P-V-T properties

In Figure 6.3, the area obtained from the m-BWR equation gives a value for 4> of 0.4386 for n-butane at T = 500 K and p = 50 MPa. Since the m-BWR equation represents the (p, V, T) properties of the gas with high accuracy, these d> values are reliable and can be used as a reference for comparing results obtained in other ways. 

We are interested in comparing the effectiveness of the various equations of state in predicting the (p. V. T) properties. We will limit our comparisons to Tr > 1 since for Tr < 1 condensations to the liquid phase occur. Prediction of (vapor + liquid) equilibrium would be of interest, but these predictions present serious problems, since in some instances the equations of state do not converge for Tr< 1. 

Take, for example (12), the problem of solving for the P-V-T properties of a real gas obeying the van der Waals equation of state. 

Poling et al. [6] also describe methods for estimation of additional properties, such as critical properties, P-V-T properties, and phase equilibria. 

Fugacity is a thermodynamic property related to the deviation of the p—V—T properties of the gas from those of an ideal gas. At very low pressures, the fugacity of a real gas tends to its partial pressure... 

Probably the real reason EOS have been painstakingly developed is to calculate p, V, T properties of real gases for practical, that is, industrial... 

Related Calculations. This illustration outlines various simple techniques for estimating P-V-T properties of gaseous mixtures. Obtain the compressibility factor from the generalized corresponding-state correlation, as shown in step 2. 

As ammonia is transported at a temperature of at least 2 °C (pressure between 22 and 100 bar) it has to be warmed up at the supplier terminal and cooled down again to -33 °C at the receiver terminal. Main branches have diameters of 200 and 250 mm (8-10 in). Exact knowledge of the p- V- T properties is important [1320]. Automatic lock valves are installed at intervals of 10 miles, so that the volume which can be released between two valves is limited to 400 t. 

Wang D.-P. and MiUero E. J. (1973) Precise representation of the P-V-T properties of water and seawater determined from sound speeds. J. Geophys. Res. 78, 7122—7128. 

Equations of state relate the p V-T properties of a pure substance (or mixtures) by theoretical or empirical relations. The simplest example of an equation of state is the ideal gas law itself. Table 3.2 lists a few of the commonly used equations of state (from among the hundreds that have been proposed) that involve two or more coefficients. In Secs. 1.7 and 4.3 and the supplementary references at the end of this chapter, you will find numerous sources of information about equations of state and... 

In an evaluation of 11 cubic equations of state, Adachi, Sugie, and Lu reported the overall ratings listed in Table 3.3 based on 21 substances for which detailed experimental data were available in the literature. You can see that some of the two parameter equations of state listed in Table 3.2 do as good a job in predicting p-V-T properties, as do some of the equations of state with more coefficients. 

Van der Waa/s equation. We will use the van der Waals equation as an example of an equation of state, not because it is the best in predicting p-V-T properties, but because it is not too bad, is simple, and can be used to explain the computation problems involving the use of equations of state. Figure 3.3 shows how well the van der Waals equation predicts the properties of CO2 in comparison with experimental data. (Note how far the ideal gas law departs from the experimental data even at 52°C.) The van der Waals equation illustrates some of the typical theoretical development that has taken place as well as some of the computational problems involved in using equations of state. Keep in mind that some of the equations listed in Table 3.2 are completely empirical, whereas others, such as the van der Waals equation, are semiempirical that is, although they were developed from theory, the constants in the equation or portions of the equation are determined by experiment for each gas. 

Let us look at some of the computational problems that may arise when using equations of state to compute p-V T properties by using van der Waals equation as an illustration. Van der Waals equation can easily be explicitly solved for p as follows ... 

Equations of state for gases are often used to predict p-V-T properties of gas. Cite two reasons. 

The idea of using the reduced variables to correlate the p-V-T properties of gases, as suggested by van der Waals, is that all substances behave alike in their reduced (i.e., their corrected) states. In particular, any substance would have the same reduced volume at the same reduced temperature and pressure. If a relationship does exist involving the reduced variables that can be used to predict pr, Tr, and K, what would the equation be Certainly, the simplest form of such an equation would be to imitate the ideal gas law, or... 

Calculation of p-V T Properties via Generalized Group Contribution Methods... 

Kay s method is known as a two-parameter rule since only pc and Tc for each component are involved in the calculation of z. If a third parameter such as the Pitzer acentric fector, or Vc is included in the determination of the mean compressibility factor, then we would have a three-parameter rule. All the pseudocritical methods do not provide equal accuracy in predicting p V-T properties, but most suffice for engineering work. Stewart et al. reviewed 21 different methods of determining the pseudoreduced parameters by three-parameter rules (see Table 3.6). Although Kay s method was not the most accurate, it was easy to use and not consid-... 

As to liquid mixtures, it is even more difficult to predict the p-V-T properties of liquid mixtures than of real gas mixtures. Probably more experimental data (especially at low temperatures) are available than for gases, but less is lcnown bburth estimation of the p-V T properties of liquid mixtures. For compounds with like molecular structures, such as hydrocarbons of similar molecular weight, called ideal liquids, the density of a liquid mixture can be approximated by assuming that the specific volumes are additive ... 

Now in Chapter 15 we also introduced virial equations of state, which were originally developed to describe he P — V — T properties of non-ideal gases around the turn of the century. An example is the equation... 

There is in this Handbook a whole Chapter 7 by Robert Orwoll on equation-of-state P-V-T properties of polymers. Briefly, confining our attention to what is pertinent for PLCs, there are basically two experimental procedures for a determination. One can measure l in a thermomechanical analysis (TMA) apparatus and then calculate a fi om Eqs. (41.4) or (41.5) [57,58]. Or, one can use an apparatus which produces full P-V-T data, that is specific volume v as a function of temperature T, v(T), plus a(T) plus also isothermal compressibility fCj-(7), where... 

LoSurdo, A. F.J, MUlero E.M. Alzola, "T-he (p, V, T) properties of concentrated aqueous electrolytes. I. Densities and apparent molar volumes of NaCl, Na2SOi,, MgCl2 and MgSOi, solutions from. 1 mol/kg to saturation and from 273.15 to 323.15K", J. Chem, Thermo., vl4, pp649-662 (1982)... 

The P-V-T properties for these substances were plotted and values were selected by interpolation. Hydrogen data came from Michels et al. [19] and from Woolley et al. [20], Deuterium data came from Michels [19] and Schkefer [21]. Methane data came from the authors experimental work [17, 22] and from Schamp et al. [23],... 

Wilson et have discussed the heat capacity, enthalpy and p, V, T properties of helium, nitrogen, ethane, neon, methane, and carbon monoxide. Zwolinski and his colleagues have prepared a compilation of the critical properties of organic substances which also includes values for oxygen, water, carbon monoxide, and carbon dioxide. 

Methods that Determine p, V, T Relations Indirectly.— There are, besides the direct methods of measuring p, V, T properties described above, several indirect methods available. Measurement of any property which is dependent on the non-ideality of a gas can in principle provide information about the equation of state. 

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