The Physics of Energy, page 43
10.4.3 Stirling Engine: Features and Practical Potential
Unlike most other engines in common use, such as internal combustion engines based on the Diesel and Otto cycles (§11), the Stirling engine can, in principle, reach the Carnot efficiency limit. Stirling engines also perform much more work per cycle than Carnot engines operating with the same set points. This can be seen immediately by comparing the cycles for given limits of p and V, as in Figure 10.13.
Figure 10.13 The Carnot and Stirling engines depicted in Figures 10.10 and 10.11 have the same minimum and maximum pressure, volume, and temperature. The work per cycle can be compared by superimposing their pV curves as in this figure. As computed in Example 10.4, the Stirling engine produces almost three times as much work per cycle.
Example 10.4 A Geothermal Stirling Cycle
Consider the design of a geothermal heat engine as proposed in Example 10.3. Rather than a Carnot engine, however, consider a Stirling engine operating between the same volume, temperature, and pressure set points. How much work is performed per cycle?
The work done by the piston in the expansion phase {34} is, analogous to the computation in Example 10.3,
where is again J but now so
Similarly, J, so the work done is
This is approximately three times as much work per cycle as is done by the equivalent Carnot engine (though with the same efficiency).
In addition to their advantage in efficiency, Stirling engines are potentially cleaner and quieter than many other types of engines in current use, and can operate on a wider range of fuels. Internal combustion engines (§11) rely on combustion of fuel added to the working fluid for the heat input. Stirling engines, by contrast, are external combustion engines, which operate with an external heat source. As a result, Stirling engines can operate on continuous combustion of fuel, allowing for much quieter operation. Furthermore, internal combustion engines are open-cycle engines and must eject the spent fuel on every cycle, often including some unburned fraction of the fuel that enters the environment as pollutants. Stirling engines, on the other hand, can burn fuel until combustion is complete. They are thus cleaner than internal combustion engines (though, of course, they still produce the CO resulting from complete combustion of the carbon when powered by fossil fuels). Finally, because the heat source is external, any heat source can be used for power. The Stirling engine can thus be powered by solar thermal energy or other non-fossil heat sources, in addition to arbitrary combustible fuels.
Despite the clear advantages of the Stirling engine in theory, for the most part Stirling engines have not seen widespread use in recent decades. While the theoretical efficiency of the Stirling cycle is very high, realizing this efficiency in practice presents substantial engineering challenges. Existing Stirling engines tend to be larger and heavier than their internal combustion counterparts, though in general they are more robust and require less repair. Some of the particular challenges to constructing an efficient Stirling engine include the following issues:
Materials Because the expansion region of a Stirling engine is constantly exposed to the high temperature, the materials must be more robust than in an internal combustion engine, where the highest temperatures are reached only briefly, and less exotic and expensive alloys can be used in the engine.
Heat exchange For a Stirling engine to cycle quickly and thus achieve maximum power (since the power achieved by an engine is given by the work done per cycle times the cycle rate), heat must move quickly into the expansion region and out of the compression region. This requires very efficient heat exchangers, which add complexity to the engine. Even with efficient heat exchangers, it is difficult to achieve truly isothermal expansion and contraction. A better model is a polytropic process with constant with α somewhere between 1 (isothermal) and γ (adiabatic).
Regenerator The Stirling engine regenerator must have a fairly high heat capacity to readily absorb heat energy during the isometric cooling process. This requires a reasonably massive material, again raising the cost of the engine.
Currently, Stirling engines are used in several niche markets. In particular they are used in reverse as miniature cryogenic refrigerators (cryocoolers) to cool infrared detectors on missile guidance systems and in night-vision equipment. Stirling engines are of particular value in these applications because of their low noise and reliable operation. Stirling engines are also used in some space applications for similar reasons.
As energy efficiency becomes a higher priority and as efforts increase to shift energy sources away from fossil fuels, advances in Stirling engine technology may enable these devices to play an increasingly important role in a wide range of systems from air conditioning to solar thermal power plants to automobile engines.
Stirling Engine
The Stirling engine is another heat engine that can, in principle reach Carnot efficiency. The Stirling engine uses a regenerator to implement isometric heating and cooling without additional entropy increase.
The Stirling cycle is composed of isothermal compression at temperature , isometric heating to temperature , isothermal expansion at , and finally isometric cooling to the starting point of the cycle at temperature .
The Stirling cycle produces significantly more work per cycle than a Carnot engine operating between the same extremes of pressure, volume, and temperature.
10.5Limitations to Efficiency of Real Engines
Due to materials and engineering limitations, no real heat engine can achieve the ideal Carnot efficiency. Issues that decrease the actual efficiency of real engines, including Carnot and Stirling engines, below the thermodynamic ideal include:
Cycles are not ideal In a real engine, the cycle is not composed of distinct idealized processes. For example, in a realistic Stirling cycle, the pistons may undergo sinusoidal motion rather than the motion described in Figure 10.12(b). The resulting thermodynamic cycle encompasses a smaller area in the pV-plane than the idealized cycle, even in the quasi-equilibrium approximation. Since not all heat is taken in at the maximum temperature nor expelled at the minimum temperature in a real cycle, the efficiency is decreased from the ideal Carnot efficiency.
Irreversibility is unavoidable We have assumed reversible quasi-equilibrium processes, where the working fluid is near equilibrium at every point in time. In a real engine, many effects lead to departures from this ideal. Heat transfer is never truly reversible, as temperature gradients are necessary for heat to flow. In every realistic working system, friction in moving parts converts mechanical energy into heat energy. This heat energy is conducted or radiated into the environment, making the mechanical process irreversible. In some real engines, there are processes that involve unconstrained expansion, such as occurs when a valve is opened (§13.2.3), which increases the entropy of the gas as it expands into a larger volume without doing work and without the addition of heat. Finally, in any practical engine, like an automobile engine, where intake, combustion, and exhaust happen thousands of times per minute (e.g. as measured on a tachometer), the gas inside the engine has a spatially inhomogeneous pressure, density, and temperature, and is far from equilibrium at any given instant. To accurately estimate the efficiency of a real engine, engineers often use sophisticated modeling tools that simulate fluid and heat flow within the engine.
Materials limitations In reality, no material is a perfect thermal insulator or conductor. Thus, truly adiabatic expansion and compression, and reversible isothermal heating and cooling are not possible in practice. Deviations from these ideals again reduce the efficiency of real engines. In particular, any part of an engine that becomes extremely hot will transfer thermal energy to the rest of the engine and the environment through conduction, convection, and radiation, leading to energy loss and a net increase in entropy that decreases the overall engine efficiency.
The idealized thermodynamic analysis described here is also inaccurate for a variety of other reasons that are less directly related to efficiency. For example, the heat capacity , adiabatic index γ, etc., are all temperature-dependent. This results in a deviation from the idealized behavior described above. Corrections to the ideal gas law may also affect the behavior of real engines.
Despite these limitations, the thermodynamic analysis of engine efficiency in the ideal limit forms an invaluable starting point for estimating the efficiency of different engine designs. In a real engineering context, however, detailed consideration of many issues including those mentioned above must be taken into account to optimize efficiency.
10.6Heat Extraction Devices: Refrigerators and Heat Pumps
Up to this point we have only considered half of the story of conversion between thermal and mechanical energy. Heat engines can be considered as devices that move thermal energy from high to low temperature and produce work. Heat extraction devices do the opposite: they employ work to move heat from low to high temperature. The same operations that are combined to make a heat engine, if applied in reverse, will act as a heat extraction device. After describing general aspects of such devices, we look at the steps in a cooling cycle based on the Carnot engine run in reverse.
If the primary goal of transferring heat from the colder region to the hotter region is to cool the region at lower temperature, the device is called a refrigerator or an air conditioner. Although they may be used over different temperature ranges, the basic thermodynamics of air conditioners and refrigerators is the same. Sometimes, for brevity, we refer to both as refrigerators. If the primary goal is to increase or maintain the temperature of the hotter region, for example to use thermal energy from the colder outside environment to warm a living space, then the device is called a heat pump. Heat extraction devices based on phase-change cycles are described in detail in §13.2 and heat pumps are discussed further in §32 (Geothermal energy).
10.6.1 Coefficients of Performance for Heat Extraction Devices
Before describing heat extraction cycles in more detail, we reconsider the notion of efficiency as it applies to this situation. The flow of energy in a heat extraction device is sketched in Figure 10.14. Note the resemblance to Figure 8.4, where the basic structure of an engine was sketched. The heat extraction device uses work to extract an amount of thermal energy from a low-temperature reservoir , and delivers heat to a high-temperature reservoir . The effectiveness of a heat pump or refrigerator is termed its coefficient of performance or CoP. In general, the coefficient of performance measures the ratio of the desired effect to the energy input. (For a heat engine, the efficiency is also sometimes referred to as a CoP.) For a heat pump, the CoP is measured by the amount of heat delivered to the warm environment divided by the work needed to deliver it,
(10.15)
For a refrigerator, the aim is to remove thermal energy from the low-temperature environment, so its CoP is measured by the amount of heat removed divided by the work needed to remove it,
(10.16)
Figure 10.14 A schematic description of a heat extraction device which uses work W to extract thermal energy from a reservoir at temperature and expels thermal energy to a reservoir at temperature .
The 1 Law relates and to ,
(10.17)
which enables us to relate the coefficients of performance for the same device working as a refrigerator or as a heat pump,
(10.18)
Either way it is used, the 2 Law limits the magnitude of the CoP of a heat extraction device. Because, as for an engine, the entropy and energy in a heat extraction device do not change over a full cycle, the same argument that led to eq. (8.26) for engines leads to
(10.19)
for heat extraction. Combining this result with the expressions for the CoP of refrigerators and heat pumps, we obtain the Carnot limit on the performance of heat extraction devices,
(10.20)
where is the Carnot limit on the efficiency of an engine based on the same thermodynamic cycle. Figure 10.15 shows the three (log scaled) coefficients of performance (with CoP) as functions of .
Figure 10.15 The thermodynamic upper limits on the CoPs of an engine, refrigerator, and heat pump as a function of . Note that refrigerators and heat pumps become more efficient as , while engines become more efficient as .
Note that the Carnot limits on CoP and CoP can both be greater than one.5 When the temperature difference is very small, engines become very inefficient. It is possible, at least in principle, however, to move large quantities of thermal energy from low to high temperature across a small temperature difference with very little work. This is the reason why heat pumps are so effective as a source of space heating, particularly in environments where the temperature difference between inside and outside is not large.
Heat Extraction Devices
In principle, any reversible engine cycle run backward becomes a heat extraction cycle that uses work to extract heat from a domain at low temperature and expel it as heat into a domain at high temperature . Such a device can function either as a refrigerator or a heat pump. The 2 Law limits the coefficient of performance (CoP) of these devices to
where .
These limits can be significantly greater than one for .
10.6.2 A Carnot Heat Extraction Cycle
Figure 10.16 shows the thermodynamic properties of an ideal Carnot device run as a heat pump under the same conditions as the Carnot engine of Figure 10.10. The detailed description of the steps in the cycle and some of their implications are left to Example 10.5. We note a few crucial features here. First, the cycle is run in the opposite direction than an engine cycle, as denoted by the directions of the arrows in Figure 10.16. Thus, the yellow shaded area in the figure signifies the work done on the device by an outside agent and the red shaded area signifies the heat transferred to the high-temperature reservoir at . The coefficient of performance is the ratio of these areas, which is the inverse of the efficiency of the engine run on the same Carnot cycle. Thus the CoP for an ideal Carnot heat pump or refrigerator becomes infinite in the limit in which , as the work needed for a fixed heat transfer goes to 0.
Figure 10.16 A Carnot cycle for a heat pump operating between and , with a compression ratio of . See Figure 10.10 for comparison.
Example 10.5 Using a Carnot Heat Pump to Boil Water
Describe the steps required for a Carnot engine to be run in reverse as a heat pump to boil water. Use thesame minimum and maximum temperatures and volumes as for the Carnot engine example in Example 10.3.Compute the CoP of a heat pump that uses mechanical energy to take in heat at 20C and output heat at 100C.
For the reverse of the Carnot cycle analyzed previously, heat is taken in at temperature , and combinedwith mechanical work to produce heat output at temperature . The Carnot heat extraction cycle shown in Figure 10.3 begins at with the gas at low temperature and low pressure. First it is necessary toheat the gas to . This is accomplished by compressing it adiabatically in step {12}. The work required tocompress the gas heats it to . Next, the hot gas is put in contact with the high-temperature thermal reservoir andcompressed isothermally from to . This squeezes heat out of the gas, which can beused to boil some water. To ready the gas to absorb heat from the low-temperature reservoir, it must be cooledto . This is accomplished by expanding the gas adiabatically in step {34}. Finally, the gas is put in contact withthe low-temperature thermal reservoir, and allowed to expand isothermally, sucking thermal energy out of theambient environment.
The coefficient of performance of this device acting as a heat pump is
With the bounds on temperature and volumes used previously, the work input per cycle is 8.6 J, and heat output is 40.2 J per cycle. Note that heat pumps are generally quite effective, and move much more thermal energy from cold to hot than the work input, as long as the temperature differential is sufficiently small.
Despite their potentially high coefficients of performance, heat pumps and refrigerators based on Carnot cycles are of little practical interest. Such devices move relatively little heat per cycle compared to other gas cycles. An even more important consideration is that devices that employ fluids that change phase over the cycle have much more desirable heat transfer properties and are much easier to run in near-equilibrium conditions. Thus, even though they usually include an intrinsically irreversible step (free expansion) and therefore cannot reach the Carnot limit, phase-change systems dominate practical applications. We postpone further discussion of heat extraction cycles until §13, after we have explored the thermodynamics of phase change in §12.
Discussion/Investigation Questions
10.1 Why does it take more work to compress a gas adiabatically than isothermally from to ? Why does it take more heat to raise the temperature of a gas isobarically (constant pressure) than isometrically from to ?
10.2 In an engine cycle like the one shown in Figure 10.3, heat is added over a range of temperatures. Explain why even if it were run reversibly, its efficiency would be less than the Carnot limit defined by the high- and low-temperature set points and shown in the figure (). Is this in itself a disadvantage of such a cycle?
