Real Gas Thermodynamics. and the isentropic behavior of substances. P. Nederstigt

Transcription

1 Real Gas Thermodynamics and the isentropic behaior of substances. Nederstigt

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3 Real Gas Thermodynamics and the isentropic behaior of substances by. Nederstigt in partial fulfillment of the requirements for the degree of Master of Science in Mechanical Engineering at the Delft Uniersity of Technology, to be defended publicly on Wednesday September 6, 017 at 11:00 AM. Superisor: Thesis committee: Dr. ir. R. ecnik, rof. dr. ir. B. J. Boersma, Dr. ir. M. ini, iii

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5 Abstract A generalized isentropic gas model is deried following earlier work by Kouremenos et al. [1 3] by replacing the traditional adiabatic exponent γ by the real exponents γ, γ T, and γ T, describing the isentropic pressure-olume, temperature-olume, and pressure-temperature relations respectiely. The real adiabatic exponents are expressed as functions of state ariables to take into account compressibility effects on the isentropic behaior of substances. Due to the implicit analytical nature of the real exponents, any equation of state or thermodynamic library can be used for their ealuation. The theoretical limits and oerall behaior of the real isentropic gas model are explored for a Van der Waals substance. In the two opposing physical limits, the model is shown to reduce to the incompressible substance model for liquid densities and the ideal gas model as the temperature increases or the pressure goes to zero. The relation of the generalized isentropic gas model with other thermodynamic properties is explored, leading to the deelopment of specific heat relations and other thermodynamic properties in terms of the real exponents γ, γ T, and γ T. Besides proiding alternatie schemes for their ealuation, special features of thermodynamic properties such as the state of maximum density and inersion temperature may be related to the alue of the isentropic exponents determined by the local compressibility of the substance. Due to the exact definitions of the real adiabatic exponents at a state point, the relations between properties is thermodynamically consistent another physical requirement. The generalized isentropic gas model is then applied to isentropic flows to derie traditional gas dynamic relations such as speed of sound, stagnation properties, and choked flow conditions for non-ideal compressible fluid flows. Exact solutions are proided for randtl-meyer expansion fans, and approximate Rankine-Hugoniot jump conditions are explored for real gases. Finally, attributes of the fundamental deriatie of gas dynamics are explored under the generalized isentropic gas model to gain new insights into its mathematical properties. Under the generalized isentropic model, the fundamental deriatie is shown to satisfy both liquid and gaseous physical limits. Non-classical behaior is attributed to higher-order deriaties of the real exponents. The application of the generalized isentropic gas model is demonstrated and alidated for use in non-ideal compressible fluid dynamic (NICFD) codes by simulation of the one-dimensional Euler equations for a standard shock tube problem. Seeral numerical schemes for the ealuation of thermodynamic properties of arying leels of accuracy are presented for ealuation of the isentropic gas model. In the application of the shock tube problem, the general equation for the speed of sound is demonstrated to be equialent to the speed of sound of a Van der Waals gas, proing the alidity of the model.

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7 Acknowledgements Before you lies Real Gas Thermodynamics and the isentropic behaior of substances, an attempt to extend the analytical structure of ideal gases to a general description of real substances. This work would neer hae come to be without my patient superisor, Dr. René ecnik. The faith you hae placed in me by granting me the academic freedom to explore this field has continued to motiate me throughout the process. I would also like to express my gratitude to Dr. Matteo ini, whose inolement towards completion of the project has added much needed practical alue to this work. I would also like to thank my friends from both inside and outside the uniersity whose personal contributions made the past few years such an enjoyable experience. Lastly, I would like to thank my parents and sister whose continued loe and support throughout the years hae helped me get to this point in life. im Nederstigt August 017 ii

8 Table of Contents Nomenclature x Introduction xii I Real Isentropic Relations 15 1 Real Isentropic Exponents ressure-volume Exponent Temperature-Volume Exponent ressure-temperature Exponent Summary of the Isentropic Functions Approximate Real Isentropes.1 ressure-volume Isentrope Temperature-Volume Isentrope ressure-temperature Isentrope Behaior of the Isentropic Exponents Van der Waals Isentropic Exponents hysical Limits of the Isentropic Exponents Contours of the Isentropic Exponents II Real roperty Relations 39 4 Real Specific Heat Functions and Compressibility Specific Heat Relation for Calorically Imperfect Gases Real Specific Heat Functions Other Thermodynamic roperties and Deriaties Deriaties of Mechanical roperties Deriaties of Internal Energy and Enthalpy Isentropic Work 48 7 Real Entropy Changes and Carnot s Theorem Carnot s Theorem Reiewed iii

9 III Real Gas Dynamics 57 8 One-Dimensional Real Isentropic Flow Relations Speed of Sound in Real Gases Real Stagnation roperties Real Characteristic Velocities Real Critical roperties Real Shock Waes randtl-meyer Expansion Fans Normal Shock Waes Fundamental Deriatie of Gas Dynamics Fundamental Deriatie and Isentropic Flows IV Numerical Simulations of Real Gas Flows Explicit Numerical Schemes The Van der Waals Equation of State Soae-Redlich-Kwong Equation of State eng-robinson Equation of State The Riemann roblem 9 Conclusion 99 Recommendations 101 Bibliography 103 Appendix 109 A Thermodynamic Identities 109 A.1 Exact Differentials A. Maxwell Relations A.3 Deriaties B Supercritical CO Compressor 111 ix

10 Nomenclature hysical Constants R k B N A uniersal gas constant Boltzmann constant Aogadro s constant a b f molecular attraction parameter molecular olume parameter molecular degrees of freedom Thermodynamic roperties T Z ρ pressure temperature compressibility factor density specific olume e h s specific internal energy specific enthalpy specific entropy c c p specific isochoric heat capacity specific isobaric heat capacity γ γ γ T ideal isentropic exponent pressure-olume isentropic exponent temperature-olume isentropic exponent x

11 γ T pressure-temperature isentropic exponent β κ K thermal expansion coefficient isothermal compressibility adiabatic bulk modulus µ JT Joule-Thomson coefficient µ J Joule coefficient Γ fundamental deriatie of gas dynamics Gas Dynamic Variables u c M elocity speed of sound mach number A ṁ flow area mass flow rate α δ ν geometrical angle deflection angle randtl-meyer angle Superscripts and Subscripts c r critical property reduced property critical flow property 0 stagnation property 1,,... thermodynamic state xi

12 Introduction The burden placed upon the enironment by the energy requirement of today s society urges humanity to deelop new sustainable technologies to meet demands. An enormous potential lies in the exploitation of geothermal reseroirs, ocean thermal gradients, concentrated solar radiation, waste heat from prime moers and industrial processes, and biomass combustion [4 7], typically consisting of small to medium size thermal reseroirs at moderate temperature leels. Conentional thermal energy conersion technologies whom oer the years hae been optimized to operate at eer higher temperature leels to improe conersion efficiency are unsuitable for efficient thermal conersion of these energy sources mostly due to the incompatibility of the working fluid and the temperature profile of the thermal reseroir [8 11]. The critical point, saturation line, specific heat capacities, and heat of eaporation associated with the choice of working fluid translates directly into the size and temperature leel of a thermal reseroir to which a system can efficiently operate [8, 10, 1]. To adapt to the temperature profiles of renewable energy sources and waste heat streams, a major deelopment in the field of power generation technologies is the selection of the working fluid as an additional degree of freedom to the design of energy conersion systems. The fluid can be chosen such that it is optimal from a thermodynamic and technical point of iew [10, 11, 13]. Examples of such technologies are the application of Organic Rankine Cycles (ORC) for low-temperature renewable sources and waste heat utility applications [8, 1, 14, 15], supercritical carbon dioxide (sco ) power cycles for medium to high temperature solar or nuclear applications [9, 16 18], and supercritical CO refrigeration cycles [19, 0]. erformance optimization of system components is of primary importance for the successful implementation of these innoations. The combination of unusual working fluid characteristics and the thermodynamic regime in which these technologies are to operate makes for fluid behaior to depart greatly from ideal behaior [1 3]. Due to the complexities of non-ideal compressible fluid dynamics (NICFD), the design and analysis of equipment operating dense gas regime is one that is mainly drien by computational fluid dynamics [4 7]. The deelopment of NICFD codes, therefore, plays a crucial role in the oerall adancement of this research field. At the same time, the conentional classification between ideal and non-ideal fluids is distinctie for the lacking means to describe the behaior of fluids in a general sense. Whereas the behaior of ideal gases is fully resoled, our capabilities are seerely limited when moing away from ideal conditions and force us to resort to thermodynamic libraries and equations of state of (semi)-empirical nature [8, 9]. Haing to rely on thermodynamic libraries and complex multiparameter equations of state (MEOS), imposes heay computational costs in the modeling of non-ideal compressible fluid flows. Moreoer, the absence or only partial aailability of experimental data in the thermodynamic region of interest introduces a leel of uncertainty that makes robust and accurate simulations of non-ideal fluid flows still a challenge [7, 30]. xii

13 In this work, an analytical framework for the ealuation of non-ideal gas behaior is constructed, based on the generalized isentropic relations introduced by Kouremenos et al. in the 1980s [1 3, 31, 3]. In a series of papers, they proposed a method to include compressibility effects into the adiabatic constant γ used in classical ideal gas dynamics a model that could potentially be ery useful in the modeling of non-ideal compressible fluid flows. Though upon its introduction, the application of the isentropic model leaned more towards empirical substitution of equations of state into compressible ideal gases relations rather than analytical arguments. Isentropic flow relations for non-ideal gases were presumed to retain their ideal form, and their exponents empirically ealuated [31]. Demonstrating these concepts based on analytical arguments will be the objectie of this work: To analytically extend concepts of classical ideal compressible gas dynamics to non-ideal gases using the generalized isentropic model. The extent to which ideal gas dynamics can be successfully extended to non-ideal fluids will be inestigated by addressing the following related topics: i. Does the mathematical behaior of the generalized isentropic gas model comply with physical limits, and to what extent can the model capture real gas behaior between those limits? ii. Is the isentropic gas model consistent with other thermodynamic properties? iii. Can isentropic flow properties be extended to non-ideal fluids using the generalized isentropic gas model, and what are the limits of its application? By adoption of the generalized isentropic model, we will consider a fluid exhibiting any gas dynamic behaior, henceforth referred to as a real fluid. The ideal gas equation will be demonstrated to be a subclass of the real model, as indeed are any other equations of state such as Van der Waals, Soae-Redlich-Kwong, eng-robinson, etc. Classical analytical ideal gas concepts such as the speed of sound, stagnation properties, choked flow conditions, and other thermodynamic identities, will be demonstrated to inherit their formulation from fundamental mass, momentum, and energy conseration equations that do not distinguish between ideal or non-ideal gases. These concepts will be shown to be shared properties of ideal gases and real gases alike. The kind of behaior that the substance exhibits is captured in the alue of the adiabatic exponents of the real isentropic model. To effectiely engage in this extensie collection of closely related topics, this work has been diided into four parts. In art I, the real isentropic model by Kouremenos is formally introduced [1 3]. Its mathematical properties and limits are explored for the underlying physics that goerns fluid behaior. In art II, the isentropic model is put into the wider perspectie of thermodynamic fluid properties concerning specific heat relations, compressibility effects and entropy itself. In art III, the isentropic model is applied to extend familiar concepts of ideal gas dynamics to real fluid flows. The speed of sound, stagnation properties, and choked flow conditions are deried for non-ideal compressible isentropic flows. An attempt is made to derie shock wae properties in real fluid flows. Finally, in art IV, the application of the isentropic model in NICFD codes is demonstrated by simulation of the shock tube problem for a real compressible gas. xiii

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15 ART I Real Isentropic Relations 1 Real Isentropic Exponents ressure-volume Exponent Temperature-Volume Exponent ressure-temperature Exponent Summary of the Isentropic Functions Approximate Real Isentropes.1 ressure-volume Isentrope Temperature-Volume Isentrope ressure-temperature Isentrope Behaior of the Isentropic Exponents Van der Waals Isentropic Exponents hysical Limits of the Isentropic Exponents Contours of the Isentropic Exponents art I of this work seres as an introduction of the generalized isentropic gas model to the reader, which will be applied to thermodynamic property relations and isentropic flows in subsequent parts. The isentropic relations of real substances will be introduced, their functions formally deried, and their mathematical properties thoroughly inestigated. The first two chapters will together substantiate the theoretical framework behind the generalized isentropic relations based on earlier work by Kouremenos et al. [1 3]. Their original deriation is expanded for clarity and completeness, whereby the equialence between the real isentropic gas model and the ideal isentropic gas model is explicitly outlined. In the final chapter of this part, the behaior of the generalized isentropic functions are inestigated for Van der Waals gasses, and their physical limits explored. 15

16 ONE Real Isentropic Exponents The isentropic relations for ideal gases will form our point of departure for the deriation of their real gas counterparts. The isentropic relations of an ideal gas are readily deried from the entropy equations [33, 34], gien as ds = c p dt T + R d, (1.1) where ds = 0 for an isentropic process. Let R = c p c for an ideal gas. Integration of Eq. (1.1) yields ln T cp c c p ln = const. (1.) The integration constant is unique for each isentrope. Taking the exponential of Eq. (1.) we arrie at the description of the pressure-temperature isentrope in its familiar form. Using the ideal gas relation = RT the pressure-olume isentrope and temperature-olume isentrope are also obtained: γ = const, T γ 1 = const, (1.3a) (1.3b) T 1 γ γ = const. (1.3c) For the deriation of the isentropic relations for real gasses a solution in the form of Eqs. (1.3a 1.3c) will be assumed for now, where the exponents are replaced by γ, γ T, and γ T. The purpose of this chapter will be to find implicit expressions for the exponents in terms of state ariables, based on earlier work by Kouremenos et al. in the 1980s [1 3]. The original deriation is expanded by explicitly underlining the equialence between the isentropic relations in the ideal and real case. Moreoer, Chapter will proide further closure of the problem by show that the assumed form of the isentropes is indeed correct. A special notice is resered for recent work by Baltadjie, who had been able to independently come up with expressions for the real gas isentropes using a slightly different choice of notation in terms of compressibility coefficients [35, 36]. The original notation introduced by Kouremenos is preferred in this work as it coneniently adheres to the ideal gas notation, thereby presering familiar notations and emphasizing the equialence between the ideal and generalized isentropic gas models. 16

17 1.1 ressure-volume Exponent The isentropic pressure-olume relationship of an ideal gas is gien by Eq. (1.3a). For the general case, we introduce the unknown adiabatic function γ which replaces the ideal adiabatic coefficient γ = c p/c. The generalized isentropic pressure-olume relation becomes γ = const. (1.4) To derie an expression for γ in terms of state ariables, let entropy be defined as a function of pressure and specific olume s = s(, ). In this case, the change in entropy is expressed as (App. A.1) ( ) ( ) s s ds = d + d = 0, (1.5) where ds = 0 for an isentropic process. Rearranging the partial deriaties yields ( ) s ( ) d = ( ). (1.6) d s s The left-hand side of Eq. (1.6) can be ealuated by differentiation of the assumed isentropic pressure-olume relation Eq. (1.4) with respect to, which becomes ( ) d = γ d. (1.7) s The right-hand side of Eq. (1.6) can be re-expressed using Maxwell relations, and subsequently expanded using the triple product (App. A.1 A.) ( ) s ( ) ( ) s s = ( ), (1.8a) s ( ) ( ) s = s ( ) s T T ( ) s, (1.8b) where ( s/ ) = c p/t and ( s/ ) = c /T (App. A.3). Using the Maxwell relations once more, the remaining partial deriaties can be expressed as ( ) s ( ) s = cp c ( ) ( ), (1.9a) 17

18 = cp c ( ). (1.9b) T exponent γ Equating the left-hand side and right-hand side results, we find an implicit function of γ in terms of state ariables based on the assumed form of the isentropic pressureolume relation [1]: γ = c p c ( ). T (1.10) To ealuate the function γ the expression must be made explicit by adopting an equation of state, or by using a thermodynamic library. This will be the subject of art IV where the real isentropic gas model is used for modeling non-ideal compressible flows. For now, we will use this concept to demonstrate that the isentropic pressureolume relation of an ideal gas is a subset of the solutions of Eq. (1.10). Ealuating the deriatie ( / ) T for the ideal gas equation yields ( ) = RT T. (1.11) Elimination of the deriatie in Eq. (1.10), we find that the function of γ reduces to the ratio of the specific heats, thereby demonstrating the equialence between the ideal gas and real gas isentropic pressure-olume relation as 1. Temperature-Volume Exponent γ = cp c. (ideal gas) (1.1) Similarly, let us assume the general temperature-olume relation to be of the from of Eq. (1.3b) where the adiabatic coefficient is replaced by the unknown function γ T. The generalized temperature-olume relation along an isentrope becomes T γ T 1 = const. (1.13) The procedure for deriing an implicit expression for the function γ T follows the same steps as preiously for the pressure-olume exponent. We start out by defining entropy as a function of temperature and olume s = s(t, ). With ds = 0 for an isentropic process, the deriaties can be related as (App. A.1) ( ) s ( ) dt T = ( ). (1.14) d s s Again, a function for γ T can be found by ealuating both sides of Eq. (1.14). The lefthand side of Eq. (1.14) can be ealuated by differentiation of the assumed temperatureolume relation with respect to 18

19 ( ) dt = (γ T 1) T d s. (1.15) The partial deriaties on the right-hand side of Eq. (1.14) can be expanded using Maxwell relations (App. A.) ( ) s ( ) s T ( ) = ( ), (1.16a) s = T ( ). (1.16b) c Equating the left-hand side and right-hand side in Eq. (1.14), the function for the exponent γ T for an real isentropic process becomes [1]: γ T = 1 + ( ). (1.17) exponent γ T c Like in the case of γ, the ideal adiabatic exponent can be shown to be a subset of the solutions of γ T by ealuating the deriatie ( / ) using the ideal gas relation which yields ( ) = R. (1.18) Elimination of the partial deriatie in Eq. (1.17) the exponent γ T is demonstrated to reduce to the ratio of the specific heats in the ideal case 1.3 ressure-temperature Exponent γ T = 1 + R c = cp c. (ideal gas) (1.19) Finally, we introduce the function γ T to relate the isentropic pressure-temperature relation in the general case. Again, the purpose is to find an implicit expression for γ T in terms of state ariables. The general pressure-temperature relation for an isentropic process becomes T 1 γ T γ T = const. (1.0) We start by expressing entropy as an exact differential in terms of pressure and temperature s = s(, T ) (App. A.1). Again, with ds = 0 the following relation between the partial deriaties is obtained 19

20 ( ) s ( ) d = ( ) dt s s T. (1.1) The left-hand side of Eq. (1.1) is ealuated by differentiating the assumed pressuretemperature relation with respect to temperature, giing ( ) d = γ T dt γ T 1 T. (1.) s The right-hand side of Eq. (1.1) can be expanded using Maxwell relations (App. A.) ( ) s ( ) s ( ) s T = ( ), (1.3a) ( ) = cp. (1.3b) T Equating the left-hand side and right-hand side in Eq. (1.1), the following expression is found for the function γ T in terms of state ariables [1]: exponent γ T γ T = 1 1 c p ( ). (1.4) In agreement with the preious sections, the isentropic pressure-temperature relation for an ideal gas can be demonstrated to be a subset of the solutions of Eq. (1.4). The partial deriatie ( / ) for an ideal gas becomes ( ) = R. (1.5) Elimination of the partial deriatie in Eq. (1.4), the exponent γ T can be shown to reduce to the ratio of the specific heats in the ideal gas case γ T = cp c p R = cp c. (ideal gas) (1.6) 0

21 1.4 Summary of the Isentropic Functions In summary, the generalized isentropic relations for real substances introduced in the preious sections for are expressed as [1] γ = const, (1.7) T γ T 1 = const, (1.8) T 1 γ T γ T = const, (1.9) where the isentropic exponents are functions of state ariables summarized as γ = c p c ( ), (1.30) T γ T = 1 + ( ), (1.31) c γ T = 1 1 c p ( ). (1.3) The isentropic functions Eqs. ( ) are state ariables that can be ealuated using any thermodynamic library or equation of state. Moreoer, the accuracy of the generalized isentropic functions is determined only by the accuracy of the equation of state using to relate the state ariables [3]. Additionally, as each of the isentropic functions has some deriatie of pressure, temperature, or olume, they can be shown to be interdependent using the chain rule. One can use the cyclic --T relation and eliminate the partial deriaties by Eqs. ( ) to relate the isentropic exponents ( ) s = ( ) s ( ). (1.33) s Elimination of the partial deriaties using Eq. (1.7), Eq. (1.15) and Eq. (1.), we find [1]: γ γ T 1 = γ T γ T 1. (1.34) relation between exponents 1

22 TWO Approximate Real Isentropes In the preious chapter, the isentropic relations for real gases were deried based on the presumed form of the pressure, olume, and temperature relation along an isentrope, Eqs. ( ). The legitimacy of this assumption will be demonstrated in this chapter. Howeer, we shall begin with a general notion on the ability of the real isentropic gas model to describe isentropic state changes. Each of the isentropic exponents introduced in the preious chapter possesses a multiariate dependency on state ariables, whose alues change continuously along an isentrope. The isentropic exponents are therefore themseles state ariables. Howeer, we may think of thermodynamic regions where the alues of the isentropic exponents are locally constant such is the case of ideal gases for example where the adiabatic ratio is assumed constant. Similarly, to ealuate isentropic state changes of real gases, the isentropic exponents γ, γ T, and γ T may assumed to be locally constant functions. This approximation is of a somewhat higher order than the assumption of constant specific heat ratio of ideal gases, as the latter indirectly imposes restrictions on the ariation of the isobaric heat capacity c p. The assumption locally constant alues of the isentropic exponents will be used throughout this work to relate states isentropic state changes. The limits of this assumption may be clarified by Figure.1, which shows the discrepancy between the isentropic constant in Eq. (1.7) calculated with ariable and constant alues of the isentropic exponent γ. As the alue of the constant in Eq. (1.7) is unique along each isentrope, the contours of the isentropic constant outline the contours of the isentropes by requirement. In Figure.1 the isentropes are approximated by their alues of γ at the critical pressure. The critical pressure leel has been chosen to demonstrate the ability of the continuous isentropic pressure-olume relation the solid lines in Figure.1 to demonstrate the ability of the model Eq. (1.7) to capture the highly non-linear fluid behaior at critical conditions. Note for example the curature of the isentropes theoretically encountered in dense gases in Figure.1b [37 41], which is the subject of Chapter 10. The polytropic Van der Waals equation of state has been used to ealuate the function of γ, Eq. (1.30). The details of this procedure are discussed in Chapter 11. Two important physical aspects can be outlined from this figure. Firstly, the approximate isentropes the dashed lines in Figure.1 form an increasingly better fit moing away from the critical point, in the limit where the solid and dashed lines will eentually coincide in the ideal limit for increasingly higher temperatures. Secondly, the discrepancy between the approximate and continuous isentropes increases with increasing molecular weight. The deiations are caused by higher order compressibility effects caused by the increased molecular complexity as the molecular size increases [37 41]. Howeer, note that Figure.1 only displays the contours of the isentropic constants

23 /c / c (a) Low molecular weight c =.5R /c / c (b) High molecular weight c = 50R Figure.1: ressure-olume isentropes Eq. (1.7) with ariable ( ) and constant ( ) exponents for a molecular light and heay Van der Waals substance. 3

24 regardless of their actual alues. In the next chapter, we will see that the ariation of the exponents γ, γ T and γ T reduces with increased molecular weight. Neertheless, the message coneyed by Figure.1 on the alidity of the assumption of constant isentropic exponents must be bared in mind throughout the subsequent chapters of this work. It presents the context in which the application of the approximated isentropic relations in later chapters are to be iewed..1 ressure-volume Isentrope In this and the following sections, the alidity of the form of the isentropic relations assumed in Chapter 1 will be demonstrated, starting with the pressure-olume relationship. The form of the pressure-olume isentrope of section 1.1, can be shown to be correct by expressing the exact differential of entropy in terms of pressure and olume ds = ( s ) d + ( s ) d. (.1) The partial differentials of Eq. (.1) can be rewritten using Maxwell relations (App. A.). The resulting deriaties ( / ) s and ( / ) s, in turn, may be ealuated as the deriatie of the isentropes in terms of temperature and olume Eq. (1.15), and pressure and temperature Eq. (1.), summarized as ( ) = 1 s γ T 1 T, ( ) = γ T s γ T 1 T. (.a) (.b) Here we introduce the assumption of a locally constant alue for the isentropic functions. From a mathematical point of iew, the isentropes deried in this chapter may be iewed as approximation of an exponential function by a power function. Substitution of the partial deriaties back into Eq. (.1) yields 1 ds = γ T 1 T d + γ T γ T 1 d. (.3) T Introducing the real gas relation = ZRT into the equation we obtain ds = 1 γ T 1 ZR d + γ T d ZR γ T 1. (.4) Rearranging terms, and introducing the relation between the isentropic exponents, Eq. (1.34), yields γ T 1 ZR ds = d + γ d. (.5) The ratio ZR/(γ T 1), will be shown to be equal to the isochoric heat capacity c in Chapter 4. With respect to Eq. (4.13), the aboe equation does not include any 4

25 deriatie terms following from the assumption of a constant alue for the isentropic exponent γ. Integration of Eq. (.5) between state zero and the arbitrary state s(, ) gies s c = ln + γ ln + C. (.6) Rearranging terms, we arrie at γ = e s c +C. (.7) When relating states along an isentrope, the change in entropy s = 0, for which the right-hand side of Eq. (.7) becomes a constant whose alue is unique for each isentrope. Moreoer, as the isentropic exponent γ reduces to the ratio of the specific heats in the ideal case, it is easily erified that the ideal isentropes are a subset of the solutions of Eq. (.7).. Temperature-Volume Isentrope Similarly, the assumed form of the temperature-olume isentrope in section 1. can be demonstrated to be correct. Let the change in entropy be expressed as an exact differential in terms of temperature and olume ( ) ( ) s s ds = dt + d, (.8) T where the partial deriatie ( s/ ) = c /T according to Maxwell relations. The right-hand side term of Eq. (.8) is expanded using the triple product (App. A.1 A.) ( ) s T = c T ( ). (.9) s Again, the partial deriatie term (/ ) s is ealuated as the deriatie of the T - isentrope, Eq. (1.15) in section 1. ( ) = (γ T 1) T. (.10) s Substitution of the deriaties into Eq. (.9), the entropy change for real gasses can be expressed as dt d ds = c + c(γt 1) T. (.11) Integration between state zero an arbitrary state s(t, ) gies s c = ln T + (γ T 1) ln + C. (.1) Rearranging terms, we finally arrie at T γ T 1 = e s c +C. (.13) 5

26 It appears that for an isentropic process, for which s = 0, the right-hand side of Eq. (.13) becomes constant along an isentrope, whose alue is uniquely determined by the integration constant C that is unique for each isentrope..3 ressure-temperature Isentrope Lastly, the form of the isentropic pressure-temperature relation assumed in section 1.3 can be demonstrated to be correct in a ery similar way as it was done for the temperature-olume isentrope. The only difference between the two will be the introduction of the isobaric specific heat capacity instead of the isochoric specific heat capacity. Again, let the entropy be a function of pressure and temperature. The change in entropy can be expressed as ( ) ( ) s s ds = dt + d, (.14) T where the partial deriatie ( s/ ) = c p/t following Maxwell relations. Like preiously, the right-hand side term of Eq. (.14) can be expanded using the triple product (App. A.1 A.) ( ) ( ) s = cp. (.15) T T s The partial differential (/ ) s can be ealuated as the deriatie of the pressureolume isentrope, Eq. (1.) ( ) dt = γ T 1 T d s γ T. (.16) Substitution of the partial deriaties back into Eq. (.14), we obtain dt ds = c p T γ T 1 cp γ T d. (.17) Integration of relation Eq. (.17) between the zero state and the arbitrary state s(, T ) gies s c p Rearranging terms, we arrie at = ln T + 1 γ T γ T ln + C. (.18) T 1 γ T γ T = e s cp +C. (.19) Again, in the isentropic case s anishes and the right-hand side of Eq. (.19) reduces to a constant, that is uniquely determined for each isentrope by the integration constant C. Therefore, Eq. (.18) reduces to the assumed form the T -isentrope in case of an isentropic process. 6

27 THREE Behaior of the Isentropic Exponents The isentropic relations for real gasses were formally introduced in the preceding chapters. In this chapter, the limits and oerall behaior of the isentropic exponents will be explored. After the initial deriation of the real exponents γ, γ T, and γ T, Kouremenos and his co-authors engaged in similar inestigations on the behaior of the exponents for different substances including dry steam, ammonia, and some refrigerants [, 3]. In this respect, their work had mostly been of semi-empirical nature rather than a rigorous analytical inestigation. Haing demonstrated the generalized application of the isentropic relations Eqs. ( ) in Chapter, we will now focus on understanding the behaior of the isentropic exponents themseles. The present study will be a general inestigation into the trends obsered in the behaior of the real isentropic exponents. The Van der Waals equation of state is of particular interest for qualitatie analysis, owing to the mathematically simple modification to the ideal gas law to incorporate fundamental molecular interactions. With the intent of a general inestigation, the Van der Waals will be expressed in reduced form, which allows comparison of the results for different substances according to the principle of corresponding states. 3.1 Van der Waals Isentropic Exponents Expressing the isentropic exponents for a Van der Waals substance is indeed ery straightforward and is mostly an exercise in algebraic manipulations. Each of the terms in the definitions of the isentropic functions are ealuated by the Van der Waals equation. The results of this procedure are summarized in Table 3.1. A summary of the Van der Waals equation is proided in Chapter 11 on numerical simulation of real gases, along with the Soae-Redlich-Kwong and eng-robinson equations of state. Note that the Van der Waals relations for the isentropic exponents in Table 3.1 can be erified independently, as they are associated according to the relation between the exponents Eq. (1.34). The combination of any two expressions will result in the third exponent. 7

28 Table 3.1: Isentropic exponents for a Van der Waals substance in reduced form, explicit in temperature and olume, and pressure and olume. γ T ( r, r) = 1 + R c -plane 3 r 3 r 1 γ ( r, r) = cp 3 3 r ( r + 3/ r ) 6(3 r 1) c r ( r + 3/ r )(3 r 1) 3(3 r 1) γ T ( r, r) = γ T (T r, r) = 1 + R c 1 1 R c p r 3 ( r + 3/ r ) 3 r r3 ( r + 3/ r ) (3 r 1) T-plane 3 r 3 r 1 γ (T r, r) = cp 4 3 r T r 6(3 r 1) c 8 r T r(3 r 1) 3(3 r 1) γ T (T r, r) = 1 1 R c p 8 r 3 T r 3 r(3 r 1) 8 r3 T r (3 r 1) Although the reduced Van der Waals equation in principle does not require any information to characterize the substance, this is no longer true if we wish to ealuate the isentropic exponents due to their dependency on the specific heat capacities c and c p. To close the problem, the isochoric and isobaric heat capacities need to be quantified. The isochoric heat capacity will be assumed constant between states and to be solely a function of the molecular structure of the substance according to the equipartition theorem [34, 4], also known as the polytropic gas model [6, 39, 43]. Under this assumption, the isochoric heat capacity is expressed as c = f R, (3.1) where f is the molecular degrees-of-freedom. The minimum d.o.f of a single molecule is 3 one along each dimension for translation through three-dimensional space. More complex molecular structures hae additional other types of internal motions such as molecular ibrations or rotations, adding to the total molecular degree-of-freedom. The dependency of the d.o.f of a molecule to the number of molecules N is presented in Table 3.. The specific heat capacities are related by Eq. (3.) by definition [34]. Under the assumption of a constant alue of the isochoric heat capacity, the ariation of the isobaric heat capacity is entirely determined by the equation of state according to 8

29 Table 3.: Relation between the molecular degrees-of-freedom and molecular size according to the equipartition theorem. monatomic N = 1 f = 3 diatomic N = f = 5 linear polyatomic N > f = 3N c p = c + T ( ) ( ). (3.) Before we continue with the real gas analysis, let us reason about the limiting alues of Eq. (3.) in the ideal case. In this case, Eq. (3.) reduces to c p c = R. As the isochoric specific heat capacity c is only a function of the molecular degree of freedom under the polytropic gas assumption, the same hold for the isobaric heat capacity c p. In the ideal gas limit we hae c = f ( R, cp = 1 + f ) R, and γ = 1 + f. (3.3) Referring to the specific heat ratio in relation to the molecular degree of freedom Table 3., the ideal adiabatic coefficient γ can be demonstrated to attain a maximum alue of 1.6 for a monatomic gas, and approaches the alue of 1 as the size of the molecule increases [33, 34]. Although this is not the general case for the real isentropic exponents, the same limit is satisfied in the ideal limit. We will now continue with the real gas analysis using the Van der Waals equation of state. The relation between the specific heats Eq. (3.) can be expressed as [44] c p = c + 1 R (3r 1) 4T r r 3. (3.4) Due to the dependency of the specific heat capacities on the molecular composition, the analysis of behaior of the real isentropic exponents will explore two molecular species representing two limiting cases regarding molecular size. A light diatomic molecule will be considered, for which c =.5R, and a much heaier molecule for which c = 50R. Although the isobaric specific heat capacity is no longer solely a function of the molecular size as in the ideal case, the limits of c p in the non-ideal case may be explored in a similar way. With an increase of the molecular size, the term c in the expression for c p aboe becomes increasingly more dominant, in the limiting case where c p c for a infinitely large molecule. The ariation of the isobaric heat capacity on the isochoric heat capacity, therefore, becomes less pronounced as the molecular size increases, in the limit where c p also becomes constant. 9

30 3. hysical Limits of the Isentropic Exponents hysical limits bind the domain of the isentropic functions, forming a natural condition for alidity of the isentropic gas model. Exploring these limits proides an intuitie way to start the inestigation of the behaior of the isentropic relations, for which an equation of state does not yet hae to be specified. Two limits can be identified in the positie pressure-olume plane, consisting off the ideal gas limit as r >> 1 and the incompressible liquid model as r << 1. These two conditions are to be satisfied as boundary condition for the generalized isentropic substance model. The ideal gas limit is attained as either the temperature goes to infinity or the pressure of the system tends to zero [33, 34]. The incompressible limit for a Van der Waals substance is attained as the reduced olume as r tends to 1/3, corresponding to the limit where the size of the system becomes equal to molecular olume denoted as coefficient b in the Van der Waals equation. The area to the left of r = 1/3 is therefore unphysical and excluded from further discussion. When the temperature tends to infinity, the behaior of any gas has been proen to approach ideal gas behaior under any circumstance [34]. The original authors also demonstrated that the real isentropic exponents tend to the alue of γ = c p/c at increasingly higher temperatures, but did not explicitly demonstrated the equialence between the two [1]. Inersely, one might of the resemblance with the ideal gas case to be a mathematical requirement, since the deriation of the real gas isentropic relations started out by the assumption of the ideal formulation. In fact, under the ideal analysis, the exponents γ, γ T, and γ T together form different terms of Mayer s relation for an ideal gas [33, 34] ideal gas limit γ = cp c, γ T = c + R c, and γ T = cp c p R. (3.5) T (a) iston (b) -diagram (c) T -diagram Figure 3.1: Compression of an incompressible liquid 30

31 Conersely, the incompressible substance model must be satisfied for liquid states. The alues for the isentropic exponents for an incompressible fluid can be obtained by letting the reduced olume r 1/3 in the Van der Waals expressions of the isentropic exponents. The incompressible limit can be explored heuristically, by considering a liquid brought under pressure in a piston arrangement Figure 3.1a. For an incompressible substance, the isentropic pressure-olume relationship reduces to an isochoric process for which γ, Figure 3.1b. Upon isentropic compression of an incompressible fluid, the internal energy, and therefore the temperature, should remain unchanged. From here follows that the temperature is completely independent of changes in either pressure or olume for an incompressible substance, Figure 3.1c. Accordingly, the exponents γ T and γ T tend to infinity and one in the incompressible limit, respectiely. The incompressible limit can be summarized as γ =, γ T =, and γ T = 1. (3.6) Between the two olumetric limits, the two-phase region predicted by the Van der Waals equation imposes another boundary of the domain of the isentropic functions. The specific heat capacities are undefined in the coexistence region, thus neither are the exponents γ, γ T, and γ T. From here we can conclude that the applicability of the generalized isentropic model is limited to stable single phase substances, as is also the case of the ideal isentropic model. We conclude our discussion on the limits of the isentropic exponents by the limit imposed by the thermodynamic singularity at the critical point. This limit may be explored by setting T r = 1 in the Van der Waals expressions for the isentropic exponents and by letting r 1. At the critical point, the isentropic exponents assume indeterminate forms, the exponent γ T being the only exception which assumes a alue of 1 + 3R/c. The limits of the isentropic exponents are summarized in Table 3.3. incompressible limit Table 3.3: Isentropic limits in the pressure-olume diagram incompressible limit r 1 3 γ γ T γ T 1 critical limit r = 1 γ = γ T = 1 + 3R c γ T = ideal gas limit r γ = cp c γ T = c + R c γ T = cp c p R 31

32 3.3 Contours of the Isentropic Exponents Now that the boundaries of the domain of the isentropic functions hae been identified, their general behaior between those limits can be studied using the Van der Waals equation of state. We will turn to contour plots of the isentropic exponents which proide a graphical way to interpret their behaior, with it also raising the question of what lines of constant γ, γ T, and γ T physically represent. To answer this underlying thought, obsere that in the definition of the isentropic exponents Eqs. ( ) all three exponents are composed of two physically different terms; either an explicit or implicit formulation of the specific heat ratio and a term inoling compressibility of the state. The isentropic exponents may, therefore, be iewed as compressibility corrected heat capacity ratios where the deriatie term is corrected according to its gas behaior and its density. The isentropic exponent γ, for example, includes the deriatie term ( / ) T which for a gien pressure is corrected for its location in the -diagram as the compressibility of low-density gasses is naturally higher than liquids which hae a much higher density /c /c / c (a) Low molecular weight c =.5R / c (b) High molecular weight c = 50R Figure 3.: Contours of γ T of a Van der Waals substance The contours of γ T in Figure 3.a and 3.b consist of logarithmically spaced ertical lines, which is not surprising gien the fact that the reduced olume is the only ariable in the polytropic Van der Waals model for γ T (see Table 3.1). The olumetric limit bounds the left-hand side at r = 1/3 for which γ T goes to infinity, causing the logarithmic spacing of the contours towards this limit. At the right-hand side of the domain, the alue of γ T tends to the ideal gas limit and approaches 1 + R/c in both cases. 3

33 By definition, γ T is always positie as none of the terms in the analytical expression Eq. (1.31) can attain a negatie sign. The deriatie term ( / ), or the pressure change by heating of a closed olume, must be positie from the condition of mechanical stability [34]. From here γ T can be reasoned to be bound by the domain: > γ T 1 + R c. (3.7) domain γ T The contour leels plotted for the light molecule c =.5R and the heay molecule c = 50R in Figure 3. are of the same order of magnitude. Although the plots of both substances possess the same general features, due to the reduced ariation of the isobaric heat capacity for large molecules, as was reasoned in section 3.1, the ariation of γ T has become more gradual, and its features are squeezed towards the incompressible limit. This trend will also be obsered in the contour plots of γ and γ T as we shall now see /c /c / c (a) Low molecular weight c =.5R / c (b) High molecular weight c = 50R Figure 3.3: Contours of γ of a Van der Waals substance Contours of the isentropic exponent γ are displayed in Figure 3.3a and 3.3b, bound by the same limits as γ T. Because the alue of γ tends to infinity as the reduced olume r 1/3, a logarithmic spacing of the contour lines toward this limit is again obsered. The latter is more pronounced for the larger molecules than the smaller molecules, due to the reduced ariations between the specific heats c p and c for larger molecules. Similar to γ T the isentropic exponent γ can be reasoned to be restricted to positie alues only, as solely the deriatie term ( / ) T in the definition of γ can change sign. The sign change predicted by the Van der Waals equation can be seen graphically by looking at the isotherms plotted in Figure

34 4 3 /c / c Figure 3.4: Van der Waals isotherms and spinodal line The deriatie ( / ) T is negatie for any stable single phase substance by the requirement of mechanical stability [33, 34]. Due to the minus sign in the definition of γ, the exponent γ, Eq. (1.30) is always positie. Moreoer, the region where the partial deriatie ( / ) T is predicted to change sign by the Van der Waals equation is enclosed by the two-phase region. The conditions of mechanical and thermal stability in the two-phase region are replaced by the requirement of equal chemical potential between the phases. The line where ( / ) T is zero or alternatiely, where γ is zero denotes the line of ultimate stability of a single phase substance. Equating the exponent γ in Table 3.1 to zero we find the analytical expression of the dashed line in Figure 3.4, better known as the Van der Waals spinodal [45, 46] expressed as: Van der Waals spinodals r = 3r (3r 1), and T 3 r =. (for γ r 4 3 = 0) r (3.8) This feature of γ is of course not resered for the Van der Waals substances only. The spinodal line of any equation of state that predicts phase transitions can be obtained by equating the exponent γ to zero. From the aboe discussion follows that the γ spans the entire domain of positie rational numbers from zero to infinity: domain γ > γ 0. (3.9) Values between zero the ideal limit c p/c are obsered in a narrow band along the spinodal line on the apor side, which perhaps might be the result of the oer prediction of the Van der Waals equation in this region. 34

35 /c /c / c (a) Low molecular weight c =.5R / c (b) High molecular weight c = 50R Figure 3.5: Contours of γ T of a Van der Waals substance We conclude this section on the behaior of the isentropic exponents by looking at the contours of γ T in Figure 3.5a and 3.5b. The contours of γ T look distinctly different from those of γ and γ T, mostly due to absence of a limit towards infinity in the incompressible limit. The limit of γ T to one in the incompressible case has another effect, shown in Figure 3.5b. As the isobaric heat capacity tend to the isochoric heat capacity for increasingly larger molecules c p c, the ideal gas limit also tend to one. As a result, the ariation of γ T becomes eer more uniform throughout the -plane fro high molecular weight substances. Only the metastable states around the spinodal lines show slight changes in γ T. Lastly, note that the partial deriatie ( / ) in the definition of γ T must be positie for any single phase substance by the condition of thermal stability [34]. Furthermore, as neither nor c p can be negatie, it follows that γ T is always positie, and cannot attain alues smaller than one: > γ T 1. (3.10) The limit where γ T is equal to 1 will be demonstrated to hae special significance in Chapter 5, denoting the state of maximum density of a substance. domain γ T The trend of reduced ariation of the isentropic exponents for high molecular weight substances was generally obsered in this section. What does this say about the behaioral differences between high and low molecular weight species? From a mathematical point of iew, a region where the isentropic exponents show only small ariations can be reasoned to hae isentropes with only small ariations in their shape. Turning to the isentropic relations Eqs. (1.7), (1.8), and (1.9) of Chapter 1, a reduced ariation of 35