PVT Behavior of Fluids

The thermodynamic principles discussed in Section 1.1 are rigorous and general. 

The system for which thermodynamic functions were derived need not be of 

The fluid behavior is commonly represented by its pressure-volume-temperature (or PVT relationship, expressed in general as 

This relationship might be available in the form of experimental data, or it could be represented by a model. Models are usually based on experimental data, but they also possess predictive capabilities. That is, they are expected not only to reproduce the correlated data, but also to generate data over reasonable ranges of conditions. Although many PVT models are semi-empirical, some are based on theoretical principles such as molecular thermodynamics and statistical thermodynamics. No single PVT correlation exists that can accurately predict all properties for diverse substances over wide ranges of temperature, pressure, density, and composition. Nevertheless, a number of models have demonstrated their usefulness for many applications. 

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For an accurate description of the PVT behavior of fluids over wide ranges of temperature and pressure, an equation of state more comprehensive than the virial equation is required. Such an equation must be sufficiently general to apply to liquids as well as to gases and vapors. Yet it must not be so complex as to present excessive numerical or analytical difficulties in application. 

We consider, after Example 8.2, the second approach to the prediction of the PVT behavior of fluids. Equations of State. 

Equations of State. Equations of state having adjustable parameters are often used to model the pressure—volume—temperature (PVT) behavior of pure fluids and mixtures (1,2). Equations that are cubic in specific volume, such as a van der Waals equation having two adjustable parameters, are the mathematically simplest forms capable of representing the two real volume roots associated with phase equiUbrium, or the three roots (vapor, Hquid, sohd) characteristic of the triple point. 

Thermodynamic properties, such as internal energy and enthalpy, from which one calculates the heat and work requirements of industrial processes, are not directly measurable. They can, however, be calculated from volumetric data. To provide part of the background for such calculations, we describe in this chapter the pressure-volume-temperature (PVT) behavior of pure fluids. Moreover, these PVT relations are important in themselves for such purposes as the metering of fluids and the sizing of vessels and pipelines. 

The coefficients of dT and dP in Eqs. (6.20) and (6.21) are evaluated from heat-capacity and J VT data. As an example of the application of these equations, we note that the PVT behavior of a fluid in the ideal-gas state is expressed by the equations ... 

The procedure is based on the group contribution equation of state by M. S. High and R. P. Danner, "A Group Contribution Equation of State for Polymer Solutions," Fluid Phase Equilibria, 53, 323 (1989) and M. S. High Prediction of Polymer-Solvent Equilibria with a Group Contribution Lattice-Fluid Equation of State, Ph.D. Thesis, The Pennsylvania State University, University Park, PA, 1990. Additional and modified group values are from V. S. Parekh Correlation and Prediction of the PVT Behavior of Pure Polymer Liquids, M.S. Thesis, The Pennsylvania State University, University Park, PA, 1991. 

The combination of Eqs. 6.2-38b and 6.6-4a is an example of a generalized equation of. state,. since we now have an equation of state that is presumed to be valid for a class of fluids with parameters ( and b) that have not been fitted to a whole collection of experimental data, but rather are obtained only from the fluid critical properties. The important content of these equations is that they permit the calculation of the PVT behavior of a fluid knowing only its critical properties, as was the case in corresponding-states theory. 

EOS theories for polymer liquids were grouped into three categories (1) lattice-fluid theory, (2) the hole model, and (3) the cell model. The Tait equation was first developed 121 years ago in order to fit compressibility data of freshwater and seawater. The Tait equation is a four-parameter representation of PVT behavior of polymers. Expressions for zero pressure isotherms and Tait parameters were provided, and values of Tait parameters for 16 commonly used polymers were tabulated. 

We proce, then, to discuss some common characteristics in the PVT behavior of all pure fluids, which are helpful in its quantitative description and estimation that follows. 

The PVT behavior of pure fluids is typically expressed in terms of the compressibility faaor z ... 

The EOS based on the lattice fluid model has also be used to describe thermodynamic properties such as pVT behaviors, vapor pressures and liquid volumes, VLE and LLE of pure normal fluids, polymers and ionic... 

As discussed in Sec. 3.5 and illustrated by Fig. 3.10, PVT equations of state that are cubic in molar volume are capable of describing the behavior of both liquid and vapor phases of pure fluids. 

As discussed in Chap. 3, equations of state provide concise descriptions of the PVT behavior for pure fluids. The only equation of state that we have used extensively is the two-term virial equation,... 

Tire isothenrrsfor the hquid phase on the left side of Fig. 3.2(b) are very steep and closely spaced. Thus both (BVfBT)p aird (dV/d P)r and hence both P and k are small. This characteristic behavior of hquids (outside the critical region) suggests an idealization, commonly employed in fluid mechanics and kirowir as the incompressible fluid, for which both and k are zero. No real fluid is traly incompressible, but the idealization is useful, because it often provides a sufficiently realistic model of liquid behavior for practical purposes. There is no PVT equation of state for an incompressible fluid, because V is independent of T and P. 

The experimentally observed similarity of the pressure-volume-temperature behavior of many fluids can be represented by the principle of corresponding states (PCS), according to which various substances behave in the same way when expressed on suitable reduced scales. Thus the pVT surfaces of different substances superimpose upon reduction with appropriate scale factors. The reducing scale factors commonly employed are the properties of the fluid at a singular point such as the critical point. On the reduced scale, one general pVT relationship is followed by a number of substances, i.e., p, is the same function of and v, where the subscript r denotes a reduced dimensionless quantity and the subscript c the quantity at the critical point ... 

We will now discuss the problem of determining effective or optimal diameters for use with the HSE theory for real fluids when both the form of the intermolecular potential and its parameters are unknown but accurate equations of state which represent the PVT behavior over an extensive range are available for the pure components. 

Evaluation of Fugacities Using an Equation of State. The fugaci-ties of the components in the fluid phases are related to the volumetric and phase behavior of the mixture while the fugacity of the solid component depends only on the PVT relationship of the pure component. Theoretically it is possible to evaluate the fugacities using experimental volumetric and/or phase equilibrium data in conjunction with Equations 3 and 6. However, these data are normally either unavailable or insufficient and an equation-of-state model has to be used to compute the fugacities. 

Since then. Dr. Woldfarth s main researeh has been related to polymer systems. Currently, his research topics are molecular thermodynamics, continuous thermodynamics, phase equilibria in polymer mixtures and solutions, polymers in supercritical fluids, PVT behavior and equations of state, and sorption properties of polymers, about which he has published approximately 100 original papers. He has written the following books Vapor-Liquid Equilibria of Binary Polymer Solutions, CRC Handbook of Thermodynamic Data of Copolymer Solutions, CRC Handbook of Thermodynamic Data of Aqueous Polymer Solutions, CRC Handbook of Thermodynamic Data of Polymer Solutions at Elevated Pressures, CRC Handbook of Enthalpy Data of Polymer-Solvent Systems, and CRC Handbook of Liquid-Liquid Equilibrium Data of Polymer Solutions. 

Adidharma and Radosz provides an engineering form for such a copolymer SAFT approach. SAFT has successfully applied to correlate thermodynamic properties and phase behavior of pure liquid polymers and polymer solutions, including gas solubility and supercritical solutions by Radosz and coworkers Sadowski et al. applied SAFT to calculate solvent activities of polycarbonate solutions in various solvents and found that it may be necessary to refit the pure-component characteristic data of the polymer to some VLE-data of one binary polymer solution to calculate correct solvent activities, because otherwise demixing was calculated. GroB and Sadowski developed a Perturbed-Chain SAFT equation of state to improve for the chain behavior within the reference term to get better calculation results for the PVT - and VLE-behavior of polymer systems. McHugh and coworkers applied SAFT extensively to calculate the phase behavior of polymers in supercritical fluids, a comprehensive summary is given in the review by Kirby and McHugh. They also state that characteristic SAFT parameters for polymers from PVT-data lead to... 

Computer simulations are sometimes used in the geochemical literature with the sole objective to predict thermodynamic PVT properties of molecular fluids at high temperatures and pressures (e.g., Belonoshko and Saxena 1991, 1992 Duan et al. 1992 Fraser and Refson 1992). However, the ability to improve our physical understanding of the complex chemical behavior of geochemical fluids and to unravel fundamental molecular-scale correlations between the structural, transport, spectroscopic, and thermodynamic properties of supercritical aqueous fluids, seems to be a much more important feature of these techniques. 

Sin(% one (rf our interests is to obtain PVT behavior, we perform isolmric-isothomal (constant NPT) simulations by setting the pressure and measuring the density and its fluctuations. As in the simulation of simple fluids, the dinmisktns the Emulation cell are changed periodically, r uiring a displacement of all the molecules in the system. For chain systems, volume fluctuations can be implemented in several ways. In the first polymer NPT MC simulation. 

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