Example 02.09 Density of Propylene from the Peng-Robinson Equation of State

problem:

general

constants

and

definitions:

input data

From Appendix A:

Solution

The Peng-Robinson EOS can be written as:

with

Now all parameters are available and the function can be defined:

At 300K, the vapor pressure of propylene is higher than the system pressure of 10 bar. The ideal gas volume can be used as starting value for the root function:

The volume at which the Peng-Robinson EOS calculates a pressure of 10 bar can be found via:

Application of Cardano's Formula

Similar to the Vieta's formula for cubic equations, Cardano'f formula allows to solve cubic equations analytically (see Appendix C, A10).
In the first step the equation should be braucht into the normal form:

In case of the Peng/Robinson EOS, the coefficients can be calculated as

The terms in Cardano's formula can be calculated via:

This yields a negative discriminant DCar (3 real solutions):

Solution for the roots yields:

Abbreviations:

Application of the Polyroots-Function

The polyroots-Function allows to calculate all roots of a polynomial via the LaGuerre or the companion matrix method:

All three methods naturally lead to the same results. The highest calculated volume is the vapor volume. As propylene is below its vapor prealculated asssure, the liquid solution is not stable. The medium value solution is always unstable. The molar density can be calculated as

For the mass density we find:

The result from a high precision equation of state is 20.06 kg/m3. Similarly good results can only be expected for vapor phase densities.