LECTURE NOTES ON THERMODYNAMICS

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LECTURENOTESON THERMODYNAMICS Joseph M. Powers Department of Aerospace and Mechanical Engineering University of Notre Dame Notre Dame, Indiana 46556-5637 USA updated 11 March 2017, 1:55pmChapter 1 Introduction Read BS, Chapter 1 1.1 Some semantics We introduce here classical thermodynamics. The word “thermo-dynamic,” used first by 1 2 Thomson (later Lord Kelvin), has Greek origin, and is translated as the combination of • θǫ´ρμη, therme: heat, and • δυ´ναμις, dynamis: power. An image of Thomson and his 1849 first use of the word is given in Fig. 1.1. Figure 1.1: William Thomson (Lord Kelvin) (1824-1907), Ulster-born Scottish scientist; im- agefromhttp://www-history.mcs.st-and.ac.uk/∼history/Biographies/Thomson.html and image giving the first use of “thermo-dynamic” extracted from his 1849 work. 1 W. Thomson (later Kelvin), 1849, “An account of Carnot’s theory of the motive power of heat; with numerical results deduced from Regnault’s experiments on steam,” Transactions of the Royal Society of Edinburgh,16:541-574. SeealsoC. W. Smith, 1977,“WilliamThomsonandthecreationofthermodynamics: 1840-1855,” Archive for History of Exact Sciences, 16(3): 231-288. 2 All Greek spellings and etymologies are drawn from the Oxford English Dictionary, 2nd edition, 1989. 1112 CHAPTER 1. INTRODUCTION The modifier “classical” is used to connote a description in which quantum mechanical effects, the molecular nature of matter, and the statistical nature of molecular behavior are not considered in any detail. These effects will not be completely ignored; however, they will be lumped into simple averaged models which are valid on the macroscale. As an example, for ordinary gases, our classical thermodynamics will be valid for systems whose characteristic length scale is larger than the mean free path between molecular collisions. −6 For air at atmospheric density, this about 0.1 μm (1 μm = 10 m). Additionally, “classical” also connotes a description in which the effects of finite time- dependency are ignored. In this sense, thermodynamics resembles the field of statics from P 2 2 Newtonian mechanics. Recall Newton’s second law of motion, m d x/dt = F, where m is the mass, x is the position vector,t is time, and F is the force vector. In the statics limit P where F =0, inertialeffectsareignored, asistime-dependency. Now, aNewtonianwould consider dynamics to imply motion, and so would consider thermodynamics to imply the time-dependent motion of heat. So a Newtonian would be more inclined to call the subject of these notes “thermostatics.” However, if we return to the earlier Greek translation of dynamics as power, we are actually truer to the classical connotation of thermodynamics. For the fundamental interplay of thermodynamics is that between so-called thermal energy (as might be thought of when considering heat) and mechanical energy (as might be thought ofwhenconsideringpower,aworkrate). Moreformally,adoptingthelanguageofBS(p.13), we will take the definition • thermodynamics: the science that deals with heat and work and those properties of matter that relate to heat and work. Oneofthemaingoalsofthesenoteswillbetoformalizetherelationshipbetweenheat, work, and energy. We close this section by noting that the concept of energy has evolved through time, but 3 has ancient origins. The word itself had its first recorded use by Aristotle. His portrait, along with an image of the relevant section of an 1818 translation of his work, is depicted in , ǫ Figs.1.2. IntheGreek,theword νǫ´ργǫια,“energeia,”connotesactivityoroperation. While the word was known to Aristotle, its modern usage was not; it was the English polymath Thomas Young who first used the word “energy,” consistent with any sort of modern usage, 4 in this case kinetic energy. A portrait of Young and an image of his text defining energy, in actuality kinetic energy, in modern terms are shown in Fig. 1.3. 3 Aristotle,∼335 BC, The Rhetoric, Poetic, and Nicomachean Ethics, Book III, Ch. XI, English transla- tion by T. Taylor, 1818, Black, London, see pp. 242-243. 4 T. Young, 1807, Lectures on Natural Philosophy, William Savage, London, p. 52. CC BY-NC-ND. 11 March 2017, J. M. Powers.1.1. SOME SEMANTICS 13 Figure 1.2: Aristotle (384 BC-322 BC), Greek philosopher who gives the first recorded use of the word “energy” and whose method of logic permeates classical thermodynamics; image from http://www-history.mcs.st-and.ac.uk/∼history/Biographies/Aristotle.html and an image of Aristotle’s usage of the word “energy” from his Nicomachean Ethics. Figure 1.3: Thomas Young (1773-1829), English natural philosopher; image from http://en.wikipedia.org/wiki/Thomas Young (scientist), and a reproduction of his more modern 1807 definition of (kinetic) energy. CC BY-NC-ND. 11 March 2017, J. M. Powers.14 CHAPTER 1. INTRODUCTION Finally, though she did not use the word “energy,” the notion of what is now known as 5 kinetic energy being related to the square of velocity was first advanced by du Chˆatelet, pictured in Fig. 1.4. ´ Figure 1.4: Gabrielle Emilie Le Tonnelier de Breteuil, marquise du Chˆatelet (1706-1749), French physicist; image from http://en.wikipedia.org/wiki/Emilie du Chatelet. 1.2 Historical milestones Thermodynamics has a long history; unfortunately, it was not blessed with the crispness of development that mechanics realized with Newton. In fact, its growth is filled with false 6 7,8 steps, errors, and debate which continues to this day. Truesdell and Mu¨ller summarize the development in their idiosyncratic histories. Some of the milestones of its development are given here: • first century AD: Hero of Alexandria documents many early thermal engines. • 1593: Galileo develops a water thermometer. • 1650: Otto von Guericke designs and builds the first vacuum pump. • 1662: Robert Boyle develops his law for isothermal ideal gases. 5 ´ E. du Chˆatelet, 1740, Institutions de Physique, Chez Prault, Paris. 6 C. Truesdell, 1980, The Tragicomical History of Thermodynamics, 1822-1854, Springer, New York. 7 I. Mul ¨ ler, 2007, A History of Thermodynamics: the Doctrine of Energy and Entropy, Springer, Berlin. 8 I. Mul ¨ ler and W. H. Mul ¨ ler, 2009, Fundamentals of Thermodynamics and Applications with Historical Annotations and Many Citations from Avogadro to Zermelo, Springer, Berlin. CC BY-NC-ND. 11 March 2017, J. M. Powers.1.2. HISTORICAL MILESTONES 15 • 1679: Denis Papin develops his steam digester, forerunner to the steam engine. • 1698: Thomas Savery patents an early steam engine. • 1710: Thomas Newcomen creates a more practical steam engine. • 1760s: Joseph Black develops calorimetry. • 1780s: James Watt improves the steam engine. • 1798: Benjamin Thompson (Count Rumford) considers the mechanical equivalent of heat from cannon boring experiments. • 1824: Nicolas L`eonard Sadi Carnot discusses idealized heat engines. • 1840: GermainHenriHessconsidersanearlyversionofthefirstlawofthermodynamics for work-free chemical reactions. • 1840s: Julius Robert von Mayer relates heat and work. • 1840s: James Prescott Joule relates heat and work. • 1847: Hermann von Helmholtz publishes his theory of energy conservation. • 1848: William Thomson (Lord Kelvin) postulates an absolute zero of temperature. • 1850: Rudolf Julius Emanuel Clausius formalizes the second law of thermodynamics. • 1865: Clausius introduces the concept of entropy. • 1871: James Clerk Maxwell develops the Maxwell relations. • 1870s: Josiah Willard Gibbs further formalizes mathematical thermodynamics. • 1870s: Maxwell and Ludwig Boltzmann develop statistical thermodynamics. • 1889: Gibbs develops statistical mechanics, giving underlying foundations for classical and statistical thermodynamics. Much development continued in the twentieth century, with pioneering work by Nobel lau- reates: • Jacobus Henricus van’t Hoff (1901), • Johannes van der Waals (1910), • Heike Kamerlingh Onnes (1913), • Max Planck (1918), CC BY-NC-ND. 11 March 2017, J. M. Powers.16 CHAPTER 1. INTRODUCTION • Walther Nernst (1920), • Albert Einstein (1921), • Erwin Schr¨odinger (1933), • Enrico Fermi (1938), • Percy Bridgman (1946), • Lars Onsager (1968), • Ilya Prigogine (1977), and • Kenneth Wilson (1982). NotethatSirIsaacNewtonalsoconsideredthesubjectmatterofthermodynamics. Much of his work is concerned with energy; however, his theories are most appropriate only for mechanical energy. The notion that thermal energy existed and that it could be equivalent tomechanicalenergywasnotpartofNewtonianmechanics. Notehowever,thattemperature was known to Newton, as was Boyle’s law. However, when he tried to apply his theories to problems of thermodynamics, such as calculation of the speed of sound in air, they notably failed. The reason for the failure required consideration of the yet-to-be-developed second law of thermodynamics. 1.3 Philosophy of science note As with science in general, thermodynamics is based on empirical observation. Moreover, it is important that those observations be repeatable. A few postulates, also known as axioms, 9 will serve as the foundation of our science. Following Occam’s razor, we shall seek as few axioms as possible to describe this behavior. We will supplement these axioms with some necessary definitions to describe nature. Then we shall use our reason to deduce from the axioms and definitions certain theorems of engineering relevance. This approach, which has its foundations in Aristotelian methods, is not unlike the approach taken by Euclid to geometry, Aquinas to theology, or Newton to mechanics. A depiction of Euclid is given in Fig. 1.5. Consider for example that Euclid defined certain entitiessuchaspoints,lines,andplanes,thenadoptedcertainaxiomssuchasparallellinesdo not meet at infinity, and went on to prove a variety of theorems. Classical thermodynamics follows the same approach. Concepts such as system and process are defined, and axioms, knownasthelawsofthermodynamics,areproposedinsuchawaythattheminimumamount of theory is able to explain the maximum amount of data. Now, in some sense science can never be formally proved; it can only be disproved. We retain our axioms as long as they are useful. When faced with empirical facts that 9 from William of Occam, (c. 1288-c. 1348) English Franciscan friar and philosopher. CC BY-NC-ND. 11 March 2017, J. M. Powers.1.4. SOME PRACTICAL APPLICATIONS 17 Figure1.5: EuclidofAlexandria(∼325BC-∼265BC),Greekmathematicianwhoserational exposition of geometry formed a model for how to present classical thermodynamics; image from http://www-history.mcs.st-and.ac.uk/∼history/Biographies/Euclid.html. unambiguouslycontradictouraxioms,wearerequiredtothrowawayouraxiomsanddevelop new ones. For example, in physics, the Michelson-Morely experiment forced Einstein to abandon the axioms of Euclid, Newton, and Clausius for his theory of general relativity. It turns out that we can still use these axioms, as long as we are considering problems in which the speed of our reference frame is far less than the speed of light. In an example 10 from biology that is the topic of a popular science book, it was noted that it was once believed that all swans were white. This working hypothesis was perfectly acceptable until 1697, when a black swan was discovered in Australia. Thus, the “theory” (though it is not a highly profound theory) that all swans were white was unambiguously discredited. It will be briefly seen in this course that non-classical thermodynamics actually has a deep relation to probability and statistics and information, a topic which transcends thermodynamics. 1.4 Some practical applications It turns out that the classical approach to thermodynamics has had success in guiding the engineering of devices. People have been building mechanical devices based on thermal energy inputs for centuries, without the benefit of a cleanly enunciated theory. Famously, Hero of Alexandria, perhaps the first recognized thermal engineer, documented a variety of 11 devices. These include an early steam engine known as the æolipile, pumps, and a device to use fire to open doors. Hero and a nineteenth century rendition of his steam engine are shown in Fig. 1.6. While Hero’s contributions are a matter of some speculation inspired by ancient artistry, the much later works of Denis Papin (1647-1712) are more certain. Papin invented the so-called steam digester, which anticipated both the pressure cooker and the 10 N. N. Taleb, 2007, The Black Swan: The Impact of the Highly Improbable, Random House, New York. 11 P. Keyser, 1990, “A new look at Heron’s steam engine,” Archive for History of Exact Sciences, 44(2): 107-124. CC BY-NC-ND. 11 March 2017, J. M. Powers.18 CHAPTER 1. INTRODUCTION Figure 1.6: Hero of Alexandria (10-70 AD), Greek engineer and mathematician who devised some early ways to convert thermal energy into mechanical energy, and his æolipile; images from http://en.wikipedia.org/wiki/Hero of Alexandria. steam engine. The device used steam power to lift a weight. Depictions of Papin and his device are found in Fig. 1.7. Significant improvements were led by James Watt (1736-1819) Figure 1.7: French-born inventor Denis Papin (1647-1712) and his steam digester; images from http://en.wikipedia.org/wiki/Denis Papin. of Scotland. An image of Watt and one of his engines is shown in Fig. 1.8. CC BY-NC-ND. 11 March 2017, J. M. Powers.1.4. SOME PRACTICAL APPLICATIONS 19 a) b) Figure 1.8: a) Scottish engineer James Watt (1736-1819); image from http://en.wikipedia.org/wiki/James Watt, b) Sketch of one of Watt’s steam en- gines; image from W. J. M. Rankine, 1859, A Manual of the Steam Engine and Other Prime Movers, First Edition, Griffin, London. CC BY-NC-ND. 11 March 2017, J. M. Powers.20 CHAPTER 1. INTRODUCTION These engines were adopted for transportation. In 1807, the American engineer Robert Fulton (1765-1815) was the first to use steam power in a commercial nautical vessel, the Clermont, which was powered by a Boulton and Watt steam engine. Soon after, in 1811 in Scotland, the first European commercial steam vessel, the Comet, embarked. We have a sketch of the Comet and its steam power plant in Fig. 1.9. On land, steam power soon Figure 1.9: Sketch of the Comet and its steam engine; image from W. J. M. Rankine, 1859, A Manual of the Steam Engine and Other Prime Movers, First Edition, Griffin, London. enabled efficient rail transportation. A famous early steam locomotive was the English engineer Robert Stephenson’s (1803-1859) Rocket, sketched in Fig. 1.10. Figure 1.10: Sketch of the Rocket; image from W. J. M. Rankine, 1859, A Manual of the Steam Engine and Other Prime Movers, First Edition, Griffin, London. CC BY-NC-ND. 11 March 2017, J. M. Powers.1.4. SOME PRACTICAL APPLICATIONS 21 The effect of steam power, a contribution driven by engineers, on the development of the world remains remarkable. It is what is commonly known as a disruptive technology as its widespread adoption displaced other well-established technologies. While it is difficult to quantify historical pronouncements, it is likely that the effect on the world was even more profound than the introduction of networked computers in the late twentieth century. In short, steam power was the linchpin for the industrial revolution. Steam power replaced animal power as a prime mover throughout much of the world and, where implemented, enabled rapid development of broad economic segments: mining, manufacturing, land and seatransportation,amongothers. Largescalepopulationmovementsensuedasopportunities inurbanmanufacturingcentersmadeindustrialworkmoreappealingthanagriculturalwork. Certainly, changes precipitated by the advent of steam power were contributing factors in widespread social unrest in the nineteenth century, ranging from labor strife to war between nation states. The text of BS has an introduction to some more modern devices, listed here: • simple steam power plant, • fuel cells, • vapor-compression refrigeration cycle, • air separation plant, • the gas turbine, and • the chemical rocket engine. As an example, the main power plant of the University of Notre Dame, depicted in Fig. 1.11, is based on a steam power cycle which will be a topic of study in this course. Additionally, one might consider the following topics to have thermodynamic relevance: • gasoline and Diesel engines, • the weather, • cooking, • heating, ventilation, air conditioning, and refrigeration (HVAC&R), or • materials processing (metals, polymers, etc.). CC BY-NC-ND. 11 March 2017, J. M. Powers.22 CHAPTER 1. INTRODUCTION Figure 1.11: University of Notre Dame Power Plant; image from Matt Cashore, University of Notre Dame. CC BY-NC-ND. 11 March 2017, J. M. Powers.1.5. EXAMPLE TO ILLUSTRATE HOMEWORK SOLUTION STYLE 23 1.5 Example to illustrate homework solution style Proper technical communication is important for engineering. Here is an example of how one might construct a homework solution. We take an example involving mechanical energy from introductory physics: Example 1.1 A mass of m = 1 kg is initially at rest and is dropped from a height of y = y = 10 m above the o 2 ground, where gravitational acceleration g = 9.81 m/s . Neglect drag forces. Find the time to reach the ground, the kinetic energy as a function of time, and the potential energy as a function of time. Plot key results. The scenario is sketched in Fig. 1.12. The principle governing the motion of the body is Newton’s m = 1 kg 2 g = 9.81 m/s y = 10 m o y y = 0 m Figure 1.12: Sketch of problem for particle motion in a gravitational field. second law, embodied in a second order differential equation. The only force is the gravitational force acting in the negative y direction. This gives the equation 2 d y m =−mg. (1.1) 2 dt Note the mass m cancels here, giving 2 d y =−g. (1.2) 2 dt Integrate once to get dy =−gt+C , (1.3) 1 dt where C is a constant. Integrate a second time to get 1 1 2 y(t) =− gt +C t+C . (1.4) 1 2 2 We need two initial conditions for this second order ordinary differential equation. At time t = 0, we know from the problem statement that dy y(0) =y , (0) = 0. (1.5) o dt CC BY-NC-ND. 11 March 2017, J. M. Powers.24 CHAPTER 1. INTRODUCTION Applying the first initial condition, we get 1 2 y =− g(0) +C (0)+C =C . (1.6) o 1 2 2 2 Thus, we have 1 2 y(t) =− gt +C t+y . (1.7) 1 o 2 Apply the second initial condition to get 0 =−g(0)+C =C . (1.8) 1 1 Thus, we have 1 2 y(t) =− gt +y . (1.9) o 2 For the velocity, we get dy =−gt. (1.10) dt When the mass reaches the ground, y = 0. Solving for the time when y = 0, we get 1 2 0 = − gt +y , (1.11) o 2 1 2 gt = y , (1.12) o 2 r 2y o t = ± . (1.13) g We are considering t going forward, so we take the positive root, giving r 2y o t = , (1.14) g s 2(10 m) = , (1.15) m 9.81 2 s = 1.43 s. (1.16) The kinetic energy, KE, is   2 1 dy KE = m , (1.17) 2 dt 1 2 2 = mg t . (1.18) 2 The gravitational potential energy, PE, is PE = mgy, (1.19) 1 2 2 = mgy − mg t . (1.20) o 2 Note that KE +PE =mgy , (1.21) o which is a constant. Thus, mechanical energy is conserved here By conserved, we mean it does not change. CC BY-NC-ND. 11 March 2017, J. M. Powers.1.5. EXAMPLE TO ILLUSTRATE HOMEWORK SOLUTION STYLE 25 Numerically, we have for y(t), KE(t), and PE(t),   1 m 2 y(t) = − 9.81 t +(10 m), (1.22) 2 2 s   m 2 = − 4.904 t +(10 m), (1.23) 2 s     2 1 m KE(t) = (1 kg) − 9.81 t , (1.24) 2 2 s   2 kg m 2 = 48.12 t , (1.25) 4 s   2 kg m J 2 = 48.12 t , (1.26) 4 2 2 s kg m /s   J 2 = 48.12 t , (1.27) 2 s     2 m 1 m 2 PE(t) = (1 kg) 9.81 (10 m)− (1 kg) 9.81 t , (1.28) 2 2 s 2 s   J 2 = (98.1 J)− 48.12 t . (1.29) 2 s ThepositionasafunctionoftimeisplottedinFig.1.13. Thekinetic,potential,andtotalmechanical y(m  ) 10 8 6 4 2 t  (s) 0.4 0.8 1.2 Figure 1.13: Position versus time for particle accelerating in a gravitational field with no drag force. energiesasfunctionsoftimeareplottedinFig.1.14. Onecantellbyinspectionthataspotentialenergy decreases, kinetic energy increases just as much, rendering the total mechanical energy to be constant. If we include drag forces, the total mechanical energy is not constant; in fact, it dissipates with time. Wewillomitthedetails, butifweincludeasimpledragforceproportionaltotheparticlevelocity, we get the equations 2 d y dy dy m =−c −mg, y(0) =y , (t = 0) = 0. (1.30) o 2 dt dt dt Skipping the details of calculation, if we take c = 0.1 N s/m, and all other parameters as before, we CC BY-NC-ND. 11 March 2017, J. M. Powers.y g r e n y e g r l e a i n t e n l e a t i o t p n e t o p y g r e n e l a c i n a h c e m l a t o t 26 CHAPTER 1. INTRODUCTION energy ( J) total mechanical energy 100 80 60 40 20 t (s) 0.4 0.8 1.2 Figure 1.14: KE(t), PE(t) and total mechanical energy for a particle accelerating in a gravitational field with no drag force. find     −t 1 y(t) = (991 m)−(981 m)exp − 98.1 t, (1.31) 10 s s       −t −t 1 KE(t)+PE(t) = (14533.5 J)+(4811.8 J)exp −(19247.2 J)exp − 962.361 t. 20 s 10 s s (1.32) The kinetic, potential, and total mechanical energies as functions of time are plotted in Fig. 1.15. energy (  J) 100 80 60 40 20 t (s) 0.4 0.8 1.2 Figure 1.15: KE(t), PE(t) and total mechanical energy for a particle accelerating in a gravitational field in the presence of a drag force. CC BY-NC-ND. 11 March 2017, J. M. Powers. k i n k e i n t i e c t i c e n e n e r e r g g y y1.5. EXAMPLE TO ILLUSTRATE HOMEWORK SOLUTION STYLE 27 When drag forces are included, we begin with the same amount of total mechanical energy and potentialenergy. Attheendofthecalculation, wehavethesameamountofpotentialenergy(zero), but lesskineticenergyandlesstotalmechanicalenergy. Wheredidthisenergygo? Infact, itistransformed into another form of energy, thermal energy, which is not accounted for in Newtonian mechanics. When we properly account for thermal energy, we will again impose a conservation of total energy, one of the main topics of this course, to be considered at the outset of Chapter 5. We close with an image of Sir Isaac Newton in Fig. 1.16, who began to study issues related to thermodynamics, and whose scientific methods imbue its development. Figure 1.16: English genius Sir Isaac Newton (1643-1727), in a 1702 portrait by Sir Godfrey Kneller, whose classical mechanics broadly influenced the development of thermodynamics; image from http://commons.wikimedia.org/wiki/File:Sir Isaac Newton by Sir Godfrey Kneller, Bt.jpg. CC BY-NC-ND. 11 March 2017, J. M. Powers.28 CHAPTER 1. INTRODUCTION CC BY-NC-ND. 11 March 2017, J. M. Powers.Chapter 2 Some concepts and definitions Read BS, Chapter 2 2.1 Thermodynamic system and control volume We take the following definitions: • Thermodynamic system: a quantity of fixed mass under investigation, • Surroundings: everything external to the system, • System boundary: interface separating system and surroundings, and • Universe: combination of system and surroundings. The system, surroundings, system-boundary for a universe are shown for a potato-shaped system in Fig. 2.1. We allow two important interactions between the system and its sur- roundings: • heat can cross into the system (our potato can get hot), and • work can cross out of the system (our potato can expand). Now, the system boundaries can change, for example the potato might expand on heating, but we can still distinguish the system and the surroundings. We now define an • isolated system: a system which is not influenced by its surroundings. Note that a potato with thick and inelastic skin will be isolated. We distinguish the system, which has constant mass, but possible variable volume, from the • ControlVolume: fixedvolumeoverwhichmasscanpassinandoutofitsboundary. The control volume is bounded by the 29

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