures (i.e., around 30 MPa) and use multiple stage reheat to reach about 48% efficiency.
• Modern Combined Cycle Gas Turbine (CCGT) plants, in which the thermodynamic cycle consists of two power plant cycles (e.g., the Brayton cycle and the Rankine cycle), can achieve a thermal efficiency of around 55%, in contrast to a single cycle steam power plant which is limited to efficiencies of around 35-45%.

## Causes of Inefficiency

As was discussed, an efficiency can range between 0 and 1. Each heat engine is somehow inefficient. This inefficiency can be attributed to three causes.

• Irreversibility of Processes. There is an overall theoretical upper limit to the efficiency of conversion of heat to work in any heat engine. This upper limit is called the Carnot efficiency. According to the Carnot principle, no engine can be more efficient than a reversible engine (Carnot heat engine) operating between the same high temperature and low-temperature reservoirs. For example, when the hot reservoir has Thot of 400°C (673K) and Tcold of about 20°C (293K), the maximum (ideal) efficiency will be: = 1 – Tcold/Thot = 1 – 293/673 = 56%. But all real thermodynamic processes are somehow irreversible, and they are not done infinitely slowly. Therefore, heat engines must have lower efficiencies than limits on their efficiency due to the inherent irreversibility of the heat engine cycle they use.
• Presence of Friction and Heat Losses. In real thermodynamic systems or real heat engines, a part of the overall cycle inefficiency is due to the losses by the individual components. In real devices (such as turbines, pumps, and compressors), mechanical friction, heat losses, and losses in the combustion process cause further efficiency losses.
• Design Inefficiency. Finally, the last and important source of inefficiencies is the compromises made by engineers when designing a heat engine (e.g., power plant). They must consider cost and other factors in the design and operation of the cycle. As an example, consider the design of the condenser in the thermal power plants. Ideally, the steam exhausted into the condenser would have no subcooling. But real condensers are designed to subcool the liquid by a few degrees of Celsius to avoid the suction cavitation in the condensate pumps. But, this subcooling increases the inefficiency of the cycle because more energy is needed to reheat the water.

## Thermal Efficiency Improvement – Steam Turbine

There are several methods, how can be the thermal efficiency of the Rankine cycle improved. Assuming that the maximum temperature is limited by the pressure inside the reactor pressure vessel, these methods are:

Boiler and Condenser Pressures
As in the Carnot, Otto, and Brayton cycle, the thermal efficiency tends to increase as the average temperature at which energy is added by heat transfer increases and/or the average temperature at which energy is rejected decreases. This is the common feature of all thermodynamic cycles.

Condenser Pressure

The case of the decrease in the average temperature at which energy is rejected requires a decrease in the pressure inside the condenser (i.e., the decrease in the saturation temperature). The lowest feasible condenser pressure is the saturation pressure corresponding to the ambient temperature (i.e., the absolute pressure of 0.008 MPa, which corresponds to 41.5°C). The goal of maintaining the lowest practical turbine exhaust pressure is a primary reason for including the condenser in a thermal power plant. The condenser provides a vacuum that maximizes the energy extracted from the steam, resulting in a significant increase in network and thermal efficiency. But also this parameter (condenser pressure) has its engineering limits:

• Decreasing the turbine exhaust pressure decreases the vapor quality (or dryness fraction). At some point, the expansion must be ended to avoid damages that could be caused to blades of steam turbine by low-quality steam.
• Decreasing the turbine exhaust pressure significantly increases the specific volume of exhausted steam, which requires huge blades in the last rows of a low-pressure stage of the steam turbine.

In typical wet steam turbines, the exhausted steam condenses in the condenser, and it is at a pressure well below atmospheric (absolute pressure of 0.008 MPa, which corresponds to 41.5°C). This steam is in a partially condensed state (point F), typically of a quality near 90%. Note that there is always a temperature difference between (around ΔT = 14°C) the condenser temperature and the ambient temperature, which originates from the finite size and efficiency of condensers.

Boiler Pressure

The case of the increase in the average temperature at which energy is added by heat transfer requires either superheating of steam produced or an increase in the pressure in the boiler (steam generator). Superheating is not typical for nuclear power plants.

Typically most nuclear power plants operate multi-stage condensing steam turbines. In these turbines, the high-pressure stage receives steam (this steam is nearly saturated steam – x = 0.995 – point C at the figure; 6 MPa; 275.6°C). Since neither the steam generator is 100% efficient, there is always a temperature difference between the saturation temperature (secondary side) and the temperature of the primary coolant.

In a typical pressurized water reactor, the hot primary coolant (water 330°C; 626°F) is pumped into the steam generator through the primary inlet. This requires maintaining very high pressures to keep the water liquid. To prevent boiling of the primary coolant and provide a subcooling margin (the difference between the pressurizer temperature and the highest temperature in the reactor core), pressures around 16 MPa are typical for PWRs. The reactor pressure vessel is the key component that limits the thermal efficiency of each nuclear power plant since the reactor vessel must withstand high pressures.

Superheat and Reheat
As for the Carnot cycle, the thermal efficiency tends to increase as the average temperature at which energy is added by heat transfer increases. This is the common feature of all thermodynamic cycles.

One possible way is to superheat or reheat the working steam. Both processes are very similar in their manner:

• Superheater – increases the steam temperature above the saturation temperature
• Reheater – removes the moisture and increases steam temperature after a partial expansion.

The superheating process is the only way to increase the peak temperature of the Rankine cycle (and to increase efficiency) without increasing the boiler pressure. This requires the addition of another type of heat exchanger called a superheater, which produces the superheated steam.

Superheated vapor or superheated steam is a vapor at a temperature higher than its boiling point at the absolute pressure where the temperature is measured.

Reheat allows delivering more of the heat at a temperature close to the peak of the cycle. This requires the addition of another type of heat exchanger called a reheater. The use of the reheater involves splitting the turbine, i.e., using a multi-stage turbine with a reheater. It was observed that more than two stages of reheating are unnecessary since the next stage increases the cycle efficiency only half as much as the preceding stage.

The high pressure and low-pressure stages of the turbine are usually on the same shaft to drive a common generator, but they have separate cases. The flow is extracted with a reheater after a partial expansion (point D), run back through the heat exchanger to heat it back up to the peak temperature (point E), and then passed to the low-pressure turbine. The expansion is then completed in the low-pressure turbine from point E to point F.

In the superheater, further heating at fixed pressure increases both temperature and specific volume. The process of superheating water vapor in the T-s diagram is provided in the figure between state E and the saturation vapor curve. As can also be seen, wet steam turbines (e.g., used in nuclear power plants) use superheated steam, especially at the inlet of low-pressure stages. Typically most nuclear power plants operate multi-stage condensing wet steam turbines (the high-pressure stage runs on saturated steam). In these turbines, the high-pressure stage receives steam (this steam is nearly saturated steam – x = 0.995 – point C at the figure) from a steam generator and exhausts it to a moisture separator-reheater (point D). The steam must be reheated or superheated to avoid damages that could be caused to the blades of the steam turbine by low-quality steam. High content of water droplets can cause rapid impingement and erosion of the blades, which occurs when condensed water is blasted onto the blades. To prevent this, condensate drains are installed in the steam piping leading to the turbine. The reheater heats the steam (point D), and then the steam is directed to the low-pressure stage of the steam turbine, where it expands (point E to F). The exhausted steam is at a pressure well below atmospheric, and, as can be seen from the picture, the steam is in a partially condensed state (point F), typically of a quality near 90%. Still, it is much higher vapor quality than it would be without reheat. Accordingly, superheating also tends to alleviate the problem of low vapor quality at the turbine exhaust.

Since the temperature of the primary coolant is limited by the pressure inside the reactor, superheaters (except a moisture separator reheater) are not used in nuclear power plants, and they usually operate a single wet steam turbine.

Heat Regeneration
Significant increases in the thermal efficiency of steam turbine power plants can be achieved by reducing the amount of fuel that must be added to the boiler. This can be done by transferring heat (partially expanded steam) from certain steam turbine sections, which is normally well above the ambient temperature, to the feedwater. This process is known as heat regeneration, and a variety of heat regenerators can be used for this purpose. Sometimes engineers use the term economizer, which is a heat exchanger intended to reduce energy consumption, especially in preheating a fluid.

As can be seen in the article “Steam Generator”, the feedwater (secondary circuit) at the inlet of the steam generator may have about ~230°C (446°F) and then is heated to the boiling point of that fluid (280°C; 536°F; 6,5MPa) and evaporated. But the condensate at the condenser outlet may have about 40°C, so the heat regeneration in typical PWR is significant and very important:

• Heat regeneration increases the thermal efficiency since more of the heat flow into the cycle occurs at a higher temperature.
• Heat regeneration causes a decrease in the mass flow rate through a low-pressure stage of the steam turbine, thus increasing LP Isentropic Turbine Efficiency. Note that the steam has a very high specific volume at the last stage of expansion.
• Heat regeneration causes an increase in working steam quality since the drains are situated at the periphery of the turbine casing, where is a higher concentration of water droplets.

Regeneration vs. Recuperation of Heat

In general, the heat exchangers used in regeneration may be classified as either regenerators or recuperators.

• A regenerator is a heat exchanger where heat from the hot fluid is intermittently stored in a thermal storage medium before it is transferred to the cold fluid. It has a single flow path where the hot and cold fluids alternately pass through.
• A recuperator is a heat exchanger with separate flow paths for each fluid along its own passages, and heat is transferred through the separating walls. Recuperators (e.g., economizers) are often used in power engineering to increase the overall efficiency of thermodynamic cycles, for example, in a gas turbine engine. The recuperator transfers some of the waste heat in the exhaust to the compressed air, thus preheating it before entering the combustion chamber. Many recuperators are designed as counterflow heat exchangers.
Supercritical Rankine Cycle

As was discussed, the thermal efficiency can be improved “simply” by increasing the temperature of the steam entering the turbine. But this temperature is restricted by metallurgical limitations imposed by the materials and design of the reactor pressure vessel and primary piping. The reactor vessel and the primary piping must withstand high pressures and great stresses at elevated temperatures. But currently, improved materials and fabrication methods have permitted significant increases in the maximum pressures, with corresponding increases in thermal efficiency. The thermal power plants are currently designed to operate on the supercritical Rankine cycle (i.e., steam pressures exceeding the critical pressure of water 22.1 MPa, and turbine inlet temperatures exceeding 600 °C). Supercritical fossil fuel power plants that are operated at supercritical pressure have efficiencies of around 43%. Most efficient and complex coal-fired power plants operate at “ultra critical” pressures (i.e., around 30 MPa) and use multiple stage reheat to reach about 48% efficiency.

Supercritical Water Reactor – SCWR

The supercritical Rankine cycle is also the thermodynamic cycle of supercritical water reactors. The supercritical water reactor (SCWR) is a concept of Generation IV reactor that is operated at supercritical pressure (i.e., greater than 22.1 MPa). The term supercritical in this context refers to the thermodynamic critical point of water (TCR = 374 °C;  pCR = 22.1 MPa) and must not be confused with the criticality of the reactor core, which describes changes in the neutron population in the reactor core.

For SCWRs, a once-through steam cycle has been envisaged, omitting any coolant recirculation inside the reactor. It is similar to boiling water reactors, steam will be supplied directly to the steam turbine, and the feed water from the steam cycle will be supplied back to the core.

As well as, the supercritical water reactor may use light water or heavy water as a neutron moderator. As can be seen, there are many SCWR designs, but all SCWRs have a key feature, which is the use of water beyond the thermodynamic critical point as primary coolant. Since this feature allows to increase the peak temperature, the supercritical water reactors are considered a promising advancement for nuclear power plants because of their high thermal efficiency (~45 % vs. ~33 % for current LWRs).

## Isentropic Efficiency – Turbine, Pump

In previous chapters, we assumed that the steam expansion is isentropic, and therefore we used T4,is  as the outlet temperature of the gas. These assumptions are only applicable with ideal cycles.

Most steady-flow devices (turbines, compressors, nozzles) operate under adiabatic conditions, but they are not truly isentropic but are rather idealized as isentropic for calculation purposes. We define parameters ηT,  ηP, ηN, as a ratio of real work done by the device to work by the device when operated under isentropic conditions (in the case of the turbine). This ratio is known as the Isentropic Turbine/Pump/Nozzle Efficiency. These parameters describe how efficiently a turbine, compressor, or nozzle approximates a corresponding isentropic device. This parameter reduces the overall efficiency and work output. For turbines, the value of ηT is typically 0.7 to 0.9 (70–90%).

## Steam Turbine – Problem with Solution

Let us assume the Rankine cycle, one of the most common thermodynamic cycles in thermal power plants. In this case, assume a simple cycle without reheat and without condensing steam turbine running on saturated steam (dry steam). In this case, the turbine operates at a steady state with inlet conditions of  6 MPa, t = 275.6°C, x = 1 (point 3). Steam leaves this turbine stage at a pressure of 0.008 MPa, 41.5°C, and x = ??? (point 4).

Calculate:

1. the vapor quality of the outlet steam
2. the enthalpy difference between these two states (3 → 4) corresponds to the work done by the steam, WT.
3. the enthalpy difference between these two states (1 → 2) corresponds to the work done by pumps, WP.
4. the enthalpy difference between these two states (2 → 3) corresponds to the net heat added to the steam generator
5. the thermodynamic efficiency of this cycle and compare this value with Carnot’s efficiency

1)

Since we do not know the same vapor quality of the outlet steam, we must determine this parameter. State 4 is fixed by the pressure p4 =  0.008 MPa and the fact that the specific entropy is constant for the isentropic expansion (s3 = s4 = 5.89 kJ/kgK for 6 MPa). The specific entropy of saturated liquid water (x=0) and dry steam (x=1) can be picked from steam tables. In the case of wet steam, the actual entropy can be calculated with the vapor quality, x, and the specific entropies of saturated liquid water and dry steam:

s4 = sv x + (1 – x ) sl

where

s4 = entropy of wet steam (J/kg K) = 5.89 kJ/kgK

sv = entropy of “dry” steam (J/kg K) = 8.227 kJ/kgK (for 0.008 MPa)

sl = entropy of saturated liquid water (J/kg K) = 0.592 kJ/kgK (for 0.008 MPa)

From this equation the vapor quality is:

x4 =  (s4 – sl) / (sv – sl) = (5.89 – 0.592) / (8.227 – 0.592) = 0.694 = 69.4%

2)

The enthalpy for the state 3 can be picked directly from steam tables, whereas the enthalpy for the state 4 must be calculated using vapor quality:

h3, v = 2785 kJ/kg

h4, wet = h4,v x + (1 – x ) h4,l  = 2576 . 0.694 + (1 – 0.694) . 174 = 1787 + 53.2 = 1840 kJ/kg

Then the work done by the steam, WT, is

WT = Δh = 945 kJ/kg

3)

Enthalpy for state 1 can be picked directly from steam tables:

h1, l = 174 kJ/kg

State 2 is fixed by the pressure p2 =  6.0 MPa and the fact that the specific entropy is constant for the isentropic compression (s1 = s2 = 0.592 kJ/kgK for 0.008 MPa). For this entropy s2 = 0.592 kJ/kgK and p2 =  6.0 MPa we find h2, subcooled in steam tables for compressed water (using interpolation between two states).

h2, subcooled = 179.7 kJ/kg

Then the work done by the pumps, WP, is

WP = Δh = 5.7 kJ/kg

4)

The enthalpy difference between (2 → 3), which corresponds to the net heat added in the steam generator, is simply:

Qadd = h3, v  – h2, subcooled = 2785 – 179.7 =  2605.3 kJ/kg

Note that there is no heat regeneration in this cycle. On the other hand, most of the heat added is for the enthalpy of vaporization (i.e., for the phase change).

5)

In this case, steam generators, steam turbine, condensers, and feedwater pumps constitute a heat engine that is subject to the efficiency limitations imposed by the second law of thermodynamics. In the ideal case (no friction, reversible processes, perfect design), this heat engine would have a Carnot efficiency of

ηCarnot = 1 – Tcold/Thot = 1 – 315/549 = 42.6%

where the temperature of the hot reservoir is 275.6°C (548.7 K), the temperature of the cold reservoir is 41.5°C (314.7K).

The thermodynamic efficiency of this cycle can be calculated by the following formula:

thus
ηth = (945 – 5.7) / 2605.3 = 0.361 = 36.1%

References:
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