PVT behavior

The pressure—volume—temperature (PVT) behavior of many natural gas mixtures can be represented over wide ranges of temperatures and pressures by the relationship... 

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. 

Whereas this two-parameter equation states the same conclusion as the van der Waals equation, this derivation extends the theory beyond just PVT behavior. Because the partition function, can also be used to derive aH the thermodynamic functions, the functional form, E, can be changed to describe this data as weH. Corresponding states equations are typicaHy written with respect to temperature and pressure because of the ambiguities of measuring volume at the critical point. 

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Although the van der Waals equation is not the best of the semi-empirical equations for predicting quantitatively the PVT behavior of real gases, it does provide excellent qualitative predictions. We have pointed out that the temperature coefficient of the fugacity function is related to the Joule-Thomson coefficient p,j x.- Let us now use the van der Waals equation to calculate p,j.T. from a fugacity equation. We will restrict our discussion to relatively low pressures. 

Pairwise relationships such as (2.3a-d) can be represented by simple 2-dimensional (2D) graphs, but the full PVT behavior of the equation of state requires a 3-dimensional (3D) representation (for fixed n = 1). Figure 2.1 illustrates some simple graphical representations of the ideal gas equation of state (2.2). Figure 2.1a illustrates Boyle s law (2.3a) in... 

In comparison to a conventional polymer and l.c. mentioned above, we will now discuss the PVT behavior of a l.c. side chain polymer, which has linked mesogenic moieties as side chains, and is very similar to the previous monomer. The experimental results are shown in Fig. 5. It is obvious, that the phase behavior of the l.c. polymer differs from that of a 1-l.c. and amorphous polymer. At high temperature we observe a transformation from the isotropic polymer melt into the l.c. phase, indicated by the jump in the V(T) curve. At low temperatures no crystallisation is observed but the bend in the curves signifies a glass transition. Obviously the phase behaviour is determined by the combination of l.c. and polymer properties. 

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... 

A widely accepted form of modeling the density or specific volume is the Tait equation. It is often used to represent the pvT-behavior of polymers and it is represented as,... 

In a numerical solution, we can include temperature dependent density and thermal conductivity. The temperature dependent density can be modeled interpolating throughout a pvT diagram. The temperature dependence of the thermal conductivity is not always available, as is the case for many properties used in modeling. Chapter 2 presents the Tait equation, which can be used to model the pvT behavior of a polymer. 

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. 

Figure 3.2 indicates the complexity of the PVT behavior of a pure substance and suggests the difficulty of its description by an equation. However, for the gas region alone relatively simple equations often suffice. For an isotherm such as Tj we note from Fig. 3.2 that as P increases V decreases. Thus the PV product for a gas or vapor should be much more nearly constant than either of its members. This suggests the representation of PV along an isotherm by a power series expansion in P ... 

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. 

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 ... 

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,... 

A, and the result is good. The only serious deviation from linearity is with H2, the PVT behavior of which at low temperatures is subject to large quantum mechanical effects. Figure 6 also shows a fair correlation, B, between the total heat and a term a/d2, where d is as defined earlier, this term corresponding to the energy involved in field-induced polarization. 

Fig. 4. Comparison of calculated [27] and measured [29] PVT behavior of PVAc. It clearly reveals the effect of pressure on Tg...

Often empirical models or correlative equations of state are used to describe the PVT behavior of polymers (Zoller, 1989). Such correlations are useful in the interpolation and extrapolation of data to the conditions of interest. When an equation of state based on statistical mechanical theory is used to correlate the data, the resulting equation parameters can also be used in mixing rules to determine the properties of polymer solutions. 

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. 

Parekh, V. S., Correlation and Prediction of the PVT Behavior of Pure Polymer Liquids, M.S. Thesis, The Pennsylvania State University, University Park, PA (1991). 

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