The Physics of Energy, page 50
12.3.2 Thermodynamic Data on Phase Transitions
When designing real devices, engineers are often interested in domains of pressure and temperature where the behavior of gases is far from ideal. For example, near the critical point of water the density of liquid and vapor are almost the same, and it is a poor approximation to treat the vapor as an ideal gas. For concrete applications it is often necessary to know exact thermodynamic properties (density, internal energy, entropy, enthalpy, latent heats, etc.) as functions of temperature and pressure. These thermodynamic properties have been determined empirically for many fluids across a wide range of temperatures and pressures, and the corresponding numerical data is available in many places. In the pre-computer era, this data was published in voluminous sets of tables. For H2O, these tables are known as steam tables, and this name is often applied to the thermodynamic data for other materials as well. While there are now simple computer programs and smartphone apps that contain this data and can produce any thermodynamic information in the format desired (see e.g. [23]), the computer data is usually used in the same way as data from the traditional steam tables, so for clarity we illustrate the approach with the use of the tables. A sample section of a steam table (for H2O) is shown in Table 12.1. The main sections of the table are labeled by the pressure, here Pa (very close to 1 atm) and Pa. The boiling point of water is given at each pressure (99.61 °C and 104.78 °C respectively). Below that, labeled by “Sat. Liq.” and “Sat. Vap.,” the specific volume, internal energy, enthalpy, and entropy of the saturated liquid and vapor are given. On the line between, labeled “Evap.,” the differences, , etc., are listed. The remaining rows of the table, labeled by the temperature in °C (in the left-hand column) give the values of thermodynamic properties for the sub-cooled liquid and superheated vapor phases at the relevant temperatures, above and below the horizontal line half-way down the table.
Thermodynamic Data
Thermodynamic properties of working fluids such as water, R-134a refrigerant, and ammonia can be found in tables and are conveniently accessed on websites or through software available for computers and smartphones. Typically, specific volume, enthalpy, and entropy for saturated liquid and vapor are available at the phase transition. Data are also available on sub-cooled liquids and superheated vapors.
Table 12.1 Section of a steam table: thermodynamic properties for water at specified temperatures and pressures. See text for more description. The thermodynamic information summarized in steam tables can be found on-line in convenient calculators and apps. Adapted from [64].
Some entries in the table are redundant. For example the changes in enthalpy, internal energy, and volume recorded in the “Evap.” line are related by , and the changes in entropy and enthalpy are related by (Problem 12.2). It is also interesting to check how far the behavior of the vapor phase differs from what would be expected for an ideal gas (Problem 12.7).
Example 12.3 Water in a Box
One kilogram of water is introduced into an initially evacuated box of volume 1 m and is then heated to 105 °C. Estimate the resulting pressure, the relative proportion of liquid water and vapor, and the enthalpy.
Table 12.1 gives data on steam at 104.78 °C, which is close enough to 105 °C for a good estimate. The specificvolume of saturated vapor at that temperature is 1.43 m/kg, and the volume of the liquid is negligible (~1000 smaller) so the fraction of the water that is converted to vapor is 1/1.43 70%. Thus the system is in phase equilibrium with . The pressure is Pa. Finally the enthalpy is obtained by applying eq. (12.3),
Repeat the question supposing that the system is heated to 200 °C.
Table 12.1 does not contain saturation information for , so we must find data elsewhere. Using an online calculator, we find that the specific volume of saturated vapor at 200 °C is 0.127 m3/kg, which is less than the volume of the container. Thus we conclude that the steam in this case must be superheated in order to fill the entire container. Given the temperature, we seek a pressure at which the specific volume of superheated steam at 200 °C is 1 m3/kg. The online calculator gives kPa, slightly above 2 atm. As a check on this result, assume that the steam is an ideal gas:
which is in good agreement with the result obtained from the online data. At this temperature and pressure the enthalpy of superheated steam provided by the online calculator is kJ/kg.
Example 12.4 Using Steam Tables to Find the Results of Thermodynamic Processes
Example 1: Suppose that we begin with superheated water vapor at a temperature and pressure of 200 °C and 105 Pa. After a device compresses it isentropically we find that its temperature at the outlet of the device hasgone up to 220 °C. What is the final pressure?
According to Table 12.1, the initial specific entropy is 7.84 kJ/kg K. The outlet temperature is 220 °C and weknow that the specific entropy at the outlet remains 7.84 kJ/kg K. Searching the table we see that at 220 °C the specific entropy is 7.83 kJ/kg K at a pressure of Pa, which therefore must be quite close to the outlet pressure.We were lucky that the value of the pressure at which the 220 °C specific entropy was kJ kg K happened to be theone for which data was provided. (Of course the problem was set up this way!) In general one would have tomake a rough interpolation, search through the tables, or use a computer application to find the pressure to highaccuracy given the final temperature and entropy.
Example 2: Suppose we are given dry steam (i.e. ) at atm, and expand it isentropically down to a pressure of1 atm. What fraction of the steam condenses to liquid water?
Table 12.1 does not provide information at 10 atm, so we consult another reference such as [23], wherewe find that the specific entropy of saturated steam at 10 atm (and 181 °C) is 6.59 kJ/kg K. Looking back at Table 12.1, we see this does indeed lie between the specific entropy of water and steam at 1 atm, so the resultof the expansion is a mixture of liquid water and steam. Taking the entropy values from the table and setting upan equation like eq. (12.3) for the specific entropy as a function of the quality,
we find , so 13% of the steam (by mass) has condensed to liquid water.
Steam tables can be used in a multitude of ways. Not only can one find thermodynamic properties as a function of temperature or pressure, but also one can work backwards to find the temperature and/or pressure at which certain values are obtained (software containing thermodynamic information can do this automatically). This is useful in exploring energy conversion cycles. Examples 12.3 and 12.4 illustrate the use of steam tables in pure and mixed phases.
Discussion/Investigation Questions
12.1 What is the role of gravity in boiling heat transfer? Do you think boiling heat transfer would work well on an Earth-orbiting satellite?
12.2 An insulated cylinder closed by a piston is initially filled with one liter of ice at –20 °C. The piston is pulled out, doubling the volume and initially creating a vacuum above the ice. Describe qualitatively what happens as the system comes into equilibrium at –20 °C. Now the system is slowly heated to 110 °C. Again describe in words, what is the state of the water in the cylinder?
12.3 Research what a pressure cooker is and describe the physical principles that make it work.
12.4 A heat pipe is a device that uses heat conductivity and the large latent heat of phase change to transfer heat very efficiently between two solid bodies. Research the subject. What is the role of gravity? Why is the heat pipe evacuated before a carefully measured amount of fluid is added. How can the same heat pipe with the same working fluid be used between quite different temperature objects? Why does a heat pipe fail when it is overheated or sub-cooled?
12.5 On the basis of the pVT-diagram, Figure 12.8, construct a sketch of the phase diagram for water in the VT-plane (specific volume on the horizontal axis, temperature on the vertical) for temperatures at which the liquid and vapor phases are relevant. Label the graph like Figure 12.7. Note that at high temperature and low pressure in the gas phase, water is almost a perfect gas so . Use this fact to help you draw some isobars.
12.6 Since the volume parameterizes the points on the phase transition line between liquid and vapor (see Figure 12.7), one could use the density instead of the quality as the thermodynamic variable to characterize the mixed phase. Why do you suppose this is not done?
12.7 Under calm, clear conditions during the summer in a temperate climate similar to Boston’s, meteorologists can predict the minimum nighttime temperature quite accurately by making measurements of the relative humidity during the day. Explain why estimates made the same way are much less accurate in the winter, when maximum daytime temperatures are close to 0 °C.
Problems
Note: for a number of the problems in this chapter you willneed to use steam table data from a source such as [23 ] or the equivalent.
12.1 The isothermal compressibility of a fluid is the relative change of volume with pressure at constant temperature . Qualitatively describe and sketch the compressibility of water as a function of temperature from 0 °C to 1000 °C, at a pressure of 1 atm and at the critical pressure MPa. What is the compressibility at the phase transition from liquid to vapor?
12.2 Check that the and quoted in Table 12.1 are consistent with and .
12.3 Using the data given in Table 12.1, estimate the enthalpy added to one kilogram of water that undergoes the following transformations: (a) from 0 °C to 90 °C at Pa; (b) from 95 °C to 105 °C at Pa; (c) from 95 °C to 105 °C at Pa.
12.4 Consider a volume of 100 L of water, initially in liquid form at temperature , to which MJ of energy is added from an external reservoir at temperature through a thermal resistance, so that the rate of energy transfer is , where T is the instantaneous temperature of the water. The energy is added in two different ways: (a) with the water at initial temperature and the external source at temperature ; (b) with °C, °C. In each case, compute the time needed to transfer and the extra entropy produced in this process. Do the comparative results match with the expectation from the discussion of energy transfer at constant temperature in §12.1? (You may assume a constant heat capacity for liquid water, independent of temperature.)
12.5 Data on the heat flux for laminar flow of liquid water and for pool boiling of water are shown in Figure 12.4. Take the bulk temperature of the fluid in the case of laminar flow to be °C and assume that the pool boiling process takes place at 1 atm. (a) Suppose the two methods are used to transfer heat at a rate of W/m. Estimate the rate of extra entropy production (W/m K) for the two methods. (b) Estimate the heat flux for pool boiling when its rate of entropy production is the same as the rate of entropy production for laminar flow in part (a).
12.6 An industrial freezer is designed to use ammonia (NH) as a working fluid. The freezer is designed so that ammonia flowing through tubes inside the freezer at vaporizes at , drawing heat out of the interior. Outside the freezer, the ammonia vapor at liquefies at a pressure dumping heat into the environment. Find thermodynamic data on ammonia and identify the pressures that enable the liquid–vapor phase transition to take place at the required temperatures.
12.7 Using the data given in Table 12.1, find the deviations from the ideal gas law for water vapor just above the boiling point at Pa. For example, does ?
12.8 Revisit Question 12.2 quantitatively. Specifically, what is the pressure in the cylinder when the system comes to equilibrium after the volume has been doubled at –20 °C? What is the pressure after the cylinder has been heated to 110 °C? What fraction of the water in the cylinder is in vapor form at –20 °C and at 110 °C?
12.9 As explained in the following chapter (§13), a steam turbine can be modeled as an isentropic expansion. The incoming steam is superheated at an initial temperature and pressure . After expansion, the exhausted steam is at temperature . A high-performance turbine cannot tolerate low-quality steam because the water droplets degrade its mechanical components. Suppose that and that the turbine requires throughout the expansion process. If the high-pressure system is rated to 5 MPa, what is the minimum possible initial temperature of the steam?
12.10 Use steam table data to estimate accurately the pressure at which water would boil at room temperature (20 °C). Similarly estimate the temperature at which water boils on top of Mt. Everest, where the air pressure is approximately 1/3 atm?
12.11 A sample of HO at a pressure of 1 atm has a specific enthalpy of 700 kJ/kg. What is its temperature? What state is it in, a sub-cooled liquid, superheated vapor, or a mixed phase? If it is in a mixed phase, what fraction is liquid?
12.12 Repeat the preceding question for a sample at 10 atm with the same specific enthalpy.
12.13 On a hot summer day in Houston, Texas, the daytime temperature is 35 °C with relative humidity . An air conditioning system cools the indoor air to 25 °C. How much water (kg/m) must be removed from this air in order to maintain a comfortable indoor humidity of 40%? Compare your answer with the result of Example 12.1.
* * *
1 The phenomenon being described here is nucleate boiling. When the temperature difference between object and fluid gets very large, a film of vapor starts to develop between the fluid and the hot object and heat transfer actually decreases with increased temperature difference. At even higher temperature difference, film boiling takes over and the heat transfer rate grows once again [27 ].
2 One way in which water is a typical is that it expands upon freezing. This plays no role in the use of the liquid and vapor phases in engines and heat extraction devices; however, it plays a major role in Earth’s climate because ice floats on water. Also, there are actually several different phases of ice with different crystal structure. That complication is ignored here.
3 Note that the liquid phase region is very narrow in the ST-plane – in fact it has been exaggerated in Figure 12.9(b) in order to make it visible. If the entropy of liquid water were independent of the pressure at fixed temperature then all the isobars in the liquid phase including the critical isobar would lie on top of the left side of the saturation dome. Using the fact that the Gibbs free energy is a function of p and T, one can show that . (This is an example of a Maxwell relation, discussed for example, in [21 ].) Since the volume of liquid water expands very slowly with increasing temperature at fixed p, the entropy of water decreases very slowly with pressure at fixed T, and the isobars almost coincide in the liquid phase.
CHAPTER 13
Thermal Power and Heat Extraction Cycles
Steam power and vapor-compression cooling have reshaped human society, by supplying cheap electricity for industry and individuals, and by providing refrigeration to retard food spoilage and efficient cooling for modern buildings. As mentioned in the previous chapter, these technologies rely on the thermodynamics of phase change, specifically between liquid and vapor, towork quickly and efficiently and at high capacity. The thermodynamic cycles that are employed by engines that transform heat to mechanical energy in electric power plants are closely related to the cycles that are used to moveheat from low to high temperatures. This chapter describes these cycles and their practical applications. This is also the appropriate place to describe gas turbines. Although they resemble internal combustion engines in their thermodynamics, gas turbines have much in common with steam turbines and are also used for large-scale stationary power generation.
Reader’s Guide
In this last chapter on thermal energy conversion, we describe the thermodynamic cycles that power steam turbines and therelated cycles at the heart of modern heat extraction devices. In both cases the key ingredient is a working fluid that changes phasein the course of the cycle. We turn first to vapor-compression cooling cycles and then to the Rankine steam cycle used in electric power plants; we describe the idealized thermodynamic cycles and briefly characterize the physical components and the limitations of real systems. We then consider the Brayton cycle, a gas combustion cycle that also plays asignificant role in electric power generation. The chapter closes with a description of combined cycle plants and cogeneration.
Prerequisites: §5 (Thermal energy), §6 (Heat transfer), §8 (Entropy and temperature), §10 (Heat engines), §11 (Internal combustion engines), §12 (Phase-change energy conversion).
Steam power cycles are used in nuclear, geothermal, and solar thermal power systems (§19, §32, §24). Gas and steam turbinesare the prime movers that supply power to the electric generators that feed the grid (§38). Heat extraction reappears in§32 (Geothermal energy) (for ground source cooling systems) and §36 (Systems).
We begin (§13.1) with a brief digression on how to describe thermodynamic processes in which material flows from one subunit to another, a standard issue in phase-change cycles. Then, in §13.2, we turn to the vapor-compression (VC) cycle that lies at the heart of air conditioners, refrigerators, and heat pumps. As we did for internal combustion engines (§11), we present ideal cycles that allow us to understand the workings of heat extraction devices at a semi-quantitative level. Since VC cycles differ in some important ways from the gas cycles of earlier chapters, we introduce their complexities in a step-wise manner. We analyze the VC cycle in the ST-plane where it is easiest to understand the thermodynamics. We then look briefly at the physical components of anair conditioner and relate them to the abstract steps in the vapor-compression cycle. To see how heat extraction can workin practice, we design a quasi-realistic air conditioner based on a slightly simplified ideal VC cycle. Finally, we comment on losses and other departures from the ideal situation, and we look briefly at some alternative refrigeration cycles.
In §13.3, we turn to the use of phase-change cycles in heat engines. Specifically, we describe the Rankine cycle that is used totransform the heat generated by nuclear, fossil fuel, geothermal, and solar thermal sources into electrical power. The fluid at the heart of the Rankine cycle is water/steam, and the workhorse of the cycle is the steam turbine. We walk through the steps in an idealized Rankine cycle in the ST-plane, describe the physical components, and specify the parameters of a Rankine cycle for a practical application – in this case a 500 MW coal-fired steam power plant. We briefly mention ways of improving the efficiency and performance of the Rankine power cycle and describe in§13.4 the use of Rankine cycles based on other working fluids to generate electricity from low-temperature heat sources.
