The Physics of Energy, page 19
An advanced nuclear power plant design uses helium to transport heat from the reactor core out to turbines that generate electricity. The efficiency for converting reactor heat to transmitted electricity is about 40%. The core temperature is about 1000∘C and the spent helium is at about 100∘C.
If the cycle time for helium in the reactor is about 5 minutes, how much helium is needed to transport heat in a 100 MWe nuclear powerplant?aHow does this compare to the world’s annual helium production?
To produce 100 MW of electric power at an efficiency of 40%, the power plant must transport 250 MW of power from the hot reactor core to the electricity-generating turbines. The helium is heated by K. With aspecific heat capacity of approximately 5.2 kJ/kg K, each kilogram of He delivers 5.2 900 4.7 MJ of thermal energy. To transport 250 MW requires (250 MJ/s) (4.7 MJ/kg) 50 kg/s of helium. The entire helium cycle through the reactor is estimated to take about 5 minutes. So the “charge” of helium in the reactor must be 300 s 50 kg/s 15 000 kg of He.
In 2010, estimated world helium production was m3 measured at 15∘C and 1 atm [25]. Each cubic meter isabout (4 g/mol) (1000 L) (24 L/mol ) 0.17 kg. So the reactor requires kg 0.17 (kg/m3) 88 000 m3 of He – a tiny fraction (%) of the world’s annual production.
aMWe refers to electric power output as opposed to the thermal power generated by the reactor, which is often denoted MWth. See §13 for further discussion.
5.4.3 Specific Heat Capacity
Since the heat capacity of a substance scales proportionally to the quantity of material, it is traditional to quote heat capacities per unit mass. Such quantities are known as specific heat capacities (specific heats for short), and are denoted or . Specific heats have units of J/kg K. Other useful definitions include molar heat capacities (in units of J/mol K), denoted and , heat capacities per molecule and as already introduced earlier, and volumetric heat capacities (in units of J/m3 K). Specific and molar heat capacities for a few common substances are given in Table 5.2. Heat capacities change with variations in temperature, pressure, etc., particularly near phase transitions (§5.6) and when quantum effects are relevant (as in Figure 5.7). Specific heat capacities are therefore usually quoted under certain standard conditions, for example at STP or at NTP.
Note that water, hydrogen, and helium have high specific heats. Water plays an important role in many energy systems in storing and transporting thermal energy, for example in radiators used for household heating. Light elements such as helium and lithium have high specific heat capacities because, while their molar heat capacities are typical, their molar masses are very small. Helium, for example, with a molar mass of 4.00, has a very large specific heat of 5.19 kJ/kg K. This, along with the fact that it is chemically neutral and cannot be made radioactive (see §16), makes it an excellent material for transporting heat in nuclear reactors (Example 5.4).
5.5Enthalpy
When heat is added to a system at constant volume, the internal energy U changes by exactly the amount of heat that is added. It proves extremely useful to define a new state function called enthalpy,
(5.33)
Example 5.5 Enthalpy Versus Internal Energy: Adding Heat at Constant Pressure or Constant Volume
A quantity of heat Q is added to a volume V of an ideal monatomic gas initially at temperature T and pressure p. Compare the results if the heat is added (a) at constant volume or (b) at constant pressure.
Constant Volume The heat increases the internal energy, , and the change in the temperature is . The pressure increases by . Because the volume remains fixed, no work is done, and the increase in enthalpy is .
Constant Pressure The heat increases the enthalpy, , and the change in the temperature is . The volume increases by . The work done is , and by the 1st Law, the increase in the internal energy of the gas is .
Thus, a given amount of heat results in a 67% larger temperature and internal energy increase if delivered at constant volume compared to constant pressure. The internal energy increases less at constant pressure because some of the added heat is converted to work as the gas expands.
Internal Energy, Enthalpy, and Heat Capacities
For heat added to a system at constant volume, no work is done, and
For heat added to a system at constant pressure,
where H is the enthalpy
For most systems, is smaller than because work is done through expansion as heat is added, so .
For an ideal gas, .
which plays the same role for processes at constant pressure. When heat is added to a system at constant pressure, the enthalpy changes by exactly the amount of heat that is added. To derive this result we need only combine the differentials
(5.34)
with the 1st Law applied at constant pressure (5.29),
(5.35)
Enthalpy is clearly a state function because it is composed of U, V, and p, all of which are state functions. The relation is exactly analogous to (5.24), and in a fashion similar to eq. (5.25) it follows that
(5.36)
Because enthalpy is most useful when pressure is the variable under external control, H is most naturally regarded as a function of T and p, where V can be determined from T and p by an equation of state such as eq. (5.13).
The concept of enthalpy plays a key role in analyzing many energy systems that involve thermal energy conversion, such as power plants (§12), geothermal (§32) and fossil (§33) energy sources, and energy storage (§37).
5.6Phase Transitions
The definition of heat capacity (5.23) suggests that a gradual flow of heat into a system results in a gradual increase in its temperature. If is finite, then implies that any increase in internal energy, no matter how small, results in a correspondingly small but nonzero increase in temperature. This relationship breaks down when materials undergo phase transitions, such as melting or boiling. When a material is at a temperature associated with a phase transition, the addition or removal of heat has the effect of changing the relative fractions of the material in different phases, without changing the temperature of the material. Thus, for example, adding heat to a cup containing ice and water melts ice, but does not raise the temperature until all the ice is melted. Such phase transitions play an important role in many energy systems – including power plants that use steam to drive turbines, as well as air conditioning and refrigeration systems that use more unusual materials with lower boiling points than water. We introduce the basics of phase transitions here and develop and use these ideas further in §9 and §12.
5.6.1 Melting and Boiling Transitions
Consider a block of ice at a pressure of 1 atm. At temperatures below 0, the ice has a heat capacity of about 2 kJ/kg K (Table 5.2). Adding thermal energy increases the temperature of the ice until it reaches 0 (at 1 atm), and then the temperature ceases to increase further. Instead, as more thermal energy is added, an increasing fraction of the once-solid block of ice turns into liquid water while the temperature remains fixed at 0. The ice melts – or fuses, to use the term preferred by chemists. (This type of fusion is not to be confused with nuclear fusion described in §16.)
One might have thought that as ice is heated, the H2O molecules would slowly become more mobile, first starting to rotate in place, and then beginning to wander about. If this were true, the ice would slowly soften and become more plastic, eventually flowing freely in a liquid state. In reality, however, solids remain quite rigid as T increases, until they abruptly turn to liquid at their melting point. Phase transitions such as the change from solid ice to liquid water, where a finite amount of energy is added while the system stays at a fixed temperature, are known as first-order phase transitions. The theory and classification of phase transitions is a rich and fascinating branch of physics that goes beyond the scope of this text.
Figure 5.8 At atmospheric pressure ice undergoes a phase transition to liquid water at 0∘C, while dry ice, solid CO2, sublimes directly into a vapor at –78.5∘C.
Once the entire block of ice has melted, the smooth relation between thermal energy and temperature resumes. As more thermal energy is added, the temperature, internal energy, and enthalpy of the water increase, though the heat capacity ( kJ/kg K) is substantially higher for liquid water than for ice. When the temperature reaches 100, water undergoes another first-order phase transition – it turns to water vapor or steam through the process of vaporization. Below 100 °C, the attraction between the molecules of H2O is sufficient to keep them close enough that they move by slithering around one another at a roughly constant separation. Above 100∘C, the kinetic energy of thermal motion is enough to overcome the intermolecular attraction, and the molecules bounce around as a gas, filling as much space as is allotted to them. Figure 5.9 gives a schematic sense of the configuration of H2O molecules in solid, liquid, and gas states.
Figure 5.9 HO molecules in solid (left), liquid (center), and gas (right) states of water. The regular, crystalline array in the solid actually has more intermolecular space than the liquid in which the molecules move around but remain in close proximity. Unlike the liquid and solid phases, in the gas phase the spacing of molecules varies strongly with the pressure and temperature.
The transitions from solid to liquid and liquid to gas are typical of many substances. The melting and boiling points of different materials depend upon the strength of their intermolecular forces and the molecular weights of the chemical compounds. As a rule of thumb, elements or molecules that are both chemically relatively inert and low in molecular weight have the lowest melting and boiling points. Compounds with strong intermolecular forces and high molecular weight melt and vaporize at relatively high temperature. Methane, which has roughly the same molecular weight (16) as water (18), melts and vaporizes at a lower temperature because water has strong intermolecular forces (hydrogen bonds) that methane does not have. Examples of these regularities are shown in Figure 5.10, where the boiling points of the noble gases (monatomic, almost non-reactive gases with closed outer electron shells – in the right-most column of the Periodic Table D.1) and the simple linear carbon-hydrogen chain molecules (paraffins, see §11.2.1) with composition CH (methane, ethane, butane, propane, etc.) are shown.
Figure 5.10 (a) Boiling points of the noble gases; (b) boiling points of the linear hydrocarbons CH as a function of n, illustrating the increase of boiling point with molecular mass. (Based on data by Techstepp reproduced under CC-BY-SA 3.0 license via Wikimedia Commons)
The temperatures at which melting (fusion) and boiling (vaporization) occur depend upon the pressure. The solid-to-liquid transition point is usually only weakly affected by pressure, but the liquid-to-gas transition is quite sensitive. At a pressure of half an atmosphere, such as one would encounter on a six kilometer high mountain, the boiling point of water is only 82∘C, while at a pressure of 10 atmospheres, water boils above 180. Information about the phase of water at different pressures and temperatures can be displayed graphically in a phase diagram as shown in Figure 5.11. The domains where water is a solid, liquid, and gas are separated by curves that mark the solid-to-liquid and liquid-to-gas transitions. At very low pressure, a solid-to-gas transition known as sublimation can also occur. (Solid CO2 sublimes at atmospheric pressure, see Figure 5.8.) Figure 5.11 is a simplified diagram: it displays the liquid and vapor phases correctly, but at very low temperatures there are many distinct solid phases of ice differentiated by their crystal structure.
Figure 5.11 The phase diagram of water showing the first-order phase transitions where ice melts and water boils. At the triple point (TP), solid, liquid, and gaseous H2O can all coexist in equilibrium. At pressures and temperatures above the critical point (CP), liquid and vapor are no longer distinguished. The normal melting (M) and boiling (B) points at 1 atm are also marked.
Phase diagrams contain much interesting and important information. From Figure 5.11 we see that solid, liquid, and vapor coexist at the triple point, where K and Pa. We also see that the melting point of water decreases slightly with pressure. This is unusual – it increases for most substances – and is associated with the fact that the density of water increases when it melts. The phase diagram of water also shows that the sharp distinction between water liquid and vapor ceases at the critical point, which occurs at 647 K and 22.1 MPa. Phase diagrams are known in varying detail for all common substances. For example, the phase diagram of CO2 is shown in Figure 5.12. Some of the problems explore other features of phase diagrams.
Figure 5.12 The phase diagram of carbon dioxide. Note that CO2 has no liquid phase at atmospheric pressure ( Pa), as anyone who has encountered dry ice can attest.
Example 5.6 Internal Energy Change When Water Boils
Given the latent heat of vaporization, how does the internal energy change when water boils? The measured latent heat of vaporization of water is kJ/kg (at 100∘C and 1 atm). can be computed from . To compute the change in volume when water boils, we need the densities of water and steam at 100∘C, which are 957 kg/m3 and 0.597 kg/m3 respectively.
This confirms that the change in internal energy is less than the heat added because some of the heat does the work of expanding the newly created steam. The amount is small, only about 7.5%, but definitely not negligible.
Note that the phase diagrams in Figures 5.11 and 5.12 do not fully specify the state of a system on the curves that mark a phase transition. A substance at its boiling point at some fixed temperature and pressure, for example, is represented by a single point on the diagram. Depending on how much energy is added to the substance, it may exist as a pure liquid, a pure vapor, or any mixture of the two. Likewise, at the melting point, the relative proportions of solid and liquid are not specified. When we study the use of fluids to convert between heat and mechanical energy in §12, we introduce other ways of looking at the change of phase that include this information as well.
5.6.2 Latent Heat and Enthalpies of Fusion and Vaporization
The amount of energy needed to turn a solid to a liquid or a liquid to a vapor is called the latent heat of fusion or vaporization. If the phase change takes place at constant pressure, then according to eq. (5.35) the added thermal energy equals the increase in enthalpy of the substance. Thus, the latent heats of fusion and vaporization are usually denoted , and are referred to as the enthalpy of fusion and the enthalpy of vaporization. Enthalpies of vaporization are characteristically much greater than enthalpies of fusion. The reasons for this are elaborated in §9, where the enthalpies of fusions and vaporization of some common and/or interesting substances can be found in Tables 9.2 and 9.3. Since p is constant and , is related to the change in internal energy, , by . Usually we deal with enthalpies, internal energies, and volumes per unit mass, known as specific enthalpies, etc., of phase change, and denoted , , and .
Water has a remarkably large enthalpy of vaporization. This makes boiling water a very effective way to store energy. The specific enthalpy density added by boiling water at one atmosphere, 2.257 MJ/kg, is greater than the density at which energy can be stored in most chemical batteries. For comparison, lithium-ion batteries store about 0.5 MJ/kg and TNT stores about 4.6 MJ/kg in chemical energy. On the other hand, water vapor is quite diffuse at ordinary pressures so the energy density per unit volume is much less than that of either a battery or TNT. Steam engines and turbines have used water and the liquid/vapor phase transition as a way of storing and transforming energy for centuries. These mechanisms dominate the production of electricity from thermal energy sources such as fossil fuels. We return to this topic in §12.
Melting and Vaporization
Melting (fusion) and vaporization are first-order phase transitions. Although heat is added, the temperature stays fixed until all material has changed phase. Enthalpies of vaporization are typically greater than enthalpies of fusion. Water has an exceptionally large enthalpy of vaporization.
Discussion/Investigation Questions
5.1 A one liter box containing 0.1 L of liquid water at NTP is lifted from the floor and placed on a table one meter high. How much has the energy of the water changed? How much has its internal energy changed? The water is now heated until it is all converted to vapor. Has all this heat gone into the internal energy of the water? (You can assume that the box itself has negligible heat capacity.)
5.2 A mixture of sodium and potassium salts is used for energy storage in certain solar energy plant designs. Such a mixture is used in a temperature range between 300 ∘C and 500∘C where it is a liquid and where its specific heat is approximately 1.5 kJ/kg K. Why would engineers choose a substance like this over steam?
5.3 CO2 is a colorless gas. Why do you suppose the CO2 vapor in Figure 5.8 streaming away from the dry ice appears white?
5.4 The phase diagram for pure iron is shown in Figure 5.13. Iron has three different solid phases, called α, γ, and δ-iron, with different crystal structures. Discuss the behavior of iron as it is heated at different pressures, at 1 atm or at atm, for example. Under what conditions can several phases of iron coexist? Does iron have a triple point similar to water's?
Figure 5.13 A phase diagram for pure iron. The α, γ, and δ phases are solids with different crystal structure. See Question 5.4.
Problems
5.1 A cylinder initially contains L of argon at temperature T0 = 0∘ and pressure atm. Suppose that the argon is somehow made to expand to a final volume L in such a way that the pressure rises proportionally to the volume, finally reaching atm. How much work has the argon done? What is its final temperature T and how much thermal energy has been added to the gas?
