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5. Equations of State SVNA Chapter 3

5. Equations of State SVNA Chapter 3. Efforts to understand and control phase equilibrium rely on accurate knowledge of the relationship between pressure, temperature and volume for pure substances and mixtures. This PT diagram details the phase boundaries of a pure substance.

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5. Equations of State SVNA Chapter 3

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  1. 5. Equations of State SVNA Chapter 3 • Efforts to understand and control phase equilibrium rely on accurate knowledge of the relationship between pressure, temperature and • volume for pure • substances and • mixtures. • This PT diagram • details the phase • boundaries of a • pure substance. • It provides no • information • regarding molar • volume. Lecture 12

  2. P-V-T Behaviour of a Pure Substance • The pure component PV-diagram shown here describes the relationship between pressure and molar volume for the various phases assumed by the the substance. Lecture 12

  3. PV Diagram for Oxygen Lecture 12

  4. Equations of State • Experimental data exist for a great many substances and mixtures over a wide range of conditions. • Tabulated P-V-T data is cumbersome to catalogue and use • Mathematical equations (Equations of State) describing P-V-T behaviour are more commonly used to represent segments of the phase diagram, usually gas-phase behaviour • Ideal Gas Equation of State • Applicable to non-polar gases at low pressure: • where Vis the molar volume (m3/mole) of the substance. • In terms of compressibility, Z=PV/RT, the ideal gas EOS gives: Lecture 12

  5. Equations of State: Non-ideal Fluids • The ideal gas equation applies • under conditions where molecular interactions are negligible and molecular volume need not be considered. • At higher pressures, the compressibility factor, Z, is not unity, but takes on a value that is different for each substance and various mixtures. • A more complex approach is • needed to describe PVT behaviour of non-ideal fluids Lecture 12

  6. Virial Equation of State for Gases • If our goal to calculate the properties of a gas (not a liquid or solid), the PVT behaviour we need to examine is relatively simple. • The product of pressure and molar volume is relatively constant, and can be approximated by a power series expansion: • from which the compressibility is readily determined: • Eq 3.11 • The coefficients B’,C’,D’ are called the first, second and third virial coefficients, respectively, and are specific to a given substance at a given temperature. • These coefficients have a basis in thermodynamic theory, but are usually empirical parameters in engineering applications. Lecture 12

  7. Cubic Equations of State: Gases and Liquids • A need to describe PVT behaviour for both gases and liquids over a wide range of conditions using an equation of minimal computational complexity led to the development of cubic equations of state. • Peng-Robinson (PR): Sauve-Redlich-Kwong (SRK): • in terms of compressibility, Z: • PR-EOS: • SRK-EOS: • where a and b (or A and B) are positive constants that are tabulated for the substance of interest, or generalized functions of P and T. • These polynomial equations are cubic in molar volume, and are the simplest relationships that are capable of representing both liquid and gas phase properties. Lecture 12

  8. Cubic Equations of State: Gases and Liquids • Given the required equation parameters (a and b in the previous cases), the system pressure can be calculated for a given temperature and molar volume. • At T > Tc, the cubic EOS has just one real, positive root for V. • At T<Tc there exists only one real, positive root at high pressure (molar volume of the liquid phase). However, at low pressures the cubic EOS can yield three real, positive roots; the minimum representing the liquid-phase molar volume, and the maximum the vapour-phase molar volume. Lecture 12

  9. Theorem of Corresponding States • The virial and cubic equations of state require parameters (B’, C’, a, b, for example) that are specific to the substance of interest. In fact, the PVT relationships for most non-polar fluids is remarkably similar when compared on the basis of reduced pressure and temperature. • Simple fluids aside (argon, xenon, etc), some empiricism is required to achieve the required degree of accuracy. The three-parameter theorem of corresponding states is: • All fluids having the same value of acentric factor, , when compared at the same Tr and Pr, have the same value of Z. • The advantage of the corresponding states, or generalized, approach is that fluid properties can be estimated using very little knowledge (Tc, Pc and ) of the substance(s). Lecture 12

  10. Theorem of Corresponding States Lecture 12

  11. Pitzer Correlations: Gases and Liquids • Pitzer developed and introduced a general correlation for the fluid compressibility factor. • 3.54 [3.57] • where Zo and Z1 are tabulated functions of reduced pressure and temperature. • This approach is equally suitable for gases and liquid, giving it a distinct advantage over the simple virial equation of state and most of the cubic equations. • Values of , Pc and Tc for a variety of substances can be found in Table B.1 of SVNA. • The Lee/Kesler generalized correlation (found in Tables E.1-E.4 of the SVNA) is accurate for non-polar, or only slightly polar, gases and liquids to about 3 percent. Lecture 12

  12. Generalized Virial-Coefficient Correlation: Gases • The tabulated compressibility information that is the basis of the generalized Pitzer-type approach can be cumbersome (especially in an exam) • the complex PVT relationship of non-ideal fluids is difficult to represent by a simple equation, necessitating the use of tables if the corresponding states approach is to be accurate. • SVNA provides a generalized virial EOS correlation that allows you to apply the virial EOS with coefficients that are based on a corresponding states approach (Page 102 SVNA, 6th & 7th ed). • where • and Lecture 12

  13. PVT Behaviour of Mixtures • Most equations of state prescribe mixing rules that allow you to calculate EOS parameters and describe the PVT behaviour of mixtures. • The Virial EOS, • the composition dependence of the virial coefficient B is: • where y represents the mole fractions in the mixture and the indices i and j identify the species. Values of Bij are determined using generalized correlations and/or formulae specifically developed for the mixture of interest. • Mixture behaviour will be examined in greater detail later in the course Lecture 12

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