Example 10.03 Vapor Pressure of a CO2 - PS Mixture Using PC-SAFT

problem:

A vessel contains a mixture consisting of polystyrene (2) (molar mass: 158 kg/mol) and CO2 (1). The liquid mole fraction is x1 = 0.02 and the liquid density is 1007 kg/m3. Which pressure will be built up in the vessel at 393.15 K?

general

constants

and

definitions:

input data

References: http://www.th.bci.tu-dortmund.de/de/forschung/pc-saft/theory/equations

J. Gross, G. Sadowski, Ind. Eng. Chem. Res. 2001, 40, 1244 - 1260


All references to equations are to the textbook from which this example was taken.

Solution

Required Parameters for Using PC-SAFT

Pure component parameters

number of segments per chain

segment diameter

(for convenience, the energy is given in Kelvin (divided by the Boltzmann-constant k)

depth of pair potential

Like other more conventional equations of state, the PC-SAFT equation in itself is only capable of describing the behaviour of pure fluids. Therefore in order to extend the equation to mixtures, appropriate mixing rules need to be employed in place of pure component parameters, and combining rules are needed to account for the cross-interactions of these same parameters. In this regards, the van der Waals one-fluid mixing rules an the Berthelot-Lorentz combining rules are often sufficiently accurate:

The binary interaction parameter, kij, is introduced to correct the segment-segment interactions of unlike chains.

Further auxiliary quantities required in the calculation:

mean number of segments

per chain in the mixture:

temperature-dependent hard
segment diameter of the components

mean molecular weight of

the mixture:

molar density

Universal model constants (J. Gross, G. Sadowski, 2001):

The PC-SAFT Equation


The PC-SAFT equations are based on the reduced Helmholtz energy contributions relative to the ideal gas. Contributions for the residual Helmholtz energy from chain of hard-spheres and dispersion are added to the ideal gas value. The complete equation of state is therefore an additive combination of

- the ideal gas contribution (id)

- a hard-chain contribution (hc) which includes ,
- a dispersive contribution (disp),

- an association contribution (assoc)

- an polar contribution (polar)

In order to calculate the pressure from the reduced Helmholtz energy, the following thermodynamic relation is employed:

, whereby the differentiation is performed at constant mole fraction and temperature.

Ideal Contribution

Hard-Chain Reference Contribution

The Helmholtz energy of the hard-sphere fluid (first term) is given on a per segment basis by

with

From Eq.10.39, the following expression for the pressure can be derived:

Using this value for the hard-sphere contribution, the following equation allows to calculate the hard-chain contribution to the pressure:

, whereby the differentiation is performed at constant mole fraction and temperature.

Calculation of the radial distribution function can be performed using Eq.10.37:

Using the function L, which takes the place of

the pressure contribution of the hard-chain term can now be calculated:

Dispersive Contribution

The dispersive contribution is given by:

with:

The variables W1 and W2 take the place of

and

Using the variable C3 and C4 to take the place of

and

Further simplifications are made using the following definitions:

Using the variable X to take the place of

The dispersion contribution to the pressure (see Eq. 10.40) can then be written as:

Combining the individual contributions leads to the total pressure: