Vapor-Liquid Equilibrium (VLE) Calculations

Classes of VLE Calculations

Depending on the information provided, one may perform one of several types of vapor-liquid equilibrium (VLE) calculations to model the vapor-liquid partitioning. These are: bubble-point pressure (BP), dew-point pressure (DP), bubble-point temperature (BT), dew-point temperature (DT), and isothermal flash (FL) and adiabatic flash (FA). The specifications of the information required and the information to be computed are tabulated below in Table 10.1. Also shown in the table are indications of the relative difficulty of each calculation. The best approach to understanding the calculations is to gain experience by plotting phase envelopes in various situations.

Table 10.1. Summary of the Types of Phase Equilibria Calculations (This Table is Independent of the VLE Model)

 The classes of VLE calculations presented here will be used through the remainder of the text, so the concepts are extremely important.

Principles of Calculations

Standard approaches to solving VLE problems utilize the ratio of vapor mole fraction to liquid mole fraction, known as the VLE K-ratio:

 VLE K-ratio.

The information available from a physical situation is combined with the K-ratio using one of the procedures shown in Table 10.1. The information available is in the second column of the table. The procedure involves combination of the known information together with a model-dependent K-ratio to calculate an objective function based on the estimated unknown compositions. For a bubble calculation, all the x_i are known, and we find the y_i by solving for the condition where  written in terms of x_i, namely . For dew calculations, all y_i are known, and we solve for the condition where  written in terms of y_i and K_i. For a flash calculation, we solve for the condition where  written in terms of the overall mole fraction z_i and K_i. The information in Table 10.1 is rigorous. The method used to calculate K_i is model-dependent and will be the focus of the next few chapters of the text. The K_i ratios generally vary with composition, pressure, and temperature, though we focus in this chapter on the use of Raoult’s law in situations where K_i ratios are dependent on only T and P.

Strategies for Solving VLE Problems

Note that there are only six different types of calculations for VLE summarized in Table 10.1. Usually the solution of the VLE problem will be relatively straightforward after deciding which row of the table to use. A crucial skill in solving VLE problems is interpreting the physical situation to decide which of the five methods should be used, and how to calculate the K-ratio.

1. Decide if the liquid, vapor, or overall composition is known from the problem statement.

2. Identify if the fluid is at a bubble or dew point. If the fluid is at a bubble point, the overall composition will be the same as the liquid composition. At the dew point, the overall composition will be the same as the vapor.

3. Identify if the P, T, or both P and T are fixed. Decide if the system is adiabatic.

4. The information collected in the first three steps can be used with the second column in Table 10.1 to identify the method.

5. Select a method to calculate the VLE K-ratio.

6. Decide if the method will be iterative, and if so, generate an initial guess for the solution. Approaches for handling iterative calculations are introduced in the following chapters and examples.

Iterative Calculations

When the K-ratios vary, VLE calculations require iterative solutions from an initial guess. For performing iterative calculations, useful aids include the Solver tool of Excel or the fzero() or fsolve() function of MATLAB. Many of the following examples summarize detailed calculations to illustrate fully the iterative procedure. In practice, the detailed calculations can be performed rapidly using a solver. Online supplements summarize the use of the iterative aids and the methods for successive substitution.