The Physics of Energy, page 47
Figure 11.12 Stages in the four-stroke compression ignition (CI) cycle. (Credit: Adapted from Encyclopaedia Britannica. Inc). See Figure 11.1 for comparison and further labeling.
Figure 11.13 An ideal Diesel cycle in the pV- and ST-planes using cold air standard assumptions. The compression ratio is 20:1 and the cutoff ratio is chosen to be , so that the heat input, , is identical to the ideal Otto cycle shown in Figure 11.3. Both cycles begin at 300 K and 1 atm at . They reach approximately the same maximum temperature. The ideal Diesel and Otto cycles are compared in the ST-plane. Although both cycles have the same heat input , the Otto cycle adds more entropy to the engine. The area of the cycles (yellow shaded) shows the work output, which is greater for the Diesel cycle. The additional efficiency of the Diesel cycle is proportional to the area of the green shaded region (Problem 11.6).
The cold air standard thermodynamic analysis of the Diesel cycle is very similar to that of the Otto cycle. The principal difference is that, because the heat addition occurs at constant pressure rather than constant volume, in eq. (11.1) is replaced with , giving
(11.12)
The efficiency of the engine then becomes
(11.13)
Defining the cutoff ratio to be , we can express in terms of and the compression ratio . We use again eq. (11.5), , the analogous relation , and the relation that follows from the ideal gas law and , to write the efficiency as
(11.14)
As in the Otto cycle, increasing the overall compression ratio r is crucial for maximizing efficiency.
Because there is no risk of knock, diesel engines can operate at much higher compression ratios than spark ignition engines. On the other hand, realizing compression ratios much above 10:1 requires a heavy duty engine with a massive piston and cylinder head in order to withstand the higher pressure. Such engines therefore are most easily accommodated in large vehicles, such as trucks, buses, and heavy equipment, as well as locomotives and large ships. Modern diesel engines generally operate at compression ratios between 15:1 and 22:1. While the theoretical efficiency of these engines can be around 75%, actual efficiency is closer to 45%.
Because knock is not a concern, diesel engines can also operate on a more flexible range of fuels than spark ignition engines. Traditional petroleum-derived diesel fuel is a mix of hydrocarbons with 10–15 carbon atoms, with approximately 25% aromatics. Other options for diesel fuel include biodiesel and organic oils (§26). In place of octane number, the combustion quality of diesel fuel is measured by cetane number, which is a measure of the time delay between ignition and combustion. A higher cetane number indicates a more easily combustible fuel; typical diesel fuel has a cetane number between 40 and 55, with higher cetane number providing more efficient combustion for high-speed engines. Despite the efficiency advantages of diesel engines, disadvantages include the facts that diesel engines must be more massive and are therefore more expensive than comparable spark ignition engines, and diesel engines can be more difficult to start in cold conditions. In addition, compression ignition can result in less complete combustion, leading to increased emission of pollutants. The high temperatures and pressures reached in CI engines can also result in the formation of oxides of nitrogen (typically NO and NO, denoted together as NO), a further source of pollution. These disadvantages have slowed the development of diesel engines for smaller passenger automobiles, though lighter and more versatile diesel engines are now commercially available.
Diesel Engines
Diesel engines use compression ignition, where the air is compressed before fuel injection to avoid knock. Compression ignition engines are modeled by the idealized thermodynamic Diesel cycle. Because higher compression ratios are possible, diesel engines are 30–50% more efficient than spark ignition engines. Diesel engines must be more massive, however, and are generally more expensive.
Discussion/Investigation Questions
11.1 Walk your way around the pV- and ST-diagrams, Figure 11.3, for the Otto cycle, explaining the functional form of the curves, and . Why is each step vertical in one of the diagrams?
11.2 Research and summarize the advantages and disadvantages of two-stroke engines for personal motor vehicles. What do you see as their future?
11.3 Would you advocate incorporating an Atkinson-like step (isobaric compression after the completion of the power stroke) in a diesel engine? Explain.
11.4 What is the rationale for modeling combustion as a constant pressure process in the Diesel cycle when it was modeled as constant volume in the Otto cycle?
11.5 Why would a diesel engine be hard to start in cold weather?
11.6 A dual cycle attempts to model an SI engine more realistically. It models the combustion process in two steps. First some heat is added at constant volume, then the rest is added at constant pressure. Draw the dual cycle in the pV- and ST-planes. How are the Otto and Diesel cycles related to the dual cycle?
Problems
11.1 Assume that an SI engine has the following parameters: displacement (): 2.4 L; compression ratio 9.5:1; air to fuel mass ratio 15:1; heating value of fuel 44 MJ/kg, pressure at start of compression 90 kPa, intake temperature 300 K. Compute the pressure, volume, and temperature at each of the points , , , and in the idealized cold air standard Otto cycle. Compute the work done per cycle, and estimate the engine efficiency in this idealized model. Compute the power at 3000 rpm.
11.2 The cold air standard value of was based on the heat capacity of a diatomic gas with no vibrational excitation (see §9), and . In reality, the heat capacity increases with increased temperature, and γ decreases accordingly. If we assume, however, that is independent of temperature through the cycle and that remains valid, what value of is required for ?
11.3 An SI engine, modeled as an ideal Otto cycle, runs at a compression ratio of 9.6:1 with a maximum cylinder volume () of 2 L and a corresponding displacement of 1.8 L. Combustion leads to a maximum temperature of K. Using the air standard value of , find the engine efficiency and the work per cycle. Compute the engine’s power at 5000 rpm. Assume that the intake temperature and pressure are 300 K and 100 kPa. Note: you will need the value of from Problem 11.2.
11.4 [H] (Requires results from Problem 11.3) Consider the same engine as in Problem 11.3 but now run as an Atkinson cycle. The volume after expansion () is still 2 L, but the volume before compression () is 1.54 L. The compression ratio is still 9.6:1, so the minimum volume is L. What is the expansion ratio? Assuming that the amount of air and fuel are both reduced by the ratio of , find the work done in expansion, , the work required by compression, , and the energy input from combustion . Compute the efficiency and the engine’s power at 5000 rpm.
11.5 (Requires results from Problem 11.1) Consider a throttled Otto cycle for an engine with the same parameters as Problem 11.1. In the throttled cycle assume that the spent fuel–air mixture is ejected at 1 atm and brought in again at 0.5 atm. Compute the work done per cycle. Compute the associated pumping loss. Subtract from the work per cycle and compute the engine efficiency with throttling.
11.6 Consider the Otto and Diesel cycles shown in Figure 11.13. The parameters have been chosen so both cycles have the same heat input and the same low temperature and pressure set point. Explain why the difference between the Diesel cycle efficiency and the Otto cycle efficiency is proportional to the area of the green shaded region.
11.7 A marine diesel engine has a compression ratio and a cutoff ratio . The intake air is at kPa and K. Assuming an ideal cold air standard Diesel cycle, what is the engine’s efficiency? Use the ideal and adiabatic gas laws to determine the temperature at points , , and . Compute the work per cycle. What is the power output of this engine if its displacement is 10 L and it runs at 2000 rpm? What is its fuel consumption per hour (assuming the energy content of diesel fuel to be 140 MJ/gal)?
11.8 Suppose two engines, one SI, the other CI, have the same temperature range, K and K. Suppose the SI engine, modeled as an ideal Otto cycle, has a compression ratio of 10:1, while the CI engine, modeled as an ideal Diesel cycle, has twice the compression ratio, 20:1. Using cold air standard assumptions, what is the efficiency of each engine? How much more efficient is the diesel engine? [Hint: the fuel cutoff ratio, can be computed as a function of r and .]
* * *
1 If the thermodynamic properties of the gas, still assumed to be air, are allowed to vary with temperature, the analysis is called air standard.
2 These conditions “600–350°” are chosen because they enable many different compounds to be compared. Critical compression ratios observed under normal engine operating conditions are usually higher.
3 For the precise definition of octane numbers above 100, see [60].
CHAPTER 12
Phase-change Energy Conversion
The Carnot and Stirling engines considered in §10 use a gas as the working fluid, as do the various internal combustion engines described in §11 that dominate transportation. In contrast, for two major other applications of thermal energy conversion, devices are used in which the working fluid changes phase from liquid to gas and back again in the course of a single thermodynamic cycle.
Large-scale electric power production is one area in which phase-change systems play a central role. The principal means of electric power generation in stationary power plants, whether nuclear or coal-fired, is the modern steam turbine that uses a phase-change thermodynamic cycle known as the Rankine power cycle. As shown in Figure 12.1, almost 80% of the electricity generated in the world is produced from thermal energy (released by fossil fuel combustion or uranium fission). The main exception is hydropower, which provided about 17% of world electricity in 2012 [12]. Wind and solar photovoltaic power generation, which do not involve thermal energy, are still very small components of total electric power generation. Of the 80% of electricity production from thermal energy, nearly all employs the Rankine steam cycle to convert heat to mechanical energy. The rest comes from natural-gas-fueled turbines that make use of a pure gas phase combustion cycle known as the Brayton cycle. Increasingly in recent years, the gas turbine is combined with a lower-temperature steam turbine fueled by exhaust gases to make a high efficiency combined cycle. This chapter focuses on the basic physics of phase-change systems, and Brayton and Rankine cycles, including the combined cycle, are described inthe following chapter. Phase change also plays a role in geothermal power plants (§32), where liquid water from underground reservoirs at high temperature and pressure vaporizes as it reaches the surface where the pressure is lower.
Reader’s Guide
This chapter and the next explain the physics of phase-change energy conversion and its application to practical thermal energy conversion devices. In this chapter we explore the thermodynamics of phase change. We first explain why devices where fluids change phase play such a central role in thermal energy conversion. We then describe systems in phase equilibrium using various combinations of thermodynamic state variables, and introduce the state variable quality, which describes the mixed liquid–vapor state. Finally, we describe how pressure and temperature are related for water in phase equilibrium.
Prerequisites include: the introduction to phase change in §5 (Thermal energy) and §9 (Energy in matter), the description of forced convection in §6 (Heat transfer), and §10 (Entropy and temperature) and §13 (Heat engines) on thermal energy transfer.
The following chapter, §13 (Power cycles), depends critically on the material presented here. Phase change also figures in thermal energy conversion processes in §19 (Nuclear reactors), §32 (Geothermal energy), and §34 (Energy and climate).
The second major application of phase-change cycles is in heat extraction devices, i.e. air conditioners, refrigerators, and heat pumps, which were introduced in §10.6. Although the scale of refrigeration and air conditioning energy use is smaller than that of electric power generation, it nevertheless accounts for a significant fraction of total electric power consumption. Figure 12.2 shows the energy consumed in US homes in 2009 subdivided by end use. Together, household air conditioning and refrigeration accountedfor 13% of all electricity consumption in the US in 2010; energy used for air conditioning and refrigeration in US commercial buildings amounted to another 6% of total electricity consumption [12]. Industrial refrigeration and actively cooled transports (i.e.tractor-trailers with refrigerated payloads) also use energy for heat extraction and add to these totals. Heat pumps, which efficiently extract heat from a cooler environment to provide space heating in cold climates, are a small, but rapidly growing energy application that is discussed further in §32.3. Heat extraction devices are described in more detail in the following chapter(§13).
Figure 12.1 2011 world electricity generation by fuel (total 75.9 EJ) [12]. Only hydro, wind, photovoltaic (which are included in “Renewables”) and a small fraction of natural gas do not involve phase-change conversion of heat to mechanical and then to electrical energy.
Figure 12.2 US household energy use for 2009. (Adapted using data from US DOE EERE Buildings Energy Data Book)
This chapter begins (§12.1) with a discussion of the advantages of phase change in energy conversion. We then review and extend the description of phases begun in §5. In §12.2, we describe the phase diagram of a pure substance from several perspectives – as a surface in the three-dimensional space parameterized by pressure, volume, and temperature, and as seen in the entropy–temperature plane. §12.3 describes how information about phase change is used in concrete applications.
Phase change and the phase structure of pure substances are described in most thermodynamics texts. A more detailed treatment than ours with many examples can be found in [19]. The dynamics of heat transfer and phase change is explored thoroughly in [27].
12.1Advantages of Phase Change in Energy Conversion Cycles
Whether used in the context of a power plant or a heat extraction device, phase change is not an incidental feature, but is essential to the design of many practical energy conversion systems. Over decades of development, different thermodynamic systems have come to dominate different applications. Internal combustion, for example, dominates transportation, where the versatility and high energy density of liquid hydrocarbon fuels, and the compactness, safety, and reliability of modern internal combustion engines are important. Phase-change cycles have come to dominate large-scale stationary power generation and energy extraction devices, where they have several significant physical advantages. In this section we describe the physics behind four principal practical advantages of phase-change energy conversion.
Energy storage potential in phase change The energy needed to change the phase of a substance is typically much larger than the energy needed to heat it up by a few degrees (§5, §9). Enthalpies of vaporization, in particular, are generally quite large, and particularly so for water (see Table 9.3). For example, it takes 2.26 MJ to vaporize one kilogram of water at 100 °C. A small amount of that energy goes into -work, and the rest is stored as internal energy in the resulting water vapor. In contrast, one would have to heat about 5.4 kg of liquid water from 0 °C to 100 °C to store the same amount of energy.
The large amount of energy stored when a fluid vaporizes does come with a cost: its volume expands significantly in the transition. In practical devices, the expansion is limited by carrying out the phase transition at high pressure, where the volume difference between liquid and vapor is not so large. The power of an engine or heat extraction system such as a refrigerator is limited by the size of pipes and other mechanisms that transport and store the working fluid and the pressures they can tolerate.
Efficient energy transfer at constant temperature As stressed at the end of §8.5, thermodynamic efficiency of a cycle is highest when heat transfer processes within the cycle occur between parts of the system that are close to the same temperature.
When heat is transferred to a system across a large temperature difference, significant entropy is added, leading to associated losses and inefficiency. In the limit of a very small temperature difference, heat transfer generates no additional entropy and is reversible. As we saw in §6, however, the rate of heat flow is proportional to the gradient of the temperature, so heat will not flow from one system to another unless there is a temperature difference between them. The laws of thermodynamics and heat transfer present us, then, with a conundrum: how can a physical system transfer heat both efficiently and quickly?
Suppose we want to transfer an amount of thermal energy from a heat reservoir at a fixed temperature to a fluid in a single phase. As shown in Figure 12.3(a), the temperature of the fluid T will rise from its initial value to a final value , where for simplicity we assume that the heat capacity C of the fluid is constant. must be greater than . Early in the process is relatively large and the heat transfer is rapid but inefficient. Near the end, is smaller, which makes the transfer more efficient, but slower. The shaded area in Figure 12.3(a) indicates the temperature difference and is a measure of the thermodynamic inefficiency of the process. To make the transfer both relatively rapid and efficient we would have to constantly adjust the temperature of the heat reservoir upwards as the fluid warms. This kind of heat transfer is accomplished in the Stirling engine (§10), but at the cost of including a massive component – the regenerator – that stores the energy transferred between stages of the cycle. In contrast, if we add thermal energy to a fluid at its boiling point from a heat source at constant temperature, then remains fixed until all of the liquid turns to vapor (Figure 12.3(b)). We can adjust to make the process as efficient or as rapid as we require.
