Technology & Engineering
Equations of State
Equations of state are mathematical relationships that describe the behavior of a substance, such as a gas or liquid, under different conditions of temperature, pressure, and volume. These equations help engineers and scientists understand and predict the properties and behavior of materials, making them essential for designing and optimizing technological processes and systems.
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11 Key excerpts on "Equations of State"
eBook - PDFThermophysical Properties Of Fluids: An Introduction To Their Prediction
An Introduction to Their Prediction
- Marc J Assael, J P Martin Trusler, Thomas F Tsolakis(Authors)
- 1996(Publication Date)
- ICP(Publisher)
6 Equations of State Equations of State are widely used in the prediction of thermodynamic properties of pure fluids and fluid mixtures, in part because they provide a thermodynamically consistent route to the configurational properties of both gaseous and liquid phases. Consequently, equation-of-state methods may be used to determine phase-equilibrium conditions as well as other properties. Indeed, the most well-known application of such methods in chemical engineering lies in the field of high-pressure vapour-liquid equilibria (VLE) where the equation-of-state approach is the method of choice for the vast majority of systems. The main exceptions are systems containing associating substances for which activity coefficient models, which will be described in Chapter 7, may provide a more accurate route to the fugacity of the liquid phase. The application of both methods to phase equilibrium problems will be discussed in detail in Chapter 8. The main objective of this chapter is to provide an introduction to the predictive capabilities of the most commonly encountered Equations of State. Following a few historical notes and a classification of the various kinds of Equations of State, the most widely used equations will be reviewed. In each case applications to both pure substances and multicomponent mixtures will be discussed. Finally the ranges of applicability will be outlined and some examples given which will illustrate the capabilities of the approach. More advanced examples involving phase equilibria will be presented in Chapter 8. 6.1. Introduction and Historical Notes The term equation of state is used to describe an empirically-derived function which provides a relation between pressure, density, temperature and (for a mixture) composition; such a relation provides a prescription for the calculation of all of the configurational and residual thermodynamic properties of the system within some
eBook - PDF- Donald Olander(Author)
- 2007(Publication Date)
- CRC Press(Publisher)
49 2 Equations of State 2.1 WHAT IS AN EQUATION OF STATE? There are two types of Equations of State , or EOS. In general, they are thermodynamic relationships between three properties of a pure (one-component) substance. In discussing the fundamental meaning of Equations of State, it is useful to eliminate the quantity of the substance in order to deal only with intensive properties (see Section 1.6.2). Extensive properties can be converted to intensive properties by dividing the former by the quantity of the substance. (e.g., on a per-mole basis v = V / n , u = U / n ). The volumetric EOS refers to the relationship of the p -v -T properties of a gas. Solids and liquids are also described by p -v -T Equations of State, although the quantitative forms are very different from those applicable to gases. In functional form, the volumetric EOS can be written as v = f ( p,T ), or v ( p,T ) for short. This form indicates that the specific volume (or molar volume) is expressed as a function of pressure and temperature. However, the EOS can be equally well written as p ( v,T ) or T ( p,v ). Because pressure and temperature are usually specified in an experiment or in a process, the form v ( p,T ) is most commonly employed. The EOS relating v , p , and T provides no information about the other thermo-dynamic properties, in particular about the internal energy u and the entropy s . However, according to the phase rule (Section 1.12), specifying any two properties of a pure substance fixes all properties. Nonetheless, the EOS in the form p ( T,V ), for example, gives no hint about the functions u ( T,v ), s ( T,v ), or of the remaining auxilliary properties h , f , and g (see Section 1.6.1). Knowledge of ( T,v ) requires information about the substance beyond that contained in its p -v -T relationship. Functional relationships such as u ( T,v ) are sometimes called thermal Equations of State .
- John Reisel(Author)
- 2021(Publication Date)
- Cengage Learning EMEA(Publisher)
Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. Chapter 3 Thermodynamic Properties and Equations of State 80 the calculations involved in engineering thermodynamic analysis. These categories are ideal gases and incompressible substances. Because many substances can be adequately modeled as one of these two types, there are many practical engineering systems where the assumption of idealized substance behavior provides sufficient accuracy to allow for an acceptable design or analysis. Next, we consider some of the Equations of State for these substances. 3.5 Equations of State FOR IDEAL GASES An ideal gas is one for which it is assumed that there are no interactions between molecules or atoms in the system for a substance in the gas phase, such as shown in Figure 3.13. Outside of systems consisting of only a single molecule, such a substance does not exist because molecules in any real system will collide and intermolecular forces will exist. However, in engineering practice, substances that are in the gas phase and not near a phase transition are usually considered to be ideal gases. (This assumption should not be made when an exceptional degree of accuracy is required for a calculation, or when the density of the gas is very high. For such situations, other Equations of State exist, some of which will be briefly introduced later.) A general rule of thumb for the use of the ideal gas assump- tion is that it is usually acceptably used for substances that are normally thought of as a gas (such as air, nitrogen, oxygen, hydro- gen, etc.), provided that the gas is at a tem- perature well above its saturation tempera- ture at its given pressure, and provided that the density of the gas is not very high. 3.5.1 The Ideal Gas Law The ideal gas law is used to relate the pressure, volume, and temperature of an ideal gas.
eBook - PDF- Kalyan Annamalai, Ishwar K. Puri, Milind A. Jog(Authors)
- 2011(Publication Date)
- CRC Press(Publisher)
Otherwise, the relations between these properties must be derived using physical principles. First, we will discuss the simplest equation of state (i.e., the ideal gas relation). This relation will be extended to consider real gases and other fluids in terms of two- and three-parameter Equations of State. Finally, approximate Equations of State for liquids and solids are discussed. THERMOLAB 306 ◾ Advanced Thermodynamics Engineering, Second Edition software is available to solve for v given P and T and vice versa for various state equations. (for which spreadsheet software is also available at www.crcpress.com, download and update page, Advanced Thermodynamics Engineering, download thermolab-1.zip) 6.2 Equations of State The ideal gas equation of state is also considered to be a thermally or mechanically perfect state equation. In Chapter 1 we presented a simple derivation of this equation by using microscopic thermodynamic considerations and neglecting intermolecular forces and the molecular body vol-ume. The resulting relation was Pv RT 0 = . (6.1) The subscript 0 in this chapter and Chapter 7 implies that the gas is ideal at the given conditions. If the measured gas volume at given P and T values is identical to that calculated by using Equation 6.1, the gas is said to be an ideal gas. However, if the measured specific volume at the same pres-sure and temperature differs from that determined using Equation 6.1, the gas is considered to be a real gas. If so, the actual specific volume v – (T, P) can be determined from experiments and tabulate them for many species of interest just like steam and R134a tables. However this requires tabulation of a vast amount of data for many species. A way to avoid tabulation is to the use com-pressibility factor Z, which is defined as Z(T, P) = v – (T, P)/v – 0 (T, P), (6.2) while v – 0 can be determined using ideal gas law.
- Elias I. Franses(Author)
- 2014(Publication Date)
- Cambridge University Press(Publisher)
5 Equations of State for one-component and multicomponent systems 5.1 INTRODUCTION AND MEASUREMENTS ................................................................................................. An “ equation of state, ” is the mathematical relationship of how the speci fi c volume V in m 3 = kg ð Þ , the molar volume, V m in m 3 = mol ð Þ , or the density, ρ (in kg/m 3 ) depends on the pressure, p , and the temperature, T ; it is the function V m p , T ð Þ . For solutions with mole fractions x 1 , x 2 , etc., there are additional composition variables, and one needs to fi nd the function V m p , T , x 1 , x 2 , . . . , x N − 1 ð Þ . Sometimes, this equation is called a “ constitutive equation. ” For non-simple thermodynamic systems, such as solids (non-hydrostatically stressed), glasses, and liquid crystals, the volume may depend on additional variables, as discussed in Section 3.9 . The typical accuracy of volume measurements can easily be 0.1% or even 0.001% for liquids using probes whose vibration frequency depends on the mass of the liquid of a given volume. We de fi ne two measures describing how V varies with T and p . 1. One measures how V varies with pressure p at fi xed temperature T . The mathematical measure is the partial derivative ∂ V = ∂ p T . Typically one de fi nes the “ isothermal compress-ibility ” as κ T − 1 V ∂ V ∂ p T , ð 5 : 1 Þ which has units of bar − 1 . Because the partial derivative is divided by V , the isothermal compressibility is the fractional volume change per unit change in pressure, or about κ T − Δ V V 1 Δ p : To determine κ T one measures and plots V p ð Þ at a given temperature, fi nds the derivative at a given p and T , and then calculates κ T from Eq.
- Kevin Dahm, Donald Visco(Authors)
- 2014(Publication Date)
- Cengage Learning EMEA(Publisher)
289 7 Equations of State (EOS) C hapter 2 introduced the need for reliable models that provide estimates of physi-cal and thermochemical properties, for situations in which experimental data is not available. The models introduced at that time (e.g., ideal gas model) were simplistic, and valid only under specific conditions. Chapter 6 introduced a broader range of modeling techniques that are applicable to any pure compound at any conditions, but using them often requires an equation of state. In Chapter 2, we defined an “equation of state” (frequently abbreviated EOS) as a math-ematical relationship among P , V , and T , such that if any two are specified, the third can be calculated. So far, the only Equations of State we have introduced are the ideal gas law and van der Waals. The motivational examples take a closer look at these and demonstrate the need for more versatile Equations of State. Chapter 6 contained some additional equa-tions of state that were contrived by the authors for illustrative purposes but have never been used in engineering practice. 7.1 MOTIVATIONAL EXAMPLES : Transportation of Natural Gas Natural gas is a widely used fuel source, composed primarily of methane. Within the continental United States, it is most commonly transported by pipeline. However, when natural gas is shipped overseas, or to remote locations, an alternative to build-ing more pipelines is transporting it by ship. This involves liquefying the natural gas, so that it can be stored in small volumes for transport. Liquefied natural gas (LNG) is hazardous because it is flammable and also because of the extreme cold at which it must be maintained; typical temperatures would be in the vicinity of 110 K.
eBook - PDFPhysics of Strength and Fracture Control
Adaptation of Engineering Materials and Structures
- Anatoly A. Komarovsky, Viktor P. Astakhov(Authors)
- 2002(Publication Date)
- CRC Press(Publisher)
As a result, researchers overlooked the natural link between mechanical and thermal properties. Alternatively, Equation 2.65 allows explanation of thermal and many other physical–mechanical effects that accompany behavior of solids in various external fields and aggressive environments. 2.5 Parameters of State, Relationship of Equivalence, and Entropy In different scientific disciplines, different physical meanings are assigned to the term “state of a solid.” 1–24 Following the definition by Frenkel, 56 we define it as “the shape of condensation of rotoses during the solidification process” and “the ability of a structure to preserve it afterwards under various external conditions.” We can use this well-established term further on without fear of misunderstanding. Because the equation of state (2.65) is of a fundamental importance for solid-state physics, strength, reliability, and durability of structural materi-als, 21 we should consider the physical meaning of its parameters in detail. As will be shown further, entropy occupies a special place among these parameters (Equation 2.83.V). Relationship 2.83.I allows Distribution 2.84 to be written as . (2.86) Therefore, the type of the bond depends exclusively upon the kinetic part of the internal energy of a body. We can assume that the motion of atoms in solids is of a collective nature. This means that any change in the individual motion of an individual atom immediately affects the others.For example, an atom that has number n , quits the equilibrium state and moves over a certain distance dr in an arbitrary direction. As a result, it starts to become affected by its closest neighbors. N kT 1 2 1 2 , / = ± F Equation of State of a Solid and Its Manifestations 97 As the interaction between atoms is of a short-range character, a distur-bance will be transferred only to atoms with numbers ( n – 1) and ( n + 1).
eBook - PDF- Alberto Patiño Douce(Author)
- 2011(Publication Date)
- Cambridge University Press(Publisher)
8 Equations of State for solids and the internal structure of terrestrial planets Inferring the internal structure of solid planetary bodies requires that we use thermodynamic theory in order to interpolate and extrapolate often sparse experimental data to very high pressures and temperatures. The constant-volume approximation that we used in Chapter 5 to calculate the Gibbs free energy of solids at high pressure leads to erroneous results at depths greater than a few km. This situation is remedied by introducing a variety of Equations of State for condensed phases, that are accurate over progressively greater pressure ranges. The study of Equations of State for condensed phases that are valid at very high pressures and temperatures will allow us not only to perform chemical equilibrium calculations relevant to deep planetary interiors but also to predict physical conditions – pressure and temperature – as a function of depth in solid planets. 8.1 An introduction to Equations of State for solids In Chapter 5 we calculated the following integral for solid phases assuming that their molar volumes remain constant: P 1 V (P , T )dP . (8.1) This is tantamount to assuming that the second derivatives of the Gibbs free energy vanish (equations (4.136) and (4.137)), which is in general not true. All materials change in volume in response to changes in pressure and temperature, and a change in volume entails a change in free energy. This energy needs to be accounted for, both when studying chemical transformations (i.e. calculation of phase equilibria) and when inferring physical conditions in planetary interiors. For example, equation (3.32) is a general differential equation for the adiabat, based exclusively on thermodynamic relations and thus independent of specific material properties.
eBook - PDFThermodynamics
From Concepts to Applications, Second Edition
- Arthur Shavit, Chaim Gutfinger(Authors)
- 2008(Publication Date)
- CRC Press(Publisher)
393 14 Equations of State and Generalized Charts In Chapter 4 we have shown that for every real pure substance in a state of stable equilib-rium there exists an equation of state, that is, a relationship among volume, pressure, and temperature f v p T ( , , ) 0 In general, this relationship is based on experimental data and may be given in the form of tables of properties, for example, steam tables. Expressing Equations of State in algebraic form is useful for computations. Hence, several algebraic relationships were proposed for Equations of State. These algebraic relationships are only approximations that agree with real data for limited ranges. The equation of state for an ideal gas, pv RT that was introduced in Chapter 5, is an example of a simple algebraic equation. The range of applicability of the ideal gas equation is, indeed, quite limited. It gives good results only for states in which the volume is much larger than the critical volume, that is, for low pressures and high temperatures. Several Equations of States have been proposed with the objective of improving the accuracy and range of applicability for real systems. This chapter examines several widely used Equations of State for pure substances and shows how one may obtain thermodynamic data from these equations. The concept of reduced properties is introduced, leading to the rule of corresponding states. The general-ized charts, which are based on this rule, are then presented. The chapter ends with the introduction of a new property, the fugacity. 14.1 van der Waals Equation An early attempt to extend the validity of an algebraic equation of state to real substances was made in 1873 by van der Waals, who in his doctoral dissertation proposed an equation of state that was an extension of the ideal gas equation p a v v b RT 2 ( ) (14.1) The volume in the ideal gas equation is replaced by ( v − b ).
eBook - PDF- Marc J. Assael, William A. Wakeham, Anthony R. H. Goodwin, Stefan Will, Michael Stamatoudis(Authors)
- 2011(Publication Date)
- CRC Press(Publisher)
While it is of course possible to avoid moisture, for example, in the packaging of electronic equip-ment, by adding some hygroscopic material, this approach is not practicable for a continuous process, because the material would have to be removed and dried in some batch process for reuse. As a consequence, in A/C applications moist air is drawn out of a room into a machine, cooled below its dew point, the condensate removed, and the air with lower moisture is reheated to the desired temperature before being ejected back into the room. To reduce energy consumption the heating is, or should be, achieved using a heat exchanger between the two air streams. 4.7.2 Which Equations of State Should Be Used in Engineering VLE Calculations? Equations of State are used in engineering to predict thermodynamic proper-ties in particular the phase behavior of pure substances and mixtures. However, since there is neither an exact statistical-mechanical solution relating the properties of dense fluids to their intermolecular potentials, nor detailed infor-mation available on intermolecular potential functions, all Equations of State are, at least partially, empirical in nature. The Equations of State in common use within both industry and academia can be arbitrarily classified as fol-lows: (1) cubic equations such as that of van der Waals that are described by Economou (2010); (2) those based on the virial equation discussed by Trusler (2010) and Chapter 2 of this volume; (3) equations based on general results obtained from statistical mechanics and computer simulations mentioned, including the many forms of statistical associating fluid theory known by the acronym SAFT as described by McCabe and Galindo (2010); and (4) those obtained by selecting, based on statistical means, terms that best represent the available measurements obtained from a broad range of experiments as outlined by Lemmon and Span (2010).
eBook - PDFExtreme Physics
Properties and Behavior of Matter at Extreme Conditions
- Jeff Colvin, Jon Larsen(Authors)
- 2013(Publication Date)
- Cambridge University Press(Publisher)
6 Equation of state As we learned in Chapter 4, the equations describing the motion of plasma are three equations that are derived from the laws of conservation of mass, momentum, and energy. These three equations express the four variables that describe the moving plasma – mass density, pressure, temperature (or energy), and velocity – as functions of spatial position and time. Since it is not possible to solve a system of three equations for four variables, we need a fourth equation relating some or all of these four variables that does not introduce another variable. The fourth equation is the equation of state. In the discussion that follows we will refer to the equation of state as the EOS. With the EOS specified, we can then solve the equations of motion for the plasma. We will learn how this solution is done numerically in Chapter 11. In this chapter we learn the basics of how to specify the EOS for matter at extreme conditions. 6.1 Basic thermodynamic relations We learned in Section 2.2.2 about the relaxation rates in dense plasma, that is, the rates at which thermodynamic equilibrium is established. In general, the variables characterizing the state of the plasma – mass density (or alternatively, particle number density), pressure, temperature – change slowly compared to these relax- ation rates. Thus, we can consider that the plasma is, at each point in space and at each instant of time, in local thermodynamic equilibrium (LTE). In LTE the particle distribution functions for each particle species comprising the plasma can be characterized by a single parameter, the temperature. Further, in the equilibrium plasma the laws of thermodynamics apply. In partic- ular, the thermodynamic free energy of the plasma can be written as F = −kT log Z, (6.1) 159 160 Equation of state where Z is the partition function, which describes how the energies are partitioned among the particles.
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