The Physics of Energy, page 20
5.2 Non-rigid airships known as blimps have occasionally been used for transportation. A blimp is essentially a balloon of volume V filled with helium. The blimp experiences a buoyancy force , where is the density of the surrounding air and m/s. The blimp maintains its shape because the pressure of the helium is kept higher than the pressure of the surrounding atmosphere . A modern blimp with a volume m can lift a total mass kg at temperature 0∘C and atm. Assuming both air and helium behave as ideal gases, estimate the pressure of the helium gas inside the blimp. Assuming the blimp's volume is kept constant, how much mass can it lift at 20℃?
5.3 When air is inhaled, its volume remains constant and its pressure increases as it is warmed to body temperature Tbody = 37∘C. Assuming that air behaves as an ideal gas and that it is initially at a pressure of 1 atm, what is the pressure of air in the lungs after inhalation if the air is initially (a) at room temperature, T = 20∘C; (b) at the temperature of a cold winter day in Boston, T = −15∘C; (c) coming from one person's mouth to another's during cardiopulmonary resuscitation (CPR). For CPR, assume the air is initially at body temperature and at atm.
5.4 [H] Everyday experience indicates that it is much easier to compress gases than liquids. This property is measured by the isothermal compressibility, , the fractional change in a substance's volume with pressure at constant temperature. What is the isothermal compressibility of an ideal gas? The smaller β is, the less work must be done to pressurize a substance. Compare the work necessary at 20∘C to raise the pressure of a kilogram of air and a kilogram of water ( Pa) from 1 atm to atm.
5.5 [HT] A cylindrical tube oriented vertically on Earth's surface, closed at the bottom and open at the top (height 100 m, cross-sectional area 1 m) initially contains air at a pressure of one atmosphere and temperature 300 K. A disc of mass m plugs the cylinder, but is free to slide up and down without friction. In equilibrium, the air in the cylinder is compressed to atm while the air outside remains at 1 atm. What is the mass of the disc? Next, the disc is displaced slightly in the vertical (z) direction and released. Assuming that the temperature remains fixed and that dissipation can be ignored, what is the frequency with which the disc oscillates?
5.6 How much energy does it take to heat 1 liter of soup from room temperature (20∘C) to 65∘C? (You may assume that the heat capacity of the soup is the same as for water.)
5.7 A “low-flow” showerhead averages 4.8 L/min. Taking other data from Example 5.3, estimate the energy savings (in J/y) if all the people in your country switched from US code to low-flow shower heads. (You will need to estimate the average number of showers per person per year for the people of your country.)
5.8 A solar thermal power plant currently under construction will focus solar rays to heat a molten salt working fluid composed of sodium nitrate and potassium nitrate. The molten salt is stored at a temperature of 300∘C, and heated in a power tower to 550∘C. The salt has a specific heat capacity of roughly 1500 J/kg K in the temperature range of interest. The system can store 6000 metric tonnes of molten salt for power generation when no solar energy is available. How much energy is stored in this system?
5.9 A new solar thermal plant being constructed in Australia will collect solar energy and store it as thermal energy, which will then be converted to electrical energy. The plant will store some of the thermal energy in graphite blocks for nighttime power distribution. According to the company constructing the plant, the plant will have a capacity of 30 MkWh/year. Graphite has a specific heat capacity of about 700 J/kg K at room temperature (see [26] for data on the temperature dependence of the specific heat of graphite). The company claims a storage capacity of 1000 kWh/tonne at 1800∘C. Is this plausible? Explain. If they are planning enough storage so that they can keep up the same power output at night as in the day, roughly how much graphite do they need? Estimate the number of people whose electrical energy needs will be supplied by this power plant.
5.10 A start-up company is marketing steel “ice cubes” to be used in place of ordinary ice cubes. How much would a liter of water, initially at 20∘C, be cooled by the addition of 10 cubes of steel, each 2.5 cm on a side, initially at −10∘C? Compare your result with the effect of 10 ordinary ice cubes of the same volume at the same initial temperature. Would you invest in this start-up? Take the density of steel to be 8.0 gm/cm and assume the heat capacities to be constants, independent of temperature.
5.11 Roughly 70% of the m of Earth's surface is covered by oceans. How much energy would it take to melt enough of the ice in Greenland and Antarctica to raise sea levels 1 meter? Suppose that 1% of energy used by humans became waste heat that melts ice. How long would it take to melt this quantity of ice? If an increase in atmospheric CO led to a net energy flux of 2000 TW of solar energy absorbed into the Earth system of which 1% melts ice, how long would it take for sea levels to rise one meter?
5.12 Carbon dioxide sublimes at pressures below roughly 5 atm (see Figure 5.12). At a pressure of 2 atm this phase transition occurs at about −69∘ with an enthalpy of sublimation of roughly 26 kJ/mol. Suppose a kilogram of solid CO2 at a temperature of −69∘C is confined in a cylinder by a piston that exerts a constant pressure of 2 atm. How much heat must be added to completely convert it to gas? Assuming CO2 to be an ideal gas, how much work was done by the CO2 in the course of vaporizing? If the ambient pressure outside the cylinder is 1 atm, how much useful work was done? Finally, by how much did the internal energy of the CO2 change when it vaporized?
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1 Avogadro's number is defined as the number of carbon atoms in 12 g of pure carbon-12, and has been measured experimentally to take approximately the value stated. The molar mass of a substance is the mass of one mole of the substance, generally quoted in units of g/mol. Molecular and molar masses, as well as the isotope carbon-12, are discussed further in §9 and §17.
2 A thermodynamic state is a macroscopic characterization of a system consistent with many possible microscopic configurations of the molecules or other constituents (microstates), as discussed further in §8.1.
3 The physics that preceded quantum mechanics is now called classical physics. So when we say that the molecules move about classically, we mean that their translational motion can be described by Newton’s Laws; their internal dynamics may nonetheless be strongly affected by quantum physics at low temperatures.
4 Rotations of an atom (and of a diatomic molecule around its axis) can occur at very high temperatures, along with other quantum excitations that eventually lead to ionization of the atom. These effects are described in §9 (Energy in matter).
5 Note that we use the scalar quantity p here and elsewhere to denote pressure, while the vector quantity p or component denotes momentum.
6 In some texts, a special notation, and , is introduced for the differential of quantities for which there is no corresponding state function. We use the simple notation and ask the reader to remember the fact that neither the “heat content” nor the “work content” of a system are meaningful concepts. The quantities Q and W similarly always refer in this text to a quantity of energy transferred through heat or by work.
7 In general, we use to denote the heat capacity of a substance per molecule, in units of .
CHAPTER 6
Heat Transfer
The transfer of heat from one system to another, or within a single system, is an issue of central importance in energy science. Depending on the circumstances, we may want to impede heat flow, to maximize it, or simply to understand how it takes place in natural settings such as the oceans or Earth’s interior. In practical situations, such as the transfer of heat from burning coal to steam flowing through pipes in a power plant boiler, the systems are far from equilibrium and consequently hard to analyze quantitatively. Although quantitative descriptions of heat transfer are difficult to derive from first principles, good phenomenological models have been developed since the days of Isaac Newton to handle many important applications. In this chapter we focus on those aspects of heat transfer that are related to energy uses, sources, and the environment. A thorough treatment of heat transfer including many applications can be found in [27].
6.1Mechanisms of Heat Transfer
Heat is transferred from one location or material to another in three primary ways: conduction – transport of heat through material by energy transfer on a molecular level, convection – transport of heat by collective motion of material, and radiation – transport by the emission or absorption of electromagnetic radiation. Before delving into these mechanisms in detail, we summarize each briefly and explain where it enters into energy systems described in later chapters.
6.1.1 Conduction
Within a solid, liquid, or gas, or between adjacent materials that share a common boundary, heat can be transferred by interactions between nearby atoms or molecules. As described in §5, at higher temperatures particles have more energy on average. When a faster-moving atom from a warmer region interacts with a slower-moving atom from a cooler region, the faster atom usually transfers some of its energy to the slower atom. This leads to a gradual propagation of thermal energy without any overall motion of the material. This is heat conduction.
Reader’s Guide
This chapter describes the transfer of thermal energy by conduction, convection, and radiation. The principal focus is on conduction, but radiation and both free and forced convection are also characterized. The basic concepts are applied in the context of heat loss from buildings. The heat equation is introduced and applied to the problem of annual ground temperature variation.
Prerequisites: §2 (Mechanics), §3 (Electromagnetism), §4 (Waves and light), §5 (Thermal energy).
Conduction is the principal mechanism for heat transfer through solids, and plays a role in many energy systems. For example, conduction determines the way that heat flows in the solid outer layers of Earth’s crust (§32). It is also responsible for heat loss through the solid walls of buildings. Heat can be conducted in gases and liquids as well, but convection is – under many circumstances – more effective at moving heat in fluids. Heat transfer by conduction is the simplest of the three mechanisms to describe quantitatively. The fundamental theory was developed by Joseph Fourier in the early 1800s, and has quite a bit in common with the theory of electricity conduction that was described in §3.
6.1.2 Convection
Collective motion of molecules within a fluid can effectively transport both mass and thermal energy. This is the mechanism of convection. Examples of convection include currents or eddies in a river, flow in a pipe, and the rising currents of air in a thermal updraft over warm land. As the constituent molecules composing a fluid move through convection, they carry their thermal energy with them. In this way, heat can be transported rapidly over large distances. Convection generally transports heat more quickly in gases than does conduction.
Figure 6.1 (a) A metal spoon used to stir boiling water quickly conducts heat making it impossible to hold without a plastic handle. (b) Convective plumes of fog and steam rising from power plant cooling towers carry heat upward quickly. (Credit: Bloomberg/Bloomberg/Getty Images)
Convection in energy applications can be divided into two general categories: forced convection and free convection.
Forced convection occurs when a fluid is forced to flow by an external agent (such as a fan). Forced convection is a very effective means of transporting thermal energy. Often heat transport through forced convection occurs in a situation where the fluid is forced to flow past a warmer or colder body, and the rate of heat transfer is driven by the temperature difference between the object and the fluid. Although the underlying fluid dynamics can be complex [27], forced convection is often described well by a simple empirical law first proposed by Newton and known as Newton’s Law of cooling. Forced convection plays a significant role in heat extraction devices such as air-conditioners and in steam engines, where heat must be transported from one location to another as quickly and as efficiently as possible. We analyze forced convection in §6.3.1 below.
Free convection – also known as natural convection – occurs when a fluid rises or sinks because its temperature differs from that of its surroundings. Convective heat transfer occurs, for example, when air directly over a surface feature (such as a large blacktop parking lot) is heated from below. The air near the surface warms, expands, and begins to rise, initiating a convective upward flow of heat. Free convection is believed to be the principal method of heat transfer in Earth’s liquid mantle and outer core (§32). Convective heat transport in the atmosphere plays an important role in global energy balance and temperature regulation (§34). Free convection usually arises spontaneously as a result of instability and is more difficult to analyze than forced convection. We touch briefly upon free convection here only to the extent that it enters a qualitative discussion of the effectiveness of insulation; we return to this topic later in the context of atmospheric physics.
6.1.3 Radiation
Finally, any material at a nonzero temperature radiates energy in the form of electromagnetic waves. These waves are generated by the microscopic motion of the charged particles (protons and electrons) composing the material. The higher the temperature, the more vigorous the motion of the particles in matter, and the more electromagnetic waves they radiate. As reviewed in §4, electromagnetic waves carry energy. Thus, the warmer an object, the more energy it radiates.
The fundamental law governing thermal radiation, the Stefan–Boltzmann law, is relatively simple, but its implementation in practice can be complex. The situation is simplest when a hot object is radiating into empty space. When matter is present, conduction and convection often dominate over radiation. Only radiation, however, can transfer energy across a vacuum. In particular, virtually all of the Sun’s energy that reaches Earth comes via radiative heat transfer. To remain in approximate thermal equilibrium, Earth must radiate the same amount of energy back into space. These applications are described in more detail in later chapters. In fact, radiation is also the primary cooling mechanism for a human body at rest: a typical human radiates at about 1000 W, though in comfortable temperature ranges most (~90%) of this energy is returned by radiation from the surroundings.
Figure 6.2 A traditional glass blower’s kiln. The interior is heated to about 1100∘C, at which temperature the kiln is filled with radiation that appears yellow to our eyes.
6.2Heat Conduction
Many aspects of heat conduction can be understood using the physical concepts developed in §2, §3, and §5. In particular, an analogy to conduction of electricity in the presence of resistance, developed in this section, helps visualize heat conduction and simplifies the analysis of practical applications.
Mechanisms of Heat Transfer
The principal methods of heat transfer are (1) conduction, where energy of molecular motion is transferred between particles through a material; (2) convection, where thermal energy is carried by the collective motion of a fluid; and (3) radiation, where the motion of hot molecules gives rise to radiation that carries away energy.
6.2.1 Heat Flux
Heat conduction takes place when energy is transferred from more energetic particles to less energetic ones through intermolecular or interatomic interactions. When molecules with differing amounts of energy interact, energy will generally be transferred from the more energetic molecule to the less. This leads to a gradual flow of thermal energy away from regions of higher temperature towards regions of lower temperature. In liquids and gases at room temperature, thermal energy resides in the kinetic energy of translation and rotation and to a lesser extent in molecular vibration. Heat is transferred by intermolecular collisions and through the diffusive propagation of individual molecules. In a solid, the thermal energy resides principally in the vibrational energy of the atoms within the solid. The same interatomic forces that hold the atoms in place also transmit the vibrational energy between neighboring particles. A model for this system might be a web of interconnected springs. If springs on one side of the web are set in motion, their energy rapidly diffuses toward the other end.
The temperature in an object through which heat is flowing can vary as a function of position x and time t, T(x, t). As discussed in §5, T at a point x refers to a local average over a small volume, a cubic micron for example, centered at x that contains a great number of individual molecules. The flow of heat through an object is described by a vector-valued function that specifies the amount of thermal energy flowing in a particular direction per unit time and per unit area. These are the characteristics of a flux as described in Box 3.1. We therefore define the heat flux to be the thermal energy per unit time per unit area, or power per unit area (W/m2 in SI units), flowing across a surface normal to the vector . Like T, q can depend on both x and t. It is also useful to define the rate that thermal energy flows across a surface , which has the units of power.
6.2.2 Fourier’s Law
Everyday experience suggests that the greater the temperature difference across an object, the greater the rate that heat flows through it. Fingers lose heat through insulating gloves, for example, much more rapidly when the ambient temperature is –40∘C than when it is 0∘C. At the molecular level, since temperature and internal energy are roughly proportional, a larger temperature gradient increases the rate at which molecular collisions transfer energy. If the temperature is locally constant, , then nearby molecules have the same average energy and molecular interactions on average transfer no energy. So it is reasonable to expect a linear relation of the form , at least when is small. In the early nineteenth century, the French mathematician and physicist Joseph Fourier proposed just such a relation on the basis of empirical observations,
