Select thermodynamic models for process simulation
A Practical Guide to a Three Steps Methodology

Example 4-2: Distribution coefficients in an ideal mixture (propane + n-pentane)

Describe the impact of pressure on the phase compositions of an ideal mixture. As an example, we use the propane + pentane mixture at 343.15 K. Following data are available (Table 1). The A, B and C parameters are those used for the Antoine equation for calculating the vapour pressure expressed in non-dimensional units ( image and image):

image
Table 1: characteristic properties of propane (C3) and n-pentane (C5)
C3 C5
Tc(K) 369.8 469.6
Pc(MPa) 4.24866 3.37662
Zc 0.281 0.262
Acentric factor 0.152 0.251
A 5.7544 6.0008
B -5.4826 -5.5603
C -0.0472 -0.0734

Analysis:

Phase equilibrium for a binary mixture is calculated from the distribution coefficients, as follows:

(1) image

yielding:

(2) image

and therefore

(3) image

If equilibrium is considered at high pressure, fugacity coefficients and Poynting corrections must be taken into account:

(4) image

Assuming an ideal mixture (i.e. the equilibrium coefficient is independent of composition), the activity coefficient equals one:

(5) image

In expression (4), the fugacity coefficients are approximated with that of the virial equation:

(6) image

and

(7) image

The Poynting correction is written as:

(8) image

Introducing (8), (7) and (6) into (4), one finds:

(9) image

which may be written as:

(10) image

or:

(11) image

where:

(12) image

and

(13) image

The distribution coefficients thus depend only on pure component properties (liquid molar volume image, vapour pressure image and second virial coefficient image). Equation (11) shows that the product of equilibrium constant and pressure should be considered. Raoult's law suggests that this product is only a function of temperature. In practice, a linear relationship with pressure is required above 0.5 MPa.

Solution:

See complete results in file (xls):

Some help on nomenclature and tips to use this file can be found here.

In order to use the above equations, values for vapour pressure, liquid molar volume and second virial coefficient are needed. Unless specific parameters are provided, we use the corresponding states methods:

for vapour pressure, the Antoine parameters are provided

for liquid molar volume, the Rackett equation is used:

(14) image

for second virial coefficient, the method of Tsonopoulos is used:

(15) image with:

with:

(16) image with:

and:

(17) image with:

Phase equilibrium can now be computed from (2) and (3) using either the basic Raoult equation or the more complete equation (11). The results are as follows:

P(MPa)= 0.5
Ki Raoult 5.3062 0.5780
x 0.0893 0.9107
y 0.4736 0.5264
Ki (11) 3.9161 0.6209
x 0.1151 0.8849
y 0.4506 0.5494

We can notice that the complete equation yields smaller distribution coefficients than the Raoult approximation if the value is larger than one, and larger distribution coefficients if it is smaller than one. This results in smaller differences in volatilities, and as a result phase compositions that are less different.

The distribution coefficients as a function of pressure for propane and n-pentane at 343.15 K are shown in figure 1. We can see that their respective slopes are such that the equilibrium coefficients converge at higher pressures. The practical meaning of this observation is that the volatility of the two components becomes closer when pressure is increased, and therefore more equilibrium stages will be needed to reach the same separation level.

image Figure 1: Change of propane and n-pentane equilibrium coefficients at 343.15K with pressure