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Mechanical Engineers’ Handbook: Energy and Power, Volume 4, Third Edition. Edited by Myer Kutz Copyright 2006 by John Wiley & Sons, Inc. CHAPTER 3 THERMODYNAMICS FUNDAMENTALS Adrian Bejan Department of Mechanical Engineering and Materials Science Duke University Durham, North Carolina 1 2 INTRODUCTION THE FIRST LAW OF THERMODYNAMICS FOR CLOSED SYSTEMS THE SECOND LAW OF THERMODYNAMICS FOR CLOSED SYSTEMS THE ENERGY MINIMUM PRINCIPLE 94 5 THE LAWS OF THERMODYNAMICS FOR OPEN SYSTEMS RELATIONS AMONG THERMODYNAMIC PROPERTIES ANALYSIS OF ENGINEERING SYSTEM COMPONENTS REFERENCES 102 97 6 3 104 113 116 99 102 7 4 1 INTRODUCTION Thermodynamics describes the relationship between mechanical work and other forms of energy. There are two facets of contemporary thermodynamics that must be stressed in a review such as this. The ﬁrst is the equivalence of work and heat as two possible forms of energy exchange. This facet is expressed by the ﬁrst law of thermodynamics. The second aspect is the one-way character, or irreversibility, of all ﬂows that occur in nature. As expressed by the second law of thermodynamics, irreversibility or entropy generation is what prevents us from extracting the most possible work from various sources; it is also what prevents us from doing the most with the work that is already at our disposal. The objective of this chapter is to review the ﬁrst and second laws of thermodynamics and their implications in mechanical engineering, particularly with respect to such issues as energy conversion and conservation. The analytical aspects (the formulas) of engineering thermodynamics are reviewed primarily in terms of the behavior of a pure substance, as would be the case of the working ﬂuid in a heat engine or in a refrigeration machine. In the next chapter we review in greater detail the newer ﬁeld of entropy generation minimization (thermodynamic optimization) and the generation of system conﬁguration (constructal theory). SYMBOLS AND UNITS c speciﬁc heat of incompressible substance, J / (kg K) speciﬁc heat at constant pressure, J / (kg K) cP constant temperature coefﬁcient, m3 / kg cT speciﬁc heat at constant volume, J / (kg K) cv COP coefﬁcient of performance E energy, J 94 1 ƒ F g g h K m m ˙ mi M M n N0 P Q ˙ Q r R s S Sgen ˙ Sgen T u U v v Introduction 95 V V W ˙ Wlost ˙ Wsh x x Z I II speciﬁc Helmholtz free energy (u Ts), J / kg force vector, N gravitational acceleration, m / s2 speciﬁc Gibbs free energy (h Ts), J / kg speciﬁc enthalpy (u Pv), J / kg isothermal compressibility, m2 / N mass of closed system, kg mass ﬂow rate, kg / s mass of component in a mixture, kg mass inventory of control volume, kg molar mass, g / mol or kg / kmol number of moles, mol Avogadro’s constant pressure inﬁnitesimal heat interaction, J heat transfer rate, W position vector, m ideal gas constant, J / (kg K) speciﬁc entropy, J / (kg K) entropy, J / K entropy generation, J / K entropy generation rate, W / K absolute temperature, K speciﬁc internal energy, J / kg internal energy, J speciﬁc volume, m3 / kg speciﬁc volume of incompressible substance, m3 / kg volume, m3 velocity, m / s inﬁnitesimal work interaction, J rate of lost available work, W rate of shaft (shear) work transfer, W linear coordinate, m quality of liquid and vapor mixture vertical coordinate, m coefﬁcient of thermal expansion, 1 / K ratio of speciﬁc heats, cP / cv ‘‘efﬁciency’’ ratio ﬁrst-law efﬁciency second-law efﬁciency relative temperature, C SUBSCRIPTS ( )ƒ saturated liquid state (f ‘‘ﬂuid’’) ( )g saturated vapor state (g ‘‘gas’’) ( )s saturated solid state (s ‘‘solid’’) ( )in inlet port ( )out outlet port ( )rev reversible path ( )H high-temperature reservoir 96 Thermodynamics Fundamentals ( ( ( ( ( ( ( ( ( ( ( )L )max )T )C )N )D )0 )1 )2 )* ) low-temperature reservoir maximum turbine compressor nozzle diffuser reference state initial state ﬁnal state moderately compressed liquid state slightly superheated vapor state Deﬁnitions Boundary: The real or imaginary surface delineating the thermodynamic system. The boundary separates the system from its environment. The boundary is an unambiguously deﬁned surface. The boundary has zero thickness and zero volume. Closed System: A thermodynamic system whose boundary is not crossed by mass ﬂow. Cycle: The special process in which the ﬁnal state coincides with the initial state. Environment: The thermodynamic system external to the thermodynamic system. Extensive Properties: Properties whose values depend on the size of the system (e.g., mass, volume, energy, enthalpy, entropy). Intensive Properties: Properties whose values do not depend on the system size (e.g., pressure, temperature). The collection of all intensive properties constitutes the intensive state. Open System: A thermodynamic system whose boundary is permeable to mass ﬂow. Open systems (ﬂow systems) have their own nomenclature: the thermodynamic system is usually referred to as the control volume, the boundary of the open system is the control surface, and the particular regions of the boundary that are crossed by mass ﬂows are the inlet and outlet ports. Phase: The collection of all system elements that have the same intensive state (e.g., the liquid droplets dispersed in a liquid–vapor mixture have the same intensive state, that is, the same pressure, temperature, speciﬁc volume, speciﬁc entropy, etc.). Process: The change of state from one initial state to a ﬁnal state. In addition to the end states, knowledge of the process implies knowledge of the interactions experienced by the system while in communication with its environment (e.g., work transfer, heat transfer, mass transfer, and entropy transfer). To know the process also means to know the path (the history, or the succession of states) followed by the system from the initial to the ﬁnal state. State: The condition (the being) of a thermodynamic system at a particular point in time, as described by an ensemble of quantities called thermodynamic properties (e.g., pressure, volume, temperature, energy, enthalpy, entropy). Thermodynamic properties are only those quantities that do not depend on the ‘‘history’’ of the system between two different states. Quantities that depend on the system evolution (path) between states are not thermodynamic properties (examples of nonproperties are the work, heat, and mass transfer; the entropy transfer; the entropy generation; and the destroyed exergy—see also the deﬁnition of process). Thermodynamic System: The region or the collection of matter in space selected for analysis. 2 The First Law of Thermodynamics for Closed Systems 97 2 THE FIRST LAW OF THERMODYNAMICS FOR CLOSED SYSTEMS The ﬁrst law of thermodynamics is a statement that brings together three concepts in thermodynamics: work transfer, heat transfer, and energy change. Of these concepts, only energy change or, simply, energy, is a thermodynamic property. We begin with a review1 of the concepts of work transfer, heat transfer, and energy change. Consider the force Fx experienced by a system at a point on its boundary. The inﬁnitesimal work transfer between system and environment is W Fx dx where the boundary displacement dx is deﬁned as positive in the direction of the force Fx. When the force F and the displacement of its point of application dr are not collinear, the general deﬁnition of inﬁnitesimal work transfer is W F dr The work-transfer interaction is considered positive when the system does work on its environment—in other words, when F and dr are oriented in opposite directions. This sign convention has its origin in heat engine engineering, because the purpose of heat engines as thermodynamic systems is to deliver work while receiving heat. For a system to experience work transfer, two things must occur: (1) a force must be present on the boundary, and (2) the point of application of this force (hence, the boundary) must move. The mere presence of forces on the boundary, without the displacement or the deformation of the boundary, does not mean work transfer. Likewise, the mere presence of boundary displacement without a force opposing or driving this motion does not mean work transfer. For example, in the free expansion of a gas into an evacuated space, the gas system does not experience work transfer because throughout the expansion the pressure at the imaginary system–environment interface is zero. If a closed system can interact with its environment only via work transfer (i.e., in the absence of heat transfer Q discussed later), then measurements show that the work transfer during a change of state from state 1 to state 2 is the same for all processes linking states 1 and 2, 2 W 1 Q 0 E2 E1 In this special case the work-transfer interaction (W1 2) Q 0 is a property of the system, because its value depends solely on the end states. This thermodynamic property is the energy change of the system, E2 E1. The statement that preceded the last equation is the ﬁrst law of thermodynamics for closed systems that do not experience heat transfer. Heat transfer is, like work transfer, an energy interaction that can take place between a system and its environment. The distinction between Q and W is made by the second law of thermodynamics discussed in the next section: Heat transfer is the energy interaction accompanied by entropy transfer, whereas work transfer is the energy interaction taking place in the absence of entropy transfer. The transfer of heat is driven by the temperature difference established between the system and its environment.2 The system temperature is measured by placing the system in thermal communication with a test system called thermometer. The result of this measurement is the relative temperature expressed in degrees Celsius, ( C), or Fahrenheit, ( F); these alternative temperature readings are related through the conversion formulas 98 Thermodynamics Fundamentals ( C) ( F) 1F 5 – [ ( F) 9 5 – ( C) 9 5 –C 9 32] 32 The boundary that prevents the transfer of heat, regardless of the magnitude of the system– environment temperature difference, is termed adiabatic. Conversely, the boundary that is crossed by heat even in the limit of a vanishingly small system–environment temperature difference is termed diathermal. Measurements also show that a closed system undergoing a change of state 1 → 2 in the absence of work transfer experiences a heat interaction whose magnitude depends solely on the end states: 2 Q 1 W 0 E2 E1 In the special case of zero work transfer, the heat-transfer interaction is a thermodynamic property of the system, which is by deﬁnition equal to the energy change experienced by the system in going from state 1 to state 2. The last equation is the ﬁrst law of thermodynamics for closed systems incapable of experiencing work transfer. Note that, unlike work transfer, the heat transfer is considered positive when it increases the energy of the system. Most thermodynamic systems do not manifest purely mechanical ( Q 0) or purely thermal ( W 0) behavior. Most systems manifest a coupled mechanical and thermal behavior. The preceding ﬁrst-law statements can be used to show that the ﬁrst law of thermodynamics for a process executed by a closed system experiencing both work transfer and heat transfer is 2 2 Q 1 heat transfer 1 W work transfer E2 E1 energy change energy interactions (nonproperties) (property) The ﬁrst law means that the net heat transfer into the system equals the work done by the system on the environment, plus the increase in the energy of the system. The ﬁrst law of thermodynamics for a cycle or for an integral number of cycles executed by a closed system is Q W 0 Note that the net change in the thermodynamic property energy is zero during a cycle or an integral number of cycles. The energy change term E2 E1 appearing on the right-hand side of the ﬁrst law can be replaced by a more general notation that distinguishes between macroscopically identiﬁable forms of energy storage (kinetic, gravitational) and energy stored internally, E2 E1 U2 U1 mV 2 2 2 mV 2 1 2 mgZ2 mgZ1 energy change internal energy change kinetic energy change gravitaional energy change 3 The Second Law of Thermodynamics for Closed Systems 99 If the closed system expands or contracts quasistatically (i.e., slowly enough, in mechanical equilibrium internally and with the environment) so that at every point in time the pressure P is uniform throughout the system, then the work-transfer term can be calculated as being equal to the work done by all the boundary pressure forces as they move with their respective points of application, 2 2 W 1 1 P dV The work-transfer integral can be evaluated provided the path of the quasistatic process, P(V), is known; this is another reminder that the work transfer is path-dependent (i.e., not a thermodynamic property). 3 THE SECOND LAW OF THERMODYNAMICS FOR CLOSED SYSTEMS A temperature reservoir is a thermodynamic system that experiences only heat transfer and whose temperature remains constant during such interactions. Consider ﬁrst a closed system executing a cycle or an integral number of cycles while in thermal communication with no more than one temperature reservoir. To state the second law for this case is to observe that the net work transfer during each cycle cannot be positive, W 0 In other words, a closed system cannot deliver work during one cycle, while in communication with one temperature reservoir or with no temperature reservoir at all. Examples of such cyclic operation are the vibration of a spring–mass system, or a ball bouncing on the pavement: for these systems to return to their respective initial heights, that is, for them to execute cycles, the environment (e.g., humans) must perform work on them. The limiting case of frictionless cyclic operation is termed reversible, because in this limit the system returns to its initial state without intervention (work transfer) from the environment. Therefore, the distinction between reversible and irreversible cycles executed by closed systems in communication with no more than one temperature reservoir is W W 0 0 (reversible) (irreversible) To summarize, the ﬁrst and second laws for closed systems operating cyclically in contact with no more than one temperature reservoir are (Fig. 1) W Q 0 This statement of the second law can be used to show1 that in the case of a closed system executing one or an integral number of cycles while in communication with two temperature reservoirs, the following inequality holds (Fig. 1) QH TH QL TL 0 where H and L denote the high-temperature and the low-temperature reservoirs, respectively. Symbols QH and QL stand for the value of the cyclic integral Q, where Q is in one case 100 Thermodynamics Fundamentals Figure 1 The ﬁrst and second laws of thermodynamics for a closed system operating cyclically while in communication with one or two heat reservoirs. exchanged only with the H reservoir, and in the other with the L reservoir. In the reversible limit, the second law reduces to TH / TL QH / QL, which serves as deﬁnition for the absolute thermodynamic temperature scale denoted by symbol T. Absolute temperatures are expressed either in kelvins, T (K), or in degrees Rankine, T ( R); the relationships between absolute and relative temperatures are T (K) 1K ( C) 1C 273.15 K T ( R) 1 R ( F) 1F 459.67 R A heat engine is a special case of a closed system operating cyclically while in thermal communication with two temperature reservoirs, a system that during each cycle receives heat and delivers work: W Q QH QL 0 The goodness of the heat engine can be described in terms of the heat engine efﬁciency or the ﬁrst-law efﬁciency W QH 1 TL TH Alternatively, the second-law efﬁciency of the heat engine is deﬁned as1,3,4 3 The Second Law of Thermodynamics for Closed Systems W I 101 II W maximum (reversible case) 1 TL / TH A refrigerating machine or a heat pump operates cyclically between two temperature reservoirs in such a way that during each cycle it receives work and delivers net heat to the environment, W Q QH QL 0 The goodness of such machines can be expressed in terms of a coefﬁcient of performance (COP) COPrefrigerator QL W COPheat QH pump 1 TH / TL 1 W 1 1 TL / TH Generalizing the second law for closed systems operating cyclically, one can show that if during each cycle the system experiences any number of heat interactions Qi with any number of temperature reservoirs whose respective absolute temperatures are Ti, then Qi Ti 0 i Note that Ti is the absolute temperature of the boundary region crossed by Qi. Another way to write the second law in this case is Q T 0 where, again, T is the temperature of the boundary pierced by Q. Of special interest is the reversible cycle limit, in which the second law states ( Q / T )rev 0. According to the deﬁnition of thermodynamic property, the second law implies that during a reversible process the quantity Q / T is the inﬁnitesimal change in a property of the system: by deﬁnition, that property is the entropy change dS Q T 2 or S2 rev S1 1 Q T rev Combining this deﬁnition with the second law for a cycle, Q/T 0, yields the second law of thermodynamics for any process executed by a closed system, 2 S2 S1 1 Q T 0 entropy change (property) entropy transfer (nonproperty) 102 Thermodynamics Fundamentals The entire left-hand side in this inequality is by deﬁnition the entropy generated by the process, 2 Sgen S2 S1 1 Q T The entropy generation is a measure of the inequality sign in the second law and hence a measure of the irreversibility of the process. As shown in the next chapter, the entropy generation is proportional to the useful work destroyed during the process.1,3,4 Note again that any heat interaction ( Q) is accompanied by entropy transfer ( Q / T ), whereas the work transfer W is not. 4 ENERGY MINIMUM PRINCIPLE Consider now a closed system that executes an inﬁnitesimally small change of state, which means that its state changes from (U, S, ...) to (U dU, S dS, ...). The ﬁrst law and the second law statements are Q dS W Q T dU 0 If the system is isolated from its environment, then W 0 and Q 0, and the two laws dictate that during any such process the energy inventory stays constant (dU 0), and the entropy inventory cannot decrease, dS 0 Isolated systems undergo processes when they experience internal changes that do not require intervention from the outside, e.g., the removal of one or more of the internal constraints plotted qualitatively in the vertical direction in Fig. 2. When all the constraints are removed, changes cease, and, according to dS 0, the entropy inventory reaches its highest possible level. This entropy maximum principle is a consequence of the ﬁrst and second laws. When all the internal constraints have disappeared, the system has reached the unconstrained equilibrium state. Alternatively, if changes occur in the absence of work transfer and at constant S, the ﬁrst law and the second law require, respectively, dU Q and Q 0, hence dU 0 The energy inventory cannot increase, and when the unconstrained equilibrium state is reached the system energy inventory is minimum. This energy minimum principle is also a consequence of the ﬁrst and second laws for closed systems. The interest in this classical formulation of the laws (e.g., Fig. 2) has been renewed by the emergence of an analogous principle of performance increase (the constructal law) in the search for optimal conﬁgurations in the design of open (ﬂow) systems.5 This analogy is based on the constructal law of maximization of ﬂow access,1,6 and is summarized in the next chapter. 5 THE LAWS OF THERMODYNAMICS FOR OPEN SYSTEMS If m represents the mass ﬂow rate through a port in the control surface, the principle of mass ˙ conservation in the control volume is 5 The Laws of Thermodynamics for Open Systems 103 Figure 2 The energy minimum principle, or the entropy maximum principle. m ˙ in out m ˙ M t } mass transfer mass change Subscripts in and out refer to summation over all the inlet and outlet ports, respectively, while M stands for the instantaneous mass inventory of the control volume. The ﬁrst law of thermodynamics is more general than the statement encountered earlier for closed systems, because this time we must account for the ﬂow of energy associated with the m streams: ˙ m h ˙ in V2 2 gZ out m h ˙ energy transfer V2 2 gZ i ˙ Qi ˙ W E t energy change On the left-hand side we have the energy interactions: heat, work, and the energy transfer associated with mass ﬂow across the control surface. The speciﬁc enthalpy h, ﬂuid velocity V, and height Z are evaluated right at the boundary. On the right-hand side, E is the instantaneous system energy integrated over the control volume. The second law of thermodynamics for an open system assumes the form ms ˙ in out ms ˙ i ˙ Qi Ti S t } entropy transfer entropy change The speciﬁc entropy s is representative of the thermodynamic state of each stream right at the system boundary. The entropy generation rate is deﬁned by 104 Thermodynamics Fundamentals ˙ Sgen S t ms ˙ out in ms ˙ i ˙ Qi Ti and is a measure of the irreversibility of open system operation. The engineering importance ˙ of Sgen stems from its proportionality to the rate of destruction of available work. If the following parameters are ﬁxed—all the mass ﬂows (m), the peripheral conditions (h, s, V, ˙ Z ), and the heat interactions (Qi, Ti) except (Q0, T0)—then one can use the ﬁrst law and the second law to show that the work-transfer rate cannot exceed a theoretical maximum.1,3,4 ˙ W in m h ˙ V2 2 gZ T0s out m h ˙ V2 2 gZ T0s t (E T0s) ˙ The right-hand side in this inequality is the maximum work transfer rate Wsh,max , which would exist only in the ideal limit of reversible operation. The rate of lost work, or the rate of exergy (availability) destruction, is deﬁned as ˙ Wlost ˙ Wmax ˙ W Again, using both laws, one can show that lost work is directly proportional to entropy generation, ˙ Wlost ˙ T0 Sgen This result is known as the Gouy-Stodola theorem.1,3,4 Conservation of useful work (exergy) in thermodynamic systems can only be achieved based on the systematic minimization of entropy generation in all the components of the system. Engineering applications of entropy generation minimization as a design optimization philosophy may be found in Refs. 1, 3, and 4, and in the next chapter. 6 RELATIONS AMONG THERMODYNAMIC PROPERTIES The analytical forms of the ﬁrst and second laws of thermodynamics contain properties such as internal energy, enthalpy, and entropy, which cannot be measured directly. The values of these properties are derived from measurements that can be carried out in the laboratory (e.g., pressure, volume, temperature, speciﬁc heat); the formulas connecting the derived properties to the measurable properties are reviewed in this section. Consider an inﬁnitesimal change of state experienced by a closed system. If kinetic and gravitational energy changes can be neglected, the ﬁrst law reads Qany path Wany path dU which emphasizes that dU is path-independent. In particular, for a reversible path (rev), the same dU is given by Qrev Wrev dU Note that from the second law for closed systems we have Qrev T dS. Reversibility (or zero entropy generation) also requires internal mechanical equilibrium at every stage during the process; hence, Wrev P dV, as for a quasistatic change in volume. The inﬁnitesimal change experienced by U is therefore T dS P dV dU 6 Relations among Thermodynamic Properties 105 Note that this formula holds for an inﬁnitesimal change of state along any path (because dU is path-independent); however, T dS matches Q and P dV matches W only if the path is reversible. In general, Q T dS and W P dV. The formula derived above for dU can be written for a unit mass: T ds P dv du. Additional identities implied by this relation are T u s v 2 u s P v u v s T v s P s v where the subscript indicates which variable is held constant during partial differentiation. Similar relations and partial derivative identities exist in conjunction with other derived functions such as enthalpy, Gibbs free energy, and Helmholtz free energy: • Enthalpy (deﬁned as h u Pv) dh T h s 2 T ds v P v dP h P v s h s P • Gibbs free energy (deﬁned as g h dg s g T P • Helmholtz free energy (deﬁned as ƒ df s ƒ T v 2 2 T P Ts) s dT g T s s P v dP v P g P v T s P u s dT ƒ T T T P Ts) P dv P ƒ v T v s v T P T v In addition to the (P, v, T ) surface, which can be determined based on measurements (Fig. 3), the following partial derivatives are furnished by special experiments1: • The speciﬁc heat at constant volume, cv ( u / T )v, follows directly from the constant volume ( W 0) heating of a unit mass of pure substance. • The speciﬁc heat at constant pressure, cP ( h / T )P, is determined during the constant-pressure heating of a unit mass of pure substance. 106 Thermodynamics Fundamentals Figure 3 The (P, v, T ) surface for a pure substance that contracts upon freezing, showing regions of ideal gas and incompressible ﬂuid behavior. In this ﬁgure, S solid, V vapor, L liquid, TP triple point. • The Joule-Thompson coefﬁcient, ( T / P)h, is measured during a throttling process, that is, during the ﬂow of a stream through an adiabatic duct with friction (see the ﬁrst law for an open system in the steady state). • The coefﬁcient of thermal expansion, (1 / v)( v / T )P. • The isothermal compressibility, K ( 1 / v)( v / P)T. • The constant temperature coefﬁcient, cT ( h / P)T. Two noteworthy relationships between some of the partial-derivative measurements are 6 cP Relations among Thermodynamic Properties cv v 107 Tv K T 2 1 T cP v P The general equations relating the derived properties (u, h, s) to measurable quantities are du dh ds cv dT T cv dT cP dT v T T P T P dv v v T v P dP v T dv or ds v cP dT T T dP P These relations also suggest the following identities: u T T v s T cv h T T P s T cP P The relationships between thermodynamic properties and the analyses associated with applying the laws of thermodynamics are simpliﬁed considerably in cases where the pure substance exhibits ideal gas behavior. As shown in Fig. 3, this behavior sets in at sufﬁciently high temperatures and low pressures; in this limit, the (P, v, T ) surface is ﬁtted closely by the simple expression Pv T R (constant) where R is the ideal gas constant of the substance of interest (Table 1). The formulas for internal energy, enthalpy, and entropy, which concluded the preceding section, assume the following form in the ideal-gas limit: du dh ds cv dT T R v cv dT cP cP dT T cv cv(T ) cv R cv dP P cP v cP dT cP(T ) dv or ds R dP or ds P dv If the coefﬁcients cv and cP are constant in the temperature domain of interest, then the changes in speciﬁc internal energy, enthalpy, and entropy relative to a reference state ( )0 are given by the formulas u h h0 cP (T s s s0 s0 u0 T0) cv ln cP ln cv (T T0) u0 v v0 (where h0 T T0 T T0 R ln R ln RT0) P P0 108 Thermodynamics Fundamentals Table 1 Values of Ideal-Gas Constant and Speciﬁc Heat at Constant Volume for Gases Encountered in Mechanical Engineering1 R J kg K 286.8 208.1 143.2 188.8 296.8 276.3 296.4 2076.7 4123.6 518.3 412.0 296.8 72.85 259.6 188.4 461.4 cP J kg K 715.9 316.5 1595.2 661.5 745.3 1511.4 1423.5 3152.7 10216.0 1687.3 618.4 741.1 1641.2 657.3 1515.6 1402.6 Ideal Gas Air Argon, Ar Butane,C4H10 Carbon dioxide, CO2 Carbon monoxide, CO Ethane, C2H6 Ethylene, C2H4 Helium, He2 Hydrogen, H Methane, CH4 Neon, Ne Nitrogen, N2 Octane, C8H18 Oxygen, O2 Propane, C3H8 Steam, H2O s s0 cv ln P P0 cp ln v v0 The ideal-gas model rests on two empirical constants, cv and cP, or cv and R, or cP and R. The ideal-gas limit is also characterized by 0 1 P K 1 P cT 0 The extent to which a thermodynamic system destroys available work is intimately tied to the system’s entropy generation, that is, to the system’s departure from the theoretical limit of reversible operation. Idealized processes that can be modeled as reversible occupy a central role in engineering thermodynamics, because they can serve as standard in assessing the goodness of real processes. Two benchmark reversible processes executed by closed ideal-gas systems are particularly simple and useful. A quasistatic adiabatic process 1 → 2 executed by a closed ideal-gas system has the following characteristics: 2 Q 1 2 0 V2 V1 1 W 1 P2V2 1 1 where • Path cP / cv PV P1V 1 P2V 2 (constant) 6 • Entropy change Relations among Thermodynamic Properties 109 S2 • Entropy generation 2 S1 0 hence the name isoentropic or isentropic for this process Q T Sgen1→2 S2 S1 0 (reversible) 1 A quasistatic isothermal process 1 → 2 executed by a closed ideal-gas system in communication with a single temperature reservoir T is characterized by • Energy interactions 2 2 Q 1 1 W m RT ln V2 V1 P2V2 (constant) • Path T • Entropy change T1 T2 (constant) or PV P1V1 V2 V1 S2 • Entropy generation S1 m R ln 2 Sgen1→2 S2 S1 1 Q T 0 (reversible) Mixtures of ideal gases also behave as ideal gases in the high-temperature, low-pressure limit. If a certain mixture of mass m contains ideal gases mixed in mass proportions mi, and if the ideal-gas constants of each component are (cvi, cPi, Ri), then the equivalent ideal gas constants of the mixture are cv cp R where m i 1 m 1 m 1 m i micvi micPi miRi i i mi. One mole is the amount of substance of a system that contains as many elementary entities (e.g., molecules) as there are in 12 g of carbon 12; the number of such entities is Avogadro’s constant, N0 6.022 1023. The mole is not a mass unit, because the mass of 1 mole is not the same for all substances. The molar mass M of a given molecular species is the mass of 1 mole of that species, so that the total mass m is equal to M times the number of moles n, m nM Thus, the ideal-gas equation of state can be written as 110 Thermodynamics Fundamentals PV nMRT where the product MR is the universal gas constant R MR 8.314 J mol K The equivalent molar mass of a mixture of ideal gases with individual molar masses Mi is M 1 n niMi ni. The molar mass of air, as a mixture of nitrogen, oxygen, and traces of where n other gases, is 28.966 g / mol (or 28.966 kg / kmol). A more useful model of the air gas mixture relies on only nitrogen and oxygen as constituents, in the proportion 3.76 moles of nitrogen to every mole of oxygen; this simple model is used frequently in the ﬁeld of combustion.1 At the opposite end of the spectrum is the incompressible substance model. At sufﬁciently high pressures and low temperatures in Fig. 3, solids and liquids behave so that their density or speciﬁc volume is practically constant. In this limit the (P, v, T ) surface is adequately represented by the equation v v (constant) The formulas for calculating changes in internal energy, enthalpy, and entropy become (see the end of the section on relations among thermodynamic properties) du dh c dT v dP c dT ds c dT T where c is the sole speciﬁc heat of the incompressible substance, c cv cP The speciﬁc heat c is a function of temperature only. In a sufﬁciently narrow temperature range where c can be regarded as constant, the ﬁnite changes in internal energy, enthalpy, and entropy relative to a reference state denoted by ( )0 are u h h0 c (T T0) u0 v (P c (T P0) c ln T0) (where h0 T T0 u0 P0 v) s s0 The incompressible substance model rests on two empirical constants, c and v. As shown in Fig. 3, the domains in which the pure substance behaves either as an ideal gas or as an incompressible substance intersect over regions where the substance exists as a mixture of two phases, liquid and vapor, solid and liquid, or solid and vapor. The two-phase regions themselves intersect along the triple point line labeled TP-TP on the middle sketch of Fig. 3. In engineering cycle calculations, the projections of the (P, v, T ) surface on the P-v plane or, through the relations reviewed earlier, on the T-s plane are useful. The terminology associated with two-phase equilibrium states is deﬁned on the P-v diagram of Fig. 4a, where we imagine the isothermal compression of a unit mass of substance (a closed 6 Relations among Thermodynamic Properties 111 Figure 4 The locus of two-phase (liquid and vapor) states, projected on (a) the P-v plane, and (b) the T-s plane. 112 Thermodynamics Fundamentals system). As the speciﬁc volume v decreases, the substance ceases to be a pure vapor at state g, where the ﬁrst droplets of liquid are formed. State g is a saturated vapor state. It is observed that isothermal compression beyond g proceeds at constant pressure up to state ƒ, where the last bubble (immersed in liquid) is suppressed. State ƒ is a saturated liquid state. Isothermal compression beyond ƒ is accompanied by a steep rise in pressure, depending on the compressibility of the liquid phase. The critical state is the intersection of the locus of saturated vapor states with the locus of saturated liquid states (Fig. 4a). The temperature and pressure corresponding to the critical state are the critical temperature and critical pressure. Table 2 contains a compilation of critical-state properties of some of the more common substances. Figure 4b shows the projection of the liquid and vapor domain on the T-s plane. On the same drawing is shown the relative positioning (the relative slopes) of the traces of various constant-property cuts through the three-dimensional surface on which all the equilibrium states are positioned. In the two-phase region, the temperature is a unique function of pressure. This one-to-one relationship is indicated also by the Clapeyron relation dP dT hg T(vg hƒ vƒ) sg vg sƒ vƒ sat where the subscript sat is a reminder that the relation holds for saturated states (such as g and ƒ) and for mixtures of two saturated phases. Subscripts g and ƒ indicate properties Table 2 Critical-State Properties1 Critical Temperature [K ( C)] 133.2 513.2 516.5 405.4 150.9 425.9 304.3 134.3 555.9 417 305.4 282.6 5.2 508.2 33.2 190.9 416.5 44.2 179.2 125.9 569.3 154.3 368.7 430.4 647 ( 140) (240) (243.3) (132.2) ( 122.2) (152.8) (31.1) ( 138.9) (282.8) (143.9) (32.2) (9.4) ( 268) (235) ( 240) ( 82.2) (143.3) ( 288.9) ( 93.9) ( 147.2) (296.1) ( 118.9) (95.6) (157.2) (373.9) Critical Pressure [MPa (atm)] 3.77 7.98 6.39 11.3 4.86 3.65 7.4 3.54 4.56 7.72 4.94 5.85 0.228 2.99 1.30 4.64 6.67 2.7 6.58 3.39 2.5 5.03 4.36 7.87 22.1 (37.2) (78.7) (63.1) (111.6) (48) (36) (73) (35) (45) (76.14) (48.8) (57.7) (2.25) (29.5) (12.79) (45.8) (65.8) (26.6) (65) (33.5) (24.63) (49.7) (43) (77.7) (218.2) Critical Speciﬁc Volume (cm3 / g) 2.9 3.7 3.6 4.25 1.88 4.4 2.2 3.2 1.81 1.75 4.75 4.6 14.4 4.25 32.3 6.2 2.7 2.1 1.94 3.25 4.25 2.3 4.4 1.94 3.1 Fluid Air Alcohol (methyl) Alcohol (ethyl) Ammonia Argon Butane Carbon dioxide Carbon monoxide Carbon tetrachloride Chlorine Ethane Ethylene Helium Hexane Hydrogen Methane Methyl chloride Neon Nitric oxide Nitrogen Octane Oxygen Propane Sulfur dioxide Water 7 Analysis of Engineering System Components 113 corresponding to the saturated vapor and liquid states found at temperature Tsat (and pressure Psat). Built into the last equation is the identity hg hƒ T (sg sƒ) which is equivalent to the statement that the Gibbs free energy is the same for the saturated states and their mixtures found at the same temperature, gg gƒ. The properties of a two-phase mixture depend on the proportion in which saturated vapor, mg, and saturated liquid, mƒ, enter the mixture. The composition of the mixture is described by the property called quality, x mg mƒ mg The quality varies between 0 at state ƒ and 1 at state g. Other properties of the mixture can be calculated in terms of the properties of the saturated states found at the same temperature, u h uƒ hƒ xuƒg xhƒg s v sƒ vƒ xsƒg xvƒg with the notation ( )ƒg ( )g ( )ƒ. Similar relations can be used to calculate the properties of two-phase states other than liquid and vapor, namely, solid and vapor or solid and liquid. For example, the enthalpy of a solid and liquid mixture is given by h hs xhsƒ, where subscript s stands for the saturated solid state found at the same temperature as for the twophase state, and where hsƒ is the latent heat of melting or solidiﬁcation. In general, the states situated immediately outside the two-phase dome sketched in Figs. 3 and 4 do not follow very well the limiting models discussed earlier in this section (ideal gas, incompressible substance). Because the properties of closely neighboring states are usually not available in tabular form, the following approximate calculation proves useful. For a moderately compressed liquid state, which is indicated by the subscript ( )*, that is, for a state situated close to the left of the dome in Fig. 4, the properties may be calculated as slight deviations from those of the saturated liquid state found at the same temperature as the compressed liquid state of interest, h* (hƒ)T* s (vƒ)T*[P* (sƒ)T* (Pƒ)T*] For a slightly superheated vapor state, that is, a state situated close to the right of the dome in Fig. 4, the properties may be estimated in terms of those of the saturated vapor state found at the same temperature: h s (sg)T (hg)T Pgvg Tg ln T (Pg)T P In these expressions, subscript ( ) indicates the properties of the slightly superheated vapor state. 7 ANALYSIS OF ENGINEERING SYSTEM COMPONENTS This section contains a summary1 of the equations obtained by applying the ﬁrst and second laws of thermodynamics to the components encountered in most engineering systems, such 114 Thermodynamics Fundamentals as power plants and refrigeration plants. It is assumed that each component operates in steady ﬂow. • Valve (throttle) or adiabatic duct with friction (Fig. 5a): First law Second law h1 ˙ Sgen h2 m (s2 ˙ s1) 0 Figure 5 Engineering system components, and their inlet and outlet states on the T-s plane, PH pressure; PL low pressure. high 7 Analysis of Engineering System Components 115 Figure 5 (Continued ) 116 Thermodynamics Fundamentals • Expander or turbine with negligible heat transfer to the ambient (Fig. 5b) First law Second law Efficiency ˙ WT ˙ Sgen T m (h1 ˙ m (s2 ˙ h2) s1) 0 h1 h2 1 h1 h2,rev • Compressor or pump with negligible heat transfer to the ambient (Fig. 5c): First law Second law Efficiency ˙ WC ˙ Sgen C m(h2 ˙ m(s2 ˙ h1) s1) 0 h2,rev h1 1 h2 h1 • Nozzle with negligible heat transfer to the ambient (Fig. 5d): First law Second law Efficiency 1 – (V 2 2 2 V 2) 1 m(s2 ˙ h1 s1) h2 0 ˙ Sgen N 2 2 V2 V1 1 2 2 V 2,rev V 1 • Diffuser with negligible heat transfer to the ambient (Fig. 5e): First law Second law Efficiency h2 ˙ Sgen D h1 1 – (V 2 2 1 V 2) 2 0 m(s2 ˙ s1) h2,rev h1 1 h2 h1 • Heat exchangers with negligible heat transfer to the ambient (Figs. 5f and g) First law Second law mhot(h1 ˙ ˙ Sgen h2) mhot(s2 ˙ mcold(h4 ˙ s1) h3) s3) 0 mcold(s4 ˙ Figures 5f and g show that a pressure drop always occurs in the direction of ﬂow, in any heat exchanger ﬂow passage. REFERENCES 1. 2. 3. 4. 5. A. Bejan, Advanced Engineering Thermodynamics, 2nd ed., Wiley, New York, 1997. A. Bejan, Heat Transfer, Wiley, New York, 1993. A. Bejan, Entropy Generation through Heat and Fluid Flow, Wiley, New York, 1982. A. Bejan, Entropy Generation Minimization, CRC Press, Boca Raton, FL, 1996. A. Bejan and S. Lorente, ‘‘The Constructal Law and the Thermodynamics of Flow Systems with Conﬁguration,’’ Int. J. Heat Mass Transfer 47, 3203–3214 (2004). 6. A. Bejan, Shape and Structure, from Engineering to Nature, Cambridge University Press, Cambridge, UK, 2000.