The Physics of Energy, page 24
This algorithm can be improved and made more efficient in a variety of ways, but this simple iterative numerical procedure is a prototype for the vast range of computer simulations of partial differential equations currently in use. Numerical solutions of partial differential equations are important for many energy systems, in particular for simulating complex fluid dynamics systems (§29), and for modeling Earth’s global climate (§34).
All aspects of the temperature profile (for fixed ω) are determined by a single physical parameter, the thermal diffusivity a, which varies according to soil type and moisture content. As an example, for dry sandy soil m2/s. Substituting in this value, with s, we find m-1. At a depth of 1 meter, then, the seasonal variation falls in amplitude by a factor of . So a few meters below the surface the temperature is more or less constant all year round. The phase lag is substantial. At a depth of 1 meter, the phase of the sinusoidal oscillation lags behind the surface temperature by radians months. (This fact is observed by gardeners in many locations.) Some temperature profiles predicted by this simple model are shown in Figure 6.16. If we were to consider diurnal temperature fluctuations, the frequency would be larger by a factor of 365, so m-1, which means that the effects of diurnal variation are damped in 10 or 20 cm of soil. Again using the linearity of the heat equation, we can superpose the effects of diurnal fluctuations with annual fluctuations by simply adding the corresponding solutions of the heat equation.
Figure 6.16 Annual temperature variation as a function of depth in the soil for Boston where T0= 10.9∘ and ΔT = 12.4∘C.
Ground Temperature Variation
The heat equation can be used to estimate the variation in ground temperature as a function of depth in response to temperature changes at the surface. If the temperature at the soil surface varies sinusoidally with frequency ω, then the temperature variation with depth z and time t is given by
where and a is the thermal diffusivity of the soil.
Discussion/Investigation Questions
6.1 People usually discuss their experience of heat and cold in terms of temperature. Closer consideration, however, suggests that it is heat conduction rather than temperature that matters. What do you think? Have you ever taken a sauna? Or touched a metal surface at –30∘C?
6.2 Look up the effective R-value of wall insulation (for example, R-11 Mineral Fiber) when the structural support for the wall is either wood studs or metal studs. Explain the difference.
6.3 Although it was not mentioned in the text, Newton’s law of cooling requires that the thermal conductivity k of the object be “high enough,” otherwise the surface of the object will cool below its bulk temperature T. Then the convective heat transfer from the surface would be suppressed and the temperature of the object would not be uniform. Suppose L characterizes the distance from the center of the object, where the temperature is T, to its surface. What dimensionless constant must be small to avoid this problem?
6.4 A water bed replaces an ordinary mattress with a plastic bag filled with water. If the ambient temperature is below body temperature, water beds must either be heated or covered with a relatively thick layer of insulating foam. Why should this be necessary for a water bed, but not for an ordinary mattress?
6.5 Sketch the magnitude and direction of the heat flux under the ground driven by annual temperature variations as a function of depth for the four times of the year shown in Figure 6.16.
Problems
6.1 Suppose a small stoneware kiln with surface area 5 m2 sits in a room that is kept at 25∘C by ventilation. The 15 cm thick walls of the kiln are made of special ceramic insulation, which has W/m K. The kiln is kept at 1300 for many hours to fire stoneware. The room ventilating system can supply 1.5 kW of cooling. Is this adequate? (You can assume that the walls of the kiln are thin compared to their area and ignore curvature, corners, etc.)
6.2 Two rigid boards of insulating material, each with area A, have thermal conductances and . Suppose they are combined in series to make a single insulator of area A. What is the heat flux across this insulator as a function of ? Now suppose they are placed side-by-side. What is the heat flux now?
6.3 The heat transfer coefficient for air flowing at 30 m/s over a 1 m long flat plate is measured to be 80 W/m2 K. Estimate the relative importance of heat transfer by convection and conduction for this situation.
6.4 Given the Sun’s power output of 384 YW and radius 695 500 km, compute its surface temperature assuming it to be a black body with emissivity one.
6.5 Humans radiate energy at a net rate of roughly 100 W; this is essentially waste heat from various chemical processes needed for bodily functioning. Consider four humans in a roughly square hut measuring 5 m × 5 m, with a flat roof at 3 m height. The exterior walls of the hut are maintained at 0 by the external environment. For each of the following construction materials for the walls and ceiling, (i) compute the R-value () for the material, (ii) compute the equilibrium temperature in the hut when the four people are in it: (a) 0.3 m (1 foot) concrete walls/ceiling; (b) 10 cm (4”) softwood walls/ceiling; and (c) 2.5 cm (1”) softwood, with 0.09 m (3.5”) fiberglass insulation on interior of walls and ceiling. Take thermal conductivities and/or R-values from Tables 6.1 and 6.3.
6.6 Consider a building with 3200 ft2 of walls. Assume the ceiling is well-insulated and compute the energy loss through the walls based on the following materials, assuming an indoor temperature of 70 ° F and an outdoor temperature of 30 ° F: (a) walls are composed of 4” thick hard wood; (b) walls composed of 3/4” of plywood on the inside and outside, with 8” of fiberglass insulation between; and (c) same as (b) but with 18 single-pane R-1 windows (US units), each with area 0.7 m2 (remaining area is surrounded by walls as in (b)). Take thermal conductivities and/or R-values from Tables 6.1 and 6.3.
6.7 Estimate the R-value of the wall shown in the figure in Example 6.2.
6.8 Assume that a 60 m2 wall of a house is insulated to an R-value of 5.4 (SI units), but suppose the insulation was omitted from a 1 m2 gap where only 6 cm of wood with an R-value of 0.37 remains. Show that the effective R-value of the wall drops to leading to a 23% increase in heat loss through the wall.
6.9 In Example 6.1, a film of still air was identified as the source of almost all of the thermal resistance of a single-pane glass window. Determine the thickness of the still air layers on both sides of the window if radiative heat transfer is ignored. It is estimated that about half the thermal conductance of the window is due to radiative heat transfer; estimate the thickness of the still air layers if this is the case.
6.10 According to [28], a double pane window with an emissivity coating and a 1/4” air gap has a measured (center of glass) U-factor of 2.32 W/m2 K. Assume that this coating is sufficient to stop all radiative heat transfer, so that the thermal resistance of this window comes from conduction in the glass, the air gap, and the still air layers on both sides of the window. Taking m2 K/W from Example 6.1, compute the U-factor of this window ignoring convection in the air gap. How does your answer compare with the value quoted in [28]? Use the same method to compute the U-factor of a similar window with a 1/4” argon-filled gap (measured value W/m K). Repeat your calculation for a 1/2” air gap (measured U-factor l.70 W/m K). Can you explain why your answer agrees less well with the measured value in this case?
6.11 A building wall is constructed as follows: starting from the inside the materials used are (a) 1/2” gypsum wallboard; (b) wall-cavity, 80% of which is occupied by a 3.5” fiberglass batt and 20% of which is occupied by wood studs, headers, etc.; (c) rigid foam insulation/sheathing ( m2 K/W); and (d) hollow-backed vinyl siding. The wall is illustrated in Figure 6.17 along with an equivalent circuit diagram. Calculate the equivalent R-value of the wall. Do not forget the layers of still air on both inside and outside.
Figure 6.17 (a) A section through a wall in a wood-frame building. (b) A useful equivalent circuit diagram.
6.12 [H] An insulated pipe carries a hot fluid. The setup is shown in Figure 6.18. The copper pipe has radius cm and carries a liquid at T0 = 100∘C. The pipe is encased in a cylindrical layer of insulation of outer radius ; the insulation has been chosen to be closed-cell polyurethane spray foam with an R-value of 1.00 m2 K/W per inch of thickness. How large must be so that the heat loss to the surrounding room (at 20∘C) is less than 10 W/m? [Hint: First explain why the heat flux must be . You may find it useful to make the analogy between heat propagation through the insulator and current flowing outward through a cylindrical conductor. Note also that .]
Figure 6.18 Cross section through an insulated copper pipe. The pipe carries fluid at 100∘C. The radius of the insulation must be 3.6 times the radius of the pipe in order to keep the power loss to less than 10 W/m.
6.13 Ignoring convective heat transfer, estimate the change in U-factor by replacing argon by krypton in the quadruple glazed windows described in Table 6.3. You can ignore radiative heat transfer since these are low-emissivity windows.
6.14 In areas where the soil routinely freezes it is essential that building foundations, conduits, and the like be buried below the frost level. Central Minnesota is a cold part of the US with yearly average temperature of T0 ∼ 5∘C and variation ΔT ∼ 20∘C. Typical peaty soils in this region have thermal diffusivity m2/s. Building regulations require foundation footings to exceed a depth of 1.5 m. Do you think this is adequate?
6.15 Consider a region with average surface temperature T0 = 10∘C, annual fluctuations of ΔT = 30∘C, and surface soil with m2/s, W/m K. If the local upward heat flux from geothermal sources is 80 mW/m2, compute the depth at which the heat flux from surface fluctuations is comparable. What is the answer if the area is geothermally active and has a geothermal heat flux of 160 mW/m2?
6.16 [T] When an object is in radiative equilibrium with its environment at temperature T, the rates at which it emits and absorbs radiant energy must be equal. Each is given by . If the object’s temperature is raised to , show that to first order in , the object loses energy to its environment at a rate .
* * *
1 These terms are defined in §29. For now it suffices to think of laminar flow as smooth and steady and turbulent flow as chaotic and changing in time, without fixed pattern.
2 In addition to materials and design efficiency, more efficient heating equipment and migration to warmer regions have contributed to this reduction.
3 Viscosity, discussed in §29, is a measure of the “thickness” of a fluid and it inhibits convection.
CHAPTER 7
Introduction to Quantum Physics
Quantum physics is one of the least intuitive and most frequently misunderstood conceptual frameworks in physical science.Nonetheless, it is the universally accepted foundation of modern physics and is supported by an enormous range of experiments performed over the past hundred years. While quantum mechanics may seem bizarre, and applies primarily to systems much smaller than those we deal with on a day-to-day basis, it has great relevance to energy systems. Almost all of the energy that we use can betraced back to nuclear processes that originally took place in the Sun – or in some other star, in the case of geothermal or nuclear energy. Such nuclear processes can only be understood using quantum mechanics. A basic grasp of quantum mechanics is also necessary to understand the emission of light from the Sun and its absorption by carbon dioxide and water molecules in Earth’s atmosphere (§22). The photovoltaic technology we use to recapture solar energy is based on the quantum mechanical behavior of semiconductors (§25). Nuclear fission reactors – currently used to generate substantial amounts of electricity (§18) – as well as nuclear fusion reactors – which may perhaps some day provide an almost unlimitedsource of clean energy – depend upon the peculiar ability of quantum particles to tunnel through regions where classical mechanics forbids them to go. The chips in your cell phone and in the computers that were used to writethis book operate by quantum mechanical principles. Finally, quantum physics provides the basis for a rigorous understanding of thermodynamics. Concepts such as entropy and temperature can be defined in a precise way for quantum systems, so that a statistical treatment of systems with many degrees of freedom naturally leads to the laws of thermodynamics.
Reader’s Guide
Quantum mechanics is a rich and complex subject that requires substantial study for a full understanding. Here we attempt to give the reader a self-contained introduction to the basic ideas and some important applications of this theory. By focusing on the role of energy, we can systematically and efficiently introduce many of the most important concepts of quantum mechanics in away that connects directly with energy applications. We do not expect the reader to fully absorb the implications of these concepts in this chapter; as the results of quantum mechanics are applied in later chapters, however, we hope that the reader will become increasingly familiar with and comfortable with this counterintuitive physical framework.
We begin this chapter (§7.1–§7.2) with an informal introduction to the main ideas of quantum mechanics and end (§7.7–7.8) with some applications. §7.3–§7.6 contain a somewhat more precise, but also more abstract introduction to the basics of the subject. Readers interested primarily in energy applications may wish to skim the middle sections on first reading and return when questions arise in later chapters. The quantum phenomenon of tunneling is of central importance in nuclear energy, and is introduced in a chapter (§15) closer to its application in that context.
Prerequisites: §2 (Mechanics), §4 (Waves and light). We do not assume that the reader has had any previous exposure toquantum mechanics. Some familiarity with basic aspects of complex numbers is needed (Appendix B.2). Although not required, readers who have some familiarity with linear algebra (reviewed in Appendix B.3) will find it useful in the middle sections of this chapter.
We begin our exploration of quantum mechanics in §7.1, with a famous pedagogical thought experiment, due to American physicist Richard Feynman, that illustrates the counter intuitive behavior of quantum particles and provides some suggestion of how the theory of quantum mechanics can be formulated. Feynman’s thought experiment leads us in two complementary directions: first to wave mechanics, which enables us to understand the quantum mechanical propagation of particles through space, and second to a space of states, where the dynamics of quantum mechanics plays out in a remarkably simple way and where energy is the organizing principle. These two ways of thinking about quantum mechanics have been around since the mid-1920s when Austrian physicist Erwin Schrödinger proposed his Schrödinger wave equation and German physicist Werner Heisenberg introduced what he called matrix mechanics. At the time, the relation between the two formalisms was obscure, but a few years later the English physicist Paul Dirac showed that the two approaches are equivalent and that they give complementary insights into the working of the quantum world.
In §7.2 we follow the thread of wave mechanics. We introduce wavefunctions and Schrödinger’s wave equation. We develop a physical interpretation for the wavefunction and describe and interpret some simple solutions to the Schrödinger equation. We then put wave mechanics aside and take a more abstract look at the results of Feynman’s thought experiment. This leads us to a description of the foundations of quantum mechanics based on the concept of discrete quantum states labeled by their energy. We present this formulation in the next four sections, §7.3–§7.6. In these sections we introduce the basic notions needed to understand modern quantum mechanics, including the concept of the state space of a system and the Hamiltonian, which both describes thetime-evolution of a quantum system and characterizes the energy of a state. Using these ideas, we can understand more deeply the structure of the Schrödinger equation, the solution of which gives us the spectrum of allowed energies of atomic and molecular systems. These quantum spectra play a fundamental role in many applications discussed later in this book.
In a sense, energy is the key to understanding the fundamental structure of quantum mechanics. In any physical quantum system, energy is quantized and only takes specific values. The set of quantum states with definite energy form a favored class of states, in terms of which all possible quantum states can be described through quantum superposition. The way in which quantum states with fixed energy evolve is extremely simple – these states are unchanged in time except for a periodic phase oscillation with a frequency that is related to their energy. Thus, by understanding the spectrum of energies and associated quantum states we can give a simple and complete characterization of any quantum system.
A concise set of axioms for quantum mechanics is developed in §7.3–§7.6. These axioms are obeyed by any quantum system and form the “rules of the game” for quantum physics. These rules provide a nearly complete definition of quantum mechanics; by focusing on energy we can simply characterize general quantum systems while avoiding some complications in the story. Although sometimes classical physics is used to motivate parts of quantum mechanics, there is no sense in which quantum mechanics can bederived from classical physics. Quantum mechanics is simply a new set of rules, more fundamental than classical physics,which govern the dynamics of microscopic systems. With the basic axioms of the theory in hand, we return to wave mechanics in §7.7 and §7.8 where we explore the quantum dynamics of free particles and particles in potentials.For readers who would like to delve more deeply into quantum mechanics we recommend the texts by French and Taylor [38], at an introductory level, and Cohen-Tannoudji, Diu, and Laloë [39], for a somewhat more complete presentation.
