Patel Teja

The Patel Teja Cubic Equation of State With the introduction of a third constant, c, into the van der Waals equation's attraction term, the equation of state proposed in 1982 by Patel and Teja allows for adjustment of the critical compressibility, Zc, instead of predicting a fixed value. (2-8) This added flexibility improves saturation property predictions for polar fluids while maintaining a somewhat simple form that doesn't require a large amount of pure component or mixture experimental data for accuracy. For these reasons plus good predictions for enthalpy and entropy departures, the Patel Teja equation of state is used for all the property modeling in this study. As with the van der Waals equation of state, the parameters a and b in the Patel-Teja equation are evaluated using the conditions at the critical point given by equations 2-4 and 2-5. The condition for the third parameter, c, is:

(2-9) Instead of having a fixed value of the critical compressibility fixed by a and b, the value is the arbitrary, substance specific, empirical parameter, . is determined by minimizing the errors between experimental saturated liquid volume and those calculated by the equation of state. Patel and Teja calculated

for 38 substances covering most of the

substances studied here (see Table 2-1). They also correlated to the acentric factor, for non-polar substances yielding the following equation (Patel and Teja, 1982).

,

(2-10) Satisfying the conditions of equations 2-4, 2-5, and 2-9 yields the following equations for the parameters a, b, and c.

(2-11) (2-12) (2-13) where, (2-14) (2-15)

and

is the smallest positive root of the following cubic equation: (2-16)

Note that the value of

for a given substance need only be solved for once. The term

is given by: (2-17a) where,

(2-17b) In equation 2-17a, F is a substance specific, empirical parameter also determined by minimizing errors between experimental saturated vapor pressures and those calculated by the equation of state. Patel and Teja calculated F for the same 38 substances and also correlated it to the acentric factor for non-polar substances (Patel and Teja, 1982): (2-18) See Table 2-1 for the values of

and F used in this study.

Table 2-1: Emperical parameters for the Patel-Teja equation of state When equation 2-2 for the compressibility is substituted into the Patel-Teja cubic equation of state, the following expression for the compressibility, Z, is obtained (Smith, 1995). (2-19) where,

(2-20) (2-21) (2-22) Equation 2-19 will yield one or three positive real roots depending upon the number of phases in the system. If a vapor and liquid phase are present, the smallest root represents the liquid's compressibility while the largest root is that of the vapor's. Appendix A provides an analytical solution for a general cubic equation. Thus, equations 2-8 thru 2-22 along with equation 2-2, provide P-v-T relations for all of the substances found in this study. However, this study also requires calculation of the equilibrium state of a vapor and a liquid (vapor-liquid equilibrium or VLE) as well as the properties of enthalpy and entropy.