The Patel-Teja Equation of State

The parameters F and c can be optimized from a set of data or they can be obtained from the following generalized equations  

The following tables summarize the parameters used for the Peng-Robinson calculations presented in this paper. The pure component properties could also be used for Soave-Redlich-Kwong calculations. Binary interaction parameters are given for both equations. 

From Knapp et al. (1982) except for H2S + CH4 which are from this work. 

The lower reduced temperature corresponds to 0°C, the lowest temperature of interest in this study. For the SRK, the following Peneloux-type volume shift parameters were used  

And for the PR equation, the following parameters were used here  

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The Patel-Teja equation of state Is able to correlate the data for the binary systems reasonably well provided a binary Interaction coefficient (kj.) is Included In the calculations. It Is Interesting to note that the binary Interaction coefficients obtained from correlation of data for the odd members of the series are an order of magnitude smaller than those obtained for the even members of the series and that they show regular behavior with carbon number. These differences are due to differences In... 

Willman and Teja" have described the composition of fossil-fuel mixtures (including gas condensates, absorber oils, crude oils, coal hquids) with a bivariant log-normal distribution that depends on the boiling-point temperature and the specific gravity. This scheme was used to determine dew-points with the Patel-Teja equation of state. 

Solubilities for the six systems were correlated by Smith and Teja [15] using the equation of state of Patel and Teja [17], The critical temperatures and pressures of the solutes, which are required by the Patel-Teja equation of state, have not been measured because the substances decompose below their critical points. Instead, these values have been estimated by averaging the results of two group contribution methods [18-20]. In both methods, critical temperature is a function of normal... 

Enthalpy and entropy of gases at low pressure can be calculated accurately from the Soave-Redlich-Kwong, Peng-Robinson or Patel-Teja equations of state at moderate and high pressure the Peng-Robinson or Patel-Teja equations of state are recommended. On the other hand, for liquid phase enthalpy and entropy none of the cubic equations of state can provide precise results. Empirical correlations, such as that proposed by Lee-Kesler, are much more precise. 

Basically, there are three equations of state used in the study 1. Soave-Redlich-Kwong (SRK), 2. Peng-Robinson (PR), and 3. Patel-Teja (PT). Also two forms of volume-shifting were examined 1. Peneloux et al. (P) and 2. Mathias et al. (M) (so PR-P means the Peng-Robinson equation of state with Peneloux et al. volume-shifting). 

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. 

The description of hydrocarbon mixture VLB at low and high pressure is of major importance to the oil industry. For such mixtures, any of the modern cubic equations of state (such as Redlich-Kwong, Peng-Robinson or Patel-Teja) provide precise predictions when used with a temperature-independent binary interaction parameter of relatively small value (in most cases in between — 0.1 and 0.1). For the case of non-polar hydrocarbon mixtures of similar size, even kij = Q.Q results in excellent prediction of VLB. 

The mixture constants for the Patel and Teja equation of state were calculated using several mixing rules. Results of the calculations using classical one- and two-parameter van der Waals mixing rules are summarized in Table 1.9. The one-parameter mixing rule (denoted by vdWl in Table 1.9) contains one adjustable... 

Experimental data were correlated using the Patel-Teja (9) equation of state. For a pure solid phase In equilibrium with a supercritical gas phase, we may write (10). 

For the Peng-Robinson (18) equation of state, b = c and d = 0. For the Redlich-Kwong-Soave equation of state (see Ref. 19), c = d = 0. For the three-parameter Trebble-Bishnoi-Salim (20) equation of state, d = Vc/3. For the Patel-Teja (21) equation, d = 0. In Eq. (8) the temperature-dependent parameter a T) is expressed as follows ... 

It is fascinating to follow the iterations that began in the seventeenth century and are still being refined. Over the years, numerous authors modified the VDW equation of state e.g., Redlich and Kwong [1949], Soave [1972], Peng and Robinson [1976], Patel and Teja [1982], or Schwartzentruber et al. [1989]. Their relations, written as P = P V, T) are known as cubic equations of state, which are algebraically simple and relatively accurate [Wei, 1998 Valderrama and Alfaro, 2000] ... 

All of the cubic equations of state presented so far are generally termed two-parameter equations of state. In this respect, each one of them predicts a constant compressibility factor at the critical point Zc=PcVd T, irrespective of the nature of the compound for van der Waals Zc = 0.375, for Redlich-Kwong and Soave-Redlich-Kwong Zc = 0.333 and for Peng-Robinson Zc = 0.307. The actual values may vary significantly, especially for polar and associating fluids for methane Zc = 0.286, for propane Zc = 0.276, for pentane and benzene Zc = 0.268 and for water Zc = 0.229. To correct for this deficiency, a number of authors have proposed three or four adjustable parameters to the cubic equation of state. The most popular three-parameter cubic equation of state was proposed by Patel and Teja is given by ... 

Since the introduction of the SRK and PR equations, many other cubic equations of state have appeared with similar modifications to the a and b parameters. Of these, the Patel-Teja (PT) equation (Patel Teja 1982) is one which is increasingly being used. The form of the PT equation is similar to the PR but introduces an extra parameter, c, and is written as... 

Repeat Example 1.2 by using the Benedict-Webb-Rubin (BWR) and the Patel-Teja (PT) equations of state. Compare the results with those obtained in Example 1,2,... 

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