The Physics of Energy, page 55
An advantage of the Brayton cycle is that the high temperature of the gases entering the turbine leads to a high Carnot limit on its efficiency even though the temperature of the exit gases is also quite high (450 °C to 650 °C). The Brayton cycle, however, has the disadvantage compared to the Rankine cycle that a great deal of the work produced by the turbine must be dedicated to powering the compressor, since pressurizing gases – unlike pressurizing liquids – requires significant work. This lowers the actual efficiency well below the Carnot limit.
An idealized Brayton cycle operating between 300 K and 1400 K with a compression ratio (see below) of 7:1 is shown in both the ST- and pV-planes in Figure 13.19. Mechanical components in a Brayton engine are shown further in Figure 13.20; the turbine and compressor are shown on the same shaft, illustrating the fact that the turbine supplies the backwork required to run the compressor. For the conditions shown in Figure 13.19, the ratio of backwork to useful work, is 37% (Problem 13.13).
Figure 13.19 Representations of a Brayton cycle with intake at 300 K and 1 atm, maximum temperature 1400 K, compression ratio 7:1, and cold air standard assumptions (). (a) In the ST-plane, with mechanical components labeled and heat flow shown; (b) in the pV-plane with isotherms and work flow shown.
Figure 13.20 Mechanical components of an open Brayton cycle.
The Brayton cycle and the Otto internal combustion cycle look very similar in the ST-plane even though they differ dramatically in the pV-plane. Like the ideal Otto cycle, the efficiency of the ideal Brayton cycle can be written (see Box 13.2) as a function of the compression ratio alone,
(13.5)
where is the ratio of specific heats of the air/gas mixture (assumed constant).
Box 13.2 Brayton Cycle Efficiency
The efficiency of the ideal Brayton cycle of Figure 13.19 is computed as follows. By definition, , but by the 1st Law, ,so
where we have used the fact that the {23} and {41} steps are both isobaric, and we have assumed that the heat capacity of the gas is constant throughout. The compression {12} and the expansion {34} are both isentropic, so is a constant, as shown in §10. If we combine this with the ideal gas law, , we find constant during the adiabatic steps {12} and {34}. Using the fact that and , we find , so that
It would seem that increasing the efficiency of a Brayton cycle requires only increasing the compression ratio r. Increasing r, however, increases the temperature of the gases entering the turbine, which in turn is limited by materials constraints. Also, it is difficult to design axially driven (as opposed to reciprocating piston) compressors with large compression ratios. As a result, the development of relatively high efficiency gas turbines based on the Brayton cycle did not take place until the advent of jet engines for airplanes during and after World War II. Advanced gas turbines now work at temperatures in excess of 1400 °C, and with compression ratios above 15:1. Since the gas in the turbine is mostly air, we can take (see §10), and from eq. (13.5) we find a theoretical efficiency of for such an advanced turbine. For a more conservative design, with a maximum operating temperature of °C, and including the inefficiencies of the compressor and turbine, actual Brayton cycle efficiencies in the range of 35–42 % are reached. The Carnot limit for a cycle operating between 1200 °C and 300 °C is 60%, so these turbines reach 60–70% of the thermodynamic limit.
13.5.2 The Combined Cycle
The combustion products exiting the gas turbine described in the previous section are still very hot. For , with a typical outlet temperature around , the gases leaving the Brayton cycle turbine are hot enough to serve as the high-temperature heat source for a separate downstream Rankine cycle, known as a bottoming cycle.
The Brayton Cycle
The Brayton cycle is a gas phase combustion cycle used in power plants and jet engines. Intake air is first compressed, then heated by combustion of natural gas or other vapor fuel. The hot, high-pressure gas is expanded adiabatically through a turbine. The output gases can be used to heat steam for a co-located Rankine engine (combined cycle). The efficiency of the ideal Brayton cycle,
is less than the efficiency of an ideal Otto cycle with the same compression ratio r.
The marriage of gas turbines with the Rankine steam cycle has produced the most efficient commercial electric power generation method: the combined cycle gas turbine shown schematically in Figure 13.21. The crucial point is that the waste heat from the top (Brayton) cycle serves as the input heat to the bottoming (Rankine) cycle. If the separate efficiencies are and respectively, then the net efficiency (net work divided by the heat provided by the heating value of the fuel), is (Problem 13.15). If the cycles are separately, say, 40% and 30% efficient, the net efficiency is 58%, a major improvement! Typical in-service CCGT power plants have efficiencies of over 50% (HHV), and state-of-the-art designs have surpassed 60%. Most modern natural gas-fired power plants are CCGT plants.
Figure 13.21 The CCGT cycle. An ideal Brayton gas turbine cycle {1234} with a compression ratio of 8:1 and K has a theoretical turbine outlet temperature of K. Although the actual temperature is somewhat lower, it is sufficient to provide heat to an ideal steam Rankine cycle {5678} with minimum and maximum pressures and temperatures of MPa, K and MPa, K. The efficiencies of the cycles are and , and the efficiency of the (ideal) combined cycle is 60%.
The efficiency of CCGT power plants has spurred renewed interest in coal gasification. Coal can be converted to a synthetic gas (syngas) that is primarily a mixture of CO and H; syngas can be used as fuel for a CCGT plant. The syngas production process and integrated gasification combined cycle (IGCC ) plants are discussed further in §33. In light of the complexities and energy and carbon cost of the syngas production process (see §33.1.7), there is a vigorous debate as to the relative advantages and disadvantages of IGCC power generation in comparison with direct combustion of coal. There is no question, however, that the efficiency and carbon footprint of the natural-gas-based CCGT are better than either coal based alternative.
13.5.3 Cogeneration
The final step in maximizing the efficiency of a power plant is to make use of the so-called “waste” heat that is rejected from a Rankine steam cycle in the form of relatively low-temperature and low-pressure steam. The concept of using a heat source to provide both mechanical or electrical energy and useful thermal energy is known as cogeneration.
Combined Cycle and Cogeneration
The combined cycle gas turbine (CCGT) power plant combines the high-temperature, gas phase Brayton cycle with a lower-temperature and lower-pressure Rankine steam cycle to obtain relatively high thermodynamic efficiency. Efficiencies exceeding 60% have been obtained.
Exhaust heat – either gases exiting a Brayton turbine or steam from a Rankine turbine – can be used for industrial processes, space heating, or cooling in combined heat and power (CHP) or cogeneration facilities. Cogeneration plants that combine a high-temperature Brayton cycle, a low-temperature Rankine cycle, and local space heating and/or cooling can utilize up to 85–90% of the combustion energy of natural gas fuel.
There are two fundamentally different motivations for cogeneration, depending on whether the steam or the mechanical/electrical energy is the byproduct. We have been discussing power plants, where electrical energy is the primary product and the steam would be the by-product. Alternatively, heat is ubiquitous in industrial processes, and mechanical or electrical power can be produced as a by-product. Metals processing, paper manufacturing, oil refining, chemicals manufacturing, are only a few of the industrial processes that make use of process heat, which is usually supplied by steam, typically at pressures upwards of 5 atm and temperatures of 150 °C or more. Steam for process heat is usually produced in a boiler fueled by fossil fuel combustion, which takes place at temperatures exceeding 1500 K. Even if all the chemical energy in the fuel ends up in the steam, using such high temperatures to produce low-temperature steam is thermodynamically inefficient.9 Instead, the combustion can be used to produce steam at much higher temperatures and pressures, which runs a turbine to produce mechanical energy. Then, the steam that emerges from the turbine outlet can be used for the originally intended industrial purposes. In this case the mechanical/electrical energy is the by-product.
The efficiency of a Rankine cycle is a sensitive function of the temperature at the turbine outlet. In §13.3.4 we pointed out that efficient power plants keep the outlet temperature as low as possible. Clearly, designing an industrial heating plant to produce, say, steam at 5 atm and 200 °C must sacrifice some efficiency in the Rankine cycle. Since the mechanical energy is a by-product (i.e. the fuel is being combusted in any event), a lower efficiency is tolerable. Also known as combined heat and power (CHP), the practice of generating mechanical power along with process heat has been implemented by efficiency-minded plant designers for years.
Returning to power plants, the idea of cogeneration is to put the exhaust heat from the turbine to useful purposes rather than dumping it into the environment. This heat can provide space heating for homes and businesses, and it can be used as the heat source for an absorption cooling cycle (see §13.2.6). The cost of insulated piping to carry the heated or chilled water to consumers limits this kind of cogeneration to plants located near relatively dense population centers. It is ideal for universities and for compact towns and cities, such as those found in many parts of Europe. Indeed, Europe has the highest incidence of cogeneration, with 11% of all electricity generated in the European Union as a whole produced in cogeneration facilities, and over 40% in Denmark and Latvia. MIT built a cogeneration plant in the 1980s to supply its campus with electricity, heating, and cooling. Its gas turbine utilizes 56 MW of thermal energy from combustion, and generates 21 MW of electricity, for an efficiency of 37.5%. Its cogeneration efficiency – the sum of electric power plus energy used for heating or cooling – is stated to be greater than 85%.
In practice, the steam input to most heating and cooling applications must be at a temperature above 100 °C and a pressure of several atmospheres, reducing the efficiency of a Rankine cycle in a cogeneration system. For this reason, cogeneration is particularly attractive when paired with a gas turbine (as MIT’s plant is), and the exhaust gases from the Brayton cycle are used directly to produce steam, or even better, when cogeneration is paired with a combined cycle so that the high-quality heat emerging from the Brayton cycle powers a Rankine steam cycle from which emerges the steam for space heating and cooling.
Discussion/Investigation Questions
13.1 Why is water not a suitable fluid for a vapor-compression kitchen refrigerator? What about for an air conditioner intended to operate between 24 °C and 40 °C?
13.2 Discuss the distinction between an adiabatic and an isentropic process. For example, how can it be that a compressor that works very quickly, allowing negligible heat transfer, can nevertheless increase the entropy of a gas?
13.3 The saturation domes for R-134a and isopentane shown in Figures 13.4 and 13.18 are quite asymmetric, with steep (R-134a) or even recurving (isopentane) saturation curves. Water, on the other hand, has a relatively symmetric saturation dome (see Figure 12.9). Explain why this feature makes R-134a and isopentane more suitable fluids for low-temperature heat extraction devices and engines than water. Why is water so widely used for higher-temperature systems such as steam turbines? What kind of shape for the saturation dome would be better for a phase-change fluid in a higher-temperature system?
13.4 As mentioned in §13.2.6, there are several other paradigms for simple refrigerators. Research and describe a gas phase Stirling refrigeration cycle or a gas phase Brayton refrigeration cycle. What does the corresponding thermodynamic cycle look like in the ST-plane? What are the applications of the cycle you study?
13.5 A vapor-compression cycle is specified by a closed, counterclockwise loop in the ST-plane. Show that if the loop is executed reversibly then the area of the loop is equal to the net work performed on the fluid in a cycle. [Hint: This is a corollary to a similar result for engines derived in §10.1.2.] Explain why the area of the ideal VC cycle {1234} in Figure 13.5(b) is not equal to the work performed on the fluid.
13.6 Describe in words how you would compute the coefficient of performance for an ideal vapor-compression cycle working between temperatures and . Assume that you have data on the thermodynamic properties of the refrigerant both at saturation and in the superheated vapor phase. Remember that (see Figure 13.5).
13.7 The efficiency of the idealized Carnot-like Rankine cycle of Figure 13.12(b) is higher than for fixed but lower than the Carnot limit for .
13.8 [T] Consider the modified Rankine cycle of Figure 13.17(a). Describe what happens to a quantity of water as it executes the cycle starting at . How many pumps and how many turbines are needed?
13.9 Explain in words what is going on in the Rankine reheat cycle of Figure 13.16(a), and, referring to the figure, explain the statements made at the end of the discussion in the text about the efficiency, power output, and turbine conditions for this cycle.
13.10 Jet engines use a variant of the Brayton cycle in which the gases are only expanded in the turbine to a pressure sufficient to drive the compressor and the plane’s mechanical and electrical systems; the resulting gases are expelled from the engine at high velocity. Research jet engines and describe their mechanical components, efficiency, and the thrust that they produce.
Problems
13.1 [T] Prove that the Carnot-like vapor-compression cooling cycle {1234} in Figure 13.5(a) has a maximum CoP that equals the Carnot limit regardless of the properties of the working fluid. Likewise, prove that the efficiency of the simplest Rankine power cycle {1234} of Figure 13.12(a) equals the Carnot limit.
13.2 [T] Sketch the Carnot-like VC cycle of Figure 13.5(a) in the pV-plane (superimposed on the saturation dome), and contrast the resulting shape with that of the Carnot cooling cycle described in §10.
13.3 A heat pump based on the ideal VC cycle of Figure 13.5(b) uses the refrigerant R-134a to heat a house. The set points are , . What is the CoP; how does it compare with the Carnot limit? What flow rate () in kg/s is required to provide heat at 50 kW? Relevant thermodynamic data for R-134a is provided in Table 13.5. Note that since this is an ideal VC cycle, as opposed to the modified cycle analyzed in §13.2.5, the maximum temperature must be determined. To find , data in the superheated region are required [23]. Alternatively you can approximate .
Table 13.5 Data for Problem 13.3.
13.4 The 5-ton AC unit designed in §13.2.5 reached 33% of the Carnot limit on CoP. Look back at the definition of the “Energy Efficiency Ratio” in Problem 10.12, which employs a different temperature range than the one we specified. Assuming that the unit’s fraction of the Carnot limit remains unchanged, compute its EER. How does this compare with the current US standard of EER 11.7?
13.5 [T] Neglecting the work done by the pump (which is a good approximation), show that the efficiency of the ideal Rankine cycle is , where is the specific enthalpy at the point j in the cycle.
13.6 [T] Sketch the ideal Rankine cycle on a pV-diagram including the saturation dome. Label the points –. As in Figure 12.7, plot the logarithm of the specific volume on the horizontal axis.
13.7 When we designed the Rankine steam cycle in §13.3.5 we ignored the circulating water’s kinetic and potential energy, having claimed earlier (see Box 13.1) that they are negligible. Using the work per unit mass done by the turbine to set the energy scale, estimate the importance of the potential energy of the fluid if the height of the boiler is 40 m? Estimate the importance of the kinetic energy if the maximum flow speed in the pipes is a few (say 5) m/s?
13.8 Suppose the condenser in the 500 MWe coal power plant we analyzed in §13.3.5 is cooled by a once-through system, where water taken from the ocean is circulated once through the plant and then returned to the ocean. If regulations permit only a 10 °C rise in the ocean water temperature, how much water (in m/s) must be used? (Assume that the heat capacity of sea-water is the same as that of pure water. Remember that the efficiency of the plant was estimated at 38%.)
13.9 Quantitatively compare the Carnot Rankine cycle of Figure 13.2. The required information for the Carnot cycle can be obtained from Table 13.6.
Table 13.6 Data for Problem 13.9.
13.10 Modify the coal plant Rankine cycle described in §13.3.5 by including a regeneration cycle similar to the one shown in Figure 13.16(b). A fraction f of the steam is removed from the turbine and returned back to a second pump at a temperature °C. Compute the fraction f needed to heat the water from the first pump to 130 °C, and compute the increase in efficiency of the resulting cycle above that computed in the text. [Hint: in Figure 13.16(b).]
13.11 The wear on a steam turbine could be decreased by raising the pressure at the turbine outlet so that the quality of the steam at the outlet is one. In the ideal Rankine cycle of Example 13.2 and Figure 13.12. To simplify the calculation, ignore the work done by the pump. Would you advise taking this step?
13.12 [T] Explain why the Brayton and Otto cycles look so different in the pV-plane even though they are quite similar in the ST-plane.
