Prediction of Saturated Liquid Volumes

We consider, next, the third requirement to be met by these EoS, the prediction of saturation liquid volumes, and then return to discuss vapor pressure predictions. 

Q is here the saturated liquid volume, the subscript cal stands for calculated and for experimental values, and the summation is over the n data points, here thirty, equidistanced on a basis, from the triple to the critical point. The results are poorer with the SRK EoS and still poorer with the vdW EoS. 

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The equations given predict vapor behavior to high degrees of accuracy but tend to give poor results near and within the liquid region. The compressibility factor can be used to accurately determine gas volumes when used in conjunction with a vitial expansion or an equation such as equation 53 (77). However, the prediction of saturated liquid volume and density requires another technique. A correlation was found in 1958 between the critical compressibility factor and reduced density, based on inert gases. From this correlation an equation for normal and polar substances was developed (78) ... 

Figure 10.1 Prediction of saturated liquid volumes for the C - C20 n-alkanes. (Experimratal data from the correlations of such data by Daubert and Danner, 1985,1986.)...

Both the t-PR and t-vdW EoS give excellent prediction of saturation liquid volumes. 

Table 14.3 Performance of the vdW-711, PR, and SRK EoS in the Correlation of VLE Data and the Prediction of Saturated Liquid Volumes (Czerwienski al)...

As expected from our discussion on the prediction of saturated liquid volumes of pure compounds in Section 10.6, the results for mixtures should vary among different EoS. The SRK EoS, thus, gives very poor results while the translated ones, such as the vdW-711, provide reasonable accuracy (Table 14.3). 

Bubble pressure found experimentally for a mixture of 50.2% COj and 49.8 % n-butane at 344.2 K is 64.8 bar. Specific volume of saturated liquid is 99.13 cm /mol. The interaction coefficients are Oj (l)/n-butane (2) A,2 = 0.143 for SRK and = 0.133 for PR EOS. Compare bubble point pressure values estimated by the two equations of state, as well as the prediction of the liquid volume, and compare it with the value calculated by Racked equation. The critical data are ... 

The SRK, PR and vdW-711 EoS, provide also good predictions of saturation vapor volumes for nonpolar compounds as Example 8.10 demonstrates. For saturated liquid volumes, however, only the vdW-711 EoS provides consistently good results because it includes the translation factor t. This is also discussed in Chapter 10. 

We proceed then with the third requirement and demonstrate how this vdW ] S is modified further, to provide successful predictions of saturated liquid molar volumes. This will lead to the vdW-711 EoS, which is typical of cubic EoS as far as the first two requirements are concerned, and somewhat novel with respect to the third one. 

To extend the good predictions of saturated liquid molar volumes of the vdW-711 EoS to high molecular weight hydrocarbons, Magoulas proposes a similar expression as that of Table 8.3 for the translation t ... 

Peneloux et al. [35] have introduced a clever method of improving the saturated liquid molar volume predictions of a cubic equation of state, by translating the calculated volumes without efffecting the prediction of phase equilibrium. The volume-translation parameter is chosen to give the correct saturated liquid volume at some temperature, usually at a reduced temperature Tr = T/Tc = 0.7, which is near the normal boiling point. It is possible to improve the liquid density predictions further by making the translation parameter temperature dependent. 

The modern cubic equations of state provide reliable predictions for pure-component thermodynamic properties at conditions where the substance is a gas, liquid or supercritical. Walas and Valderrama provided a thorough evaluation and recommendations on the use of cubic equation of state for primary and derivative properties. Vapour pressures for non-polar and slightly polar fluids can be calculated precisely from any of the modem cubic equations of state presented above (Soave-Redlich-Kwong, Peng-Robinson or Patel-Teja). The use of a complex funetion for a (such as those proposed by Twu and co-workers ) results in a significant improvement in uncertainty of the predicted values. For associating fluids (such as water and alcohols), a higher-order equation of state with explicit account for association, such as either the Elliott-Suresh-Donohue or CPA equations of state, are preferred. For saturated liquid volumes, a three-parameter cubic equation of state (such as Patel-Teja) should be used, whereas for saturated vapour volumes any modern cubic equation of state can be used. 

Figure 1 illustrates the performance of the two equations for predicting the molal volume of saturated liquids and vapors for pure n-pentane. At reduced temperatures above about 0.8, the average error in liquid density has been reduced by a factor of about 4 by using the new equation. At lower reduced temperatures, the predictions by the new equation are better by a factor of about 2. Both equations give acceptable predictions of the vapor density. 

The ability of the new equation to predict the specific volume of saturated liquids in multicomponent systems is clearly illustrated in Figures 2 and 3. Figure 2 shows the calculated liquid volume percent in the retrograde region for a 6 component paraffin hydrocarbon mixture containing components from methane through n-decane. Figure 3 shows the same kind of information for a 9 component system... 

The most successful correlation is that of Hankinsonand Thomson (1979) which provides excellent predictions of liquid densities. The authors report an average absolute error of 0.375% for 4508 data points on 190 different nonpolar and polar compounds and in the range 0.25 < 7).< 0.98. The vdW-711 EoS, original and modified for polar compounds, provides also very good results for saturated liquid volumes. 

This form of the vdW EoS (Eq. 10.4.1 combined with Eq. 10.4.2), the SRK, and PR EoS, while providing reliable predictions of vapor pressures for nonpolar compounds, lead to poor results for saturated liquid volumes. This is demonstrated with the results for the Cj to C20 n-alkanes, obtained with the PR EoS and shown in Figure 10.1, where the average absolute error AAE) is defined by ... 

The quality of the obtained saturated liquid volume predictions is demonstrated in Figure 10.1. Notice that very good results are obtained except for high o values n-alkanes). 

To improve, in addition, the saturated liquid volume predictions of this PR EoS, he presents a translated form ... 

Prediction results for saturated liquid volumes with the t-PR and t-vdW EoS are shown in Figure 10.1. Notice the significant improvement of the t-PR over the PR EoS, while for the t-vdW EoS the improvement over the vdW-711 EoS occurs only at high

Examine the effect of the uncertainty in the and cj values on the predicted vapor pressures and saturated liquid volumes for a compound of your choice using the SRK, t-PR, and t-vdW EoS. 

The density of a material is a function of temperature and pressure but its value at some standard condition (for example, 293.15 K or 298.15 K at either atmospheric pressure or at the vapor pressure of the compound) often is used to characterize a compound and to ascertain its purity. Accurate density measurements as a function of temperature are important for custody transfer of materials when the volume of the material transferred at a specific temperature is known but contracts specify the mass of material transferred. Engineering applications utilize the density of a substance widely, frequently for the efficient design and safe operation of chemical plants and equipment. The density and the vapor pressure are the most often-quoted properties of a substance, and the properties most often required for prediction of other properties of the substance. In this volume, we do not report the density of gases, but rather the densities of solids as a function of temperature at atmospheric pressure and the densities of liquids either at atmospheric pressure or along the saturation line up to the critical temperature. 

Despite many years of interest in the phase distribution of POPs, few predictive models are available. A Langmuir-type relationship, which Junge (1977) first proposed and Pankow (1987) later reviewed and critically evaluated, is the most popular model for estimating adsorption onto aerosols. The Junge-Pankow equation relates the fraction of particulate POPs (c >) to the saturation liquid-phase vapor pressure of the compound ( P Pa) and the surface area of particles per unit volume of air (8, cm2 aerosol / cm3 air). 

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