Do We Really Know Ki?

The Need For More Advanced Tools

In the previous development, we made one crucial assumption. We assumed that, somehow, we knew all the equilibrium ratios. The fact is, however, that we usually don’t. If we do not know all equilibrium ratios, then all of the previous discussion is meaningless. So far, the only conclusion we can draw is that if we happen to know K_i’s, the VLE problem is solvable.

The K_i value of each component in a real hydrocarbon mixture is a function of the pressure, temperature, and also of the composition of each of the phases. Since the compositions of the phases are not known beforehand, equilibrium constants are not known, either. If they were known, the VLE calculation would be performed in a straightforward manner. This is because once the equilibrium constants of each component of the mixture are known for the given pressure and temperature of the system, both gas and liquid molar fractions, ${\alpha }_g$ and al can be calculated by solving the Rachford-Rice Objective Function.

Nevertheless, the good news is that sometimes K_i’s are fairly independent of the phase’s composition. This is true at pressure and temperature conditions away from the critical point of the mixture. Therefore, numerous correlations have been developed throughout the years to estimate the values of K_i for each hydrocarbon component as a function of the pressure and temperature of the system.

To illustrate how K_i may be calculated as a function of pressure and temperature, let us take the case of an ideal mixture. For a mixture to behave ideally, it must be far removed from critical conditions. The fact of the matter is, in an ideal mixture, the partial pressure of a component in the vapor phase (p_i) is proportional to the vapor pressure (P_sat) of that component when in its pure form, at the given temperature. The constant of proportionality is the molar fraction of that component in the liquid (x_i). Then, we have:

$$p_i=x_i{P}^{sat}$$ (13.10)

Equation (13.10) is known as Raoult’s law. Additionally, if the vapor phase behaves ideally, Dalton’s law of partial pressures applies. Dalton’s law of partial pressures says that the total pressure in a vapor mixture is equal to the sum of the individual contributions (partial pressures) of each component. The partial pressure of each component is a function of the composition of that component in the vapor phase:

$$p_i=y_{i}P$$ (13.11)

Equalizing equation (13.10) and (13.11),

$$y_{i}P=x_i{P}^{sat}$$ (13.12)

We rearrange equation (13.12) to show:

$$\frac{y_i}{x_i}=\frac{P^{sat}}{P}$$ (13.13)

If we recall the definition of equilibrium ratios, $$K_i=y_i/x_i$$ , we readily see:

$$K_i=\frac{P^{sat}}{P}$$ (13.14)

Since the vapor pressure of a pure substance (P_sat) is a function of temperature, we have just shown with equation (13.14) that the equilibrium ratios K_i are functions of pressure and temperature — and not of composition — when we are dealing with ideal substances. Vapor pressure can be calculated by a correlation, such as that of Lee and Kesler.

The estimation of equilibrium ratios (K_i) has been a very intensely researched subject in vapor-liquid equilibria. A number of methods have been proposed in the literature. In the early years, the most common way of estimating equilibrium ratios was with the aid of charts and graphs that provided K_i values as a function of pressure and temperature for various components. Charts provided better estimations of K_i’s than what came from the direct application of Raoult-Dalton’s derivation (equation 13.14). However, the compositional dependency could not be fully captured by the use of charts. Charts remained popular until the advent of computers, when the application of more rigorous thermodynamic models became possible.

Most of the K_i-charts available have been represented by empirical mathematical correlations to make them amenable for computer calculations. You may find a large number of correlations in the literature that would allow you to estimate K_i’s for a range of conditions. A very popular empirical correlation that is very often used in the petroleum and natural gas industry is Wilson’s empirical correlation. This correlation gives the value of K_i as a function of reduced conditions (P_ri, T_ri: reduced pressure and temperature respectively) and Pitzer’s acentric factor and is written as:

$$K_i=\frac{1}{P_{ri}}EXP\left[5.37\left(1+{\omega }_i\right)\left(1-\frac{1}{T_{ri}}\right)\right]$$ (13.15)

Wilson’s correlation is based on Raoult-Dalton’s derivation (equation 13.14). Therefore, it does not provide any compositional dependency for K_i’s and, as such, it is only applicable at low pressures (away from the critical conditions). We have included this correlation not because of its accuracy, but because it will become the initial guess that is needed to start the K_i-prediction procedure, which we will develop later.

In the following modules, we will develop a rigorous thermodynamic model for predicting equilibrium ratios. As you may anticipate, we will be taking advantage of what we have learned about equations of state in order to build up a phase behavior predictor model. This approach is known as the equation of state approach, or the “fugacity” approach. Even though the “fugacity” approach is the one that we will be covering in detail, you may at some point encounter the fact that equilibrium ratios can also be estimated by using the solution theory or “activity” approach. We will not be discussing the latter, as the former represents the most popular K_i-prediction method in the petroleum and natural gas business.

In order to guarantee a complete understanding of the thermodynamic model which we are about to apply, we need to review the basic concepts of classical thermodynamics. This is the goal for the next couple of modules (Modules 14 through 16). After all of the thermodynamic tools have been given to the reader, we will resume our discussion of Vapor-Liquid Equilibria (Module 17), along with discussing how equations of state come into the picture. Our final goal will be to tie the equilibrium ratio, K_i, to the thermodynamic concepts of chemical potential, fugacity, and equilibrium.