The Physics of Energy, page 5
Energy at the global scale Energy quantities at the global scale are measured in large units like exajoules (1 EJ = 1018 J) or quads (1 quad = 1015 Btu = 1.055 EJ), and power is measured in terawatts (1 TW = 1012 W). Total world oil consumption in 2014, for example, was about 196 EJ. The total energy used by humanity in that year was close to 576 EJ. This represents a sustained power usage of around 17 TW. Energy flow through many natural systems at the global scale is conveniently measured in units of terawatts as well. For example, solar energy hits the Earth at a rate of roughly 173 000 TW (§22). The total rate at which wave energy hits all the world’s shores is only a few (roughly 3) TW (§31.2).
Energy at the micro scale To understand many energy systems, such as photovoltaic cells, chemical fuels, and nuclear power plants, it is helpful to understand the physics of microscopic processes involving individual molecules, atoms, electrons, or photons of light. The electron volt (1.12) is the standard unit for the micro world. When an atom of carbon in coal combines with oxygen to form CO2, for example, about 4 eV ( J) of energy is liberated. Another example at the electron volt scale is a single photon of green light, which carries energy 2.5 eV. Photovoltaic cells capture energy from individual photons of light, as we describe in detail in §25.
Scales of Energy
Some useful energy numbers to remember are (in round numbers, circa 2010):
10 MJ: daily human food intake (2400 Cal)
200 MJ: average daily human energy use
500 EJ: yearly global energy use
15 TW: average global power use
Discussion/Investigation Questions
1.1 Given that energy is everywhere, and cannot be destroyed, try to articulate some reasons why it is so hard to get useful energy from natural systems in a clean and affordable way. (This question may be worth revisiting occasionally as you make your way through the book.)
1.2 A residential photovoltaic installation is described as producing “5000 kilowatt hours per year.” What is a kilowatt hour per year in SI units? What might be the purpose of using kWh/y rather than the equivalent SI unit?
1.3 Try to describe the flow of energy through various systems before and after you use it in a light bulb in your house. Which of the various forms of energy discussed in the chapter does the energy pass through?
1.4 Give examples of each of the types of energy described in §1.2.
1.5 Discuss some possible answers, depending on the context, to the question posed in the text, “What is the energy of a bucket of water?”
1.6 Compare the global average rate of energy use per person to typical human food energy consumption. What does this say about the viability of biologically produced energy as a principal energy solution for the future?
Problems
1.1 Confirm eq. (1.14) and the estimate for the rhinoceros’s kinetic energy below eq. (1.11) by explicit calculation.
1.2 How much energy would a 100 W light bulb consume if left on for 10 hours?
1.3 In a typical mid-latitude location, incident solar energy averages around 200 W/m2 over a 24-hour cycle. Compute the land area needed to supply the average person’s energy use of 200 MJ/day if solar panels convert a net 5% of the energy incident over a large area of land to useful electrical energy. Multiply by world population to get an estimate of total land area needed to supply the world energy needs from solar power under these assumptions. Compare this land area to some relevant reference areas – the Sahara Desert, your native country, etc.
1.4 The US total energy consumption in 2014 was 98.0 quads. What is this quantity in joules? About 83% of US energy comes from fossil fuels. If the whole 83% came from oil, how many tons of oil equivalent (toe) would this have been? How many barrels of oil (bbl) is this equivalent to?
1.5 The gravitational potential energy of an object of mass m at a distance h above ground is given by . Use dimensional analysis to compute the units of g. What does this result suggest about the behavior of objects near Earth’s surface?
1.6 The energy emitted or absorbed in chemical processes is often quoted in kilojoules per mole (abbreviated mol) of reactants, where a mole contains (Avogadro’s number) molecules (§5). Derive the conversion factor from eV/molecule to kJ/mol.
1.7 The energy available from one kilogram of is 82 TJ. Energy is released when each uranium nucleus splits, or fissions. (The fission process is described in §16, but you do not need to know anything about the process for this problem.) 235 grams of contain approximately Avagadro’s number (see Problem 1.6) atoms of . How many millions of electron volts (MeV) of energy are released, on average, when a nucleus fissions? How many kilograms of gasoline have the same energy content as one kilogram of ?
1.8 The US total electrical power consumption in 2010 was 3.9 TkWh. Utilities try to maintain a capacity that is twice the average power consumption to allow for high demand on hot summer days. What installed generating capacity does this imply?
CHAPTER 2
Mechanical Energy
The systematic study of physical laws begins both logically and historically with the basic notions of classical mechanics. The laws of classical mechanics, as formulated by the English physicist and mathematician Isaac Newton, describe the motion of macroscopic physical objects and their interaction through forces (see Box 2.1). The mathematical framework of calculus provides the necessary tools with which to analyze classical mechanical systems. In this chapter, we review the fundamental principles of mechanics – including kinetic energy, forces, potential energy, and frictional energy loss – all in the context of the use of energy in transport.
Box 2.1. Newton’s Laws
Newton’s three laws of motion are:
1. An object in motion remains in motion with no change in velocity unless acted on by an external force.
2. The acceleration a of an object of mass m under the influence of a force F is given by .
3. For every action there is an equal and opposite reaction.
We assume that the reader has previously encountered these laws in the context of introductory physics. In this chapter we describe how these laws can be understood in the context of energy. A more detailed discussion of the inertial reference frames in which Newton’s laws hold and analysis of apparent forces in accelerating reference frames are given in §27 (Ocean Energy Flow).
In 2016, approximately 29% of the energy used in the US – roughly 29 EJ – was used for transportation [12], including personal and commercial, land, water, and air transport (Figure 2.1). This energy use led to CO2 emissions of almost half a gigaton, or about one third of total US emissions [12]. Transport of people, as well as food, raw materials, and other goods, presents a particular challenge for clean and efficient energy systems. Because cars, airplanes, and trucks are all mobile and not (at least currently) directly connected to any kind of energy grid, they must carry their fuel with them. Historically this has favored the use of fossil fuels such as gasoline, which have high energy density and are easily combusted. In later chapters we examine other options for transportation energy sources. Here we focus on how energy is actually used in transport. Studying how energy is used to put a vehicle in motion (kinetic energy), take a vehicle up and down hills (potential energy), and keep a vehicle in motion in our atmosphere (air resistance), gives insight into how energy needs for transport might be reduced, independent of the fuel option used.
Figure 2.1 2016 US energy use by economic sector. Roughly 29% of US energy use is for transport, including air, land, and water transport of people and goods [12].
Reader’s Guide
This chapter contains a concise review of the basic elements of classical mechanics that are needed for the rest of the book. The core principle of energy conservation serves as a guide in developing and connecting the key concepts of mechanics. While this chapter is fairly self-contained, we assume that the reader has some previous exposure to classical mechanics and to elementary calculus.
The centerpiece of the chapter is a study of energy use in transport.
To introduce the principles of mechanics in the context of energy usage, we analyze a specific example throughout much of this chapter. Imagine that four friends plan to drive from MIT (Massachusetts Institute of Technology) in Cambridge, Massachusetts to New York City in a typical gasoline-powered automobile, such as a Toyota Camry. The distance from Cambridge to New York is approximately 210 miles (330 km).1 The Toyota Camry gets about 30 miles per gallon (30 mpg ≅ 13 km/L) highway mileage, so the trip requires about 7 gallons (27 L) of gasoline. The energy content of gasoline is approximately 120 MJ/gallon (32 MJ/L) [13], so the energy needed for the trip amounts to (7 gallons) × (120 MJ/gallon) ≅ 840 MJ. This is a lot of energy, compared for example to the typical daily human food energy requirement of 10 MJ. Where does it all go? As we describe in more detail in later chapters, automobile engines are far from perfectly efficient. A typical auto engine only manages to convert about 25% of its gasoline fuel energy into mechanical energy when driving long distances on a highway. Thus, we can only expect about 210 MJ of delivered mechanical energy from the 7 gallons of gasoline used in the trip. But 210 MJ is still a substantial amount of energy.
Figure 2.2 In this chapter we illustrate the basics of mechanics by analyzing the energy used by an automobile driving from Cambridge, Massachusetts to New York City.
The principles of mechanics developed in this chapter enable us to give a rough “back-of-the-envelope” estimation of how the 210 MJ of energy is used. In addition to the basic elements of kinetic energy, potential energy, forces, work, and power, we also need to address questions of friction and air resistance. The drag coefficient that arises in studying air resistance of a moving vehicle is an example of a phenomenological parameter that captures complicated details of a physical system in a single number. Such parameters can be estimated or measured even though they may be prohibitively difficult to compute from first principles. Parameters of this type are common in science and engineering and appear frequently throughout this book.
We assume that the reader has had prior exposure to the basics of classical mechanics and integral calculus. A brief review of some aspects of multi-variable calculus is given in Appendix B. Readers who are interested in a more comprehensive pedagogical introduction to elementary mechanics and/or calculus should consult an introductory textbook such as [14] or [15] for mechanics or [16] for calculus.
2.1Kinetic Energy
The simplest manifestation of energy in physics is the kinetic energy associated with an object in motion. For an object of mass m, moving at a speed υ, the motional or kinetic energy is given by2
(2.1)
In the context of the road trip from Cambridge to New York, the kinetic energy of the automobile carrying three passengers and the driver, at a total mass of 1800 kg ( 4000 lbs), moving at a speed of 100 km/h, or 62 mph (miles per hour), is
(2.2)
If we assume that the road is roughly flat from Cambridge to New York, and if we neglect friction and air resistance as is often done in elementary mechanics courses, it seems that only 0.7 MJ of energy would be needed to make the trip. The driver needs only to get the car rolling at 100 km/h, which takes 0.7 MJ, and the vehicle could then coast all the way with no further energy expenditure. We should include the effect of driving on city streets to reach the freeway, where it is occasionally necessary to stop at a red light. After every stop the car must get back to full speed again. This is why fuel efficiency in the city is very different from highway driving. But even assuming a dozen stops with acceleration to 50 km/h between each stop, we only need about 2 MJ of additional energy (kinetic energy at 50 km/h is 1/4 of that at 100 km/h). This is far less than the 210 MJ of fuel energy we are using. So what do we need the other 208 MJ for?
Kinetic Energy
The kinetic energy of an object of mass m moving at a speed υ is given by
Example 2.1Kinetic Energy of a Boeing 777–300ER
Consider a Boeing 777–300ER aircraft, loaded with fuel and passengers to a mass of 350 000 kg, at a cruising speed of 900 km/h. The plane has a kinetic energy of
This represents the daily food energy intake of a thousand people, or the full average daily energy consumption of over 50 people.
Of course, the road is not really completely flat between Cambridge and New York City. There is some moderately hilly terrain along the way that we must account for (see Figure 2.3). (For small hills, the car regains the energy used to go up when the car goes down again, but for large hills energy is lost to braking on the way down.) To include the effects of hills, we need to review the notion of mechanical potential energy.
Figure 2.3 As the car goes up hill, kinetic energy is changed into potential energy. Some of this energy is returned to kinetic form as the car goes back down the hill, but some is lost in braking to moderate the car’s speed.
2.2Potential Energy
As mentioned in §1, energy can be transformed from one form to another by various physical processes, but never created or destroyed. As a ball that is thrown directly upward slows due to the force of gravity, it loses kinetic energy. The lost kinetic energy does not disappear; rather, it is stored in the form of gravitational potential energy. We can use the fundamental principle of conservation of energy to understand the relationship between kinetic and potential energy in a precise quantitative fashion.
Potential energy is energy stored in a configuration of objects that interact through forces. To understand potential energy we begin by reviewing the nature of forces in Newtonian physics. The exchange of energy between potential and kinetic leads us to the concepts of work and power. Describing forces and potential energy in three-dimensional systems leads naturally to a review of vectors and to momentum. In the end we find a more significant contribution to the energy cost of the road trip.
Potential energy plays an important role in many aspects of practical energy systems. For example, mechanical energy, such as that produced by a windmill, can be stored as potential energy by using the mechanical energy to pump water uphill into a reservoir.
2.2.1 Forces and Potential Energy
Newton’s second law of motion describes the action of a force F on an object of mass m. For the moment we assume that the object of mass m is moving in a single dimension along a trajectory , describing the position x of the object as a function of the time t. Newton’s law in this context is
(2.3)
(We often use the shorthand notations and .)
According to eq. (2.3), an object that is acted on by any force will experience an acceleration. Recall that the velocity of an object is the rate of change of position (e.g. , and the acceleration is the rate of change of velocity (e.g. ). A force is an influence on an object from another object or field. In Newtonian physics, eq. (2.3) can be taken as the definition as well as a description of the effect of a force. The four fundamental forces of nature are described in §14, but for most of this book we are only concerned with gravitational and electromagnetic forces. As an example of Newton’s second law, consider a ball thrown up in the air near Earth’s surface. The ball experiences a constant downward acceleration of magnitude towards Earth, indicating the presence of a force of magnitude . If we denote the height of the ball at time t by , then
(2.4)
Example 2.2 Airplane at Altitude
Recall that a 777–300ER flying at 900 km/h has 11 GJ of kinetic energy. How much potential energy does it have when flying at an altitude of 12 000 meters (39 000 feet)?
Using (2.9), we have
Getting the plane up to cruising altitude takes about four times as much energy as getting it to cruising speed.
As the ball rises, the force of gravity reduces the speed, and therefore also the kinetic energy, of the ball. Here the principle of energy conservation comes into play: this energy is not lost. Rather, it is stored in the potential energy associated with the position of the ball relative to earth. After the ball reaches the top of its trajectory and begins to fall, this potential energy is converted back into kinetic energy as the force (2.4) accelerates the ball downward again.
With this example in mind, we can find the general expression for potential energy simply by applying conservation of energy to an object subject to a known force or set of forces. First, how does a force cause kinetic energy to change?
(2.5)
This change can represent a transfer of kinetic energy to or from potential energy. We denote potential energy by V. In the absence of other forms of energy, conservation of energy requires . The change in potential energy in time dt is then given by
(2.6)
If the force acts in the same direction as the particle is moving, the kinetic energy increases and the potential energy must decrease by the same amount. If the force is acting in the opposite direction, the kinetic energy decreases and the potential energy increases by the same amount. Integrating eq. (2.6), the potential energy change when moving from an initial position to another position can be described by a function V(x) that satisfies
(2.7)
We have thus defined the potential energy of a system subject to known forces using the principle of conservation of energy. A potential energy function V can be defined in this way for any force that depends only on the coordinates describing the physical configuration of the system (e.g. in a one-dimensional system F depends only on x and not on or t).
