*The whole is simpler than the sum of its parts.*

*J.W. Gibbs*

Suppose it was required to estimate the vapor-liquid *K*-ratio of methane in a mixture at room temperature. For an initial guess, we might assume it follows ideal-solution behavior. It is a relatively simple molecule (e.g., no polar moments, no hydrogen bonding). But we cannot use Raoult’s law because the required temperature is well above the critical temperature. We could use Henry’s law, or the SCVP+ model (Section. 11.12), but the assumption of low concentrations may be inappropriate at very high pressures. The equation of state method discussed here is an attractive alternative.

We begin this chapter with a review of the mixing rules introduced in Section 12.1. Then we show how the mixing rule leads to the fugacities and *K*-ratios needed for VLE calculations. We then provide algorithms and illustrate how VLE calculations are programmed using an equation of state (EOS). Finally we provide some insight into how critical behavior in mixtures differs from critical behavior in pure components, and that some “counterintuitive” behavior can exist, such as quality that decreases when pressure is increased.

**1.** Compute VLE phase diagrams using an EOS.

**2.** Characterize adjustable parameters in EOS models using experimental data.

**3.** Derive an expression for a fugacity coefficient given an arbitrary EOS and mixing rules.

**4.** Comment critically on the merits and limitations of the PREOS relative to the activity models of Chapters 11–13, including the ability to suggest ways that the PREOS can be systematically improved.

The virial equation was introduced for pure fluids in Section 7.4. Previously, we have also given a strategy for relating parameters to composition in Section 12.1. If we extend this mixing rule to the virial equation,

which for a binary mixture becomes

Similar to our previous discussion, it is understood that *B*_{12} is equivalent to *B*_{21}. *The cross coefficient B*_{12} *is not the virial coefficient for the mixture.*

To obtain the cross coefficient, *B*_{12}, we must create a **combining rule** to propose how the cross coefficient depends on the properties of the pure components 1 and 2. For the virial coefficient, the relationship between the pair potential and the virial coefficient was given in Section 7.11. However, a less rigorous method is often used in engineering applications. Rather, combining rules are created to use the **corresponding state** correlations developed for pure components in terms of *T _{c}*

Binary interaction parameters are used to adjust the combining rule to better fit experimental data, if available.

The parameter *k*′_{12} is an adjustable parameter (called the **binary interaction parameter**) to force the combining rules to more accurately represent the cross coefficients found by experiment.^{1} However, in the absence of experimental data, it is customary to set *k*′_{12} = 0.

and

The first three of these combining rules lead to:

Then, *T*_{c12}, *P _{c}*

Example 15.1. The virial equation for vapor mixtures

Calculate the molar volume for a 60 mole% mixture of neopentane(1) in CO_{2}(2) at 310 K and 0.2 MPa.

Solution

The conditions are entered in the spreadsheet Virialmx.xlsx, with the following results:

The original spreadsheet is modified slightly for this solution. Cells J9 and J10 are programmed with a rearranged form of Eqn. 7.10, *T _{r}* – 0.686 – 0.439

The virial coefficient for the mixture is given by Eqn. 15.1,*B* = 0.6^{2}·(–846.06) + 2(0.6)(0.4)(–330.5) + 0.4^{2}·(–113.39) = –481.36 cm^{3}/mol*V* = *RT/P* + *B* = 8.314 · 310/0.2 – 481.36 = 12,405 cm^{3}/mol

The volumetric behavior of the mixture depends on composition. The mixture volume differs from an ideal solution, . The difference *V* – *V ^{is}* is called the excess volume,

The molar volume of pure neopentane is

*V* = *RT/P* + *B* = 8.314 · 310/0.2 – 846.1 = 12,041 cm^{3}/mol

The molar volume of pure CO_{2} is

*V* = *RT*/*P* + *B* = 8.314 · 310/0.2 – 113.4 = 12,773 cm^{3}/mol

The molar volume of an ideal solution of a 60 mole% neopentane mixture is

*V ^{is}* = 0.6(12,041) + 0.4(12,773) = 12,334 cm

and the excess volume is

*V ^{E}* = 12,405 – 12,334 = 71.2 cm

The molar volume and excess volume can be determined across the composition range by changing *y*’s in the formulas.

The customary mixing rules for cubic equations of state have been introduced in Section 12.1:

Note the mathematical similarity of the mixing rule for *a* with the mixing rule used for the virial coefficient. All of the compositional dependence of the equation of state is incorporated into the two relations. A combining rule is not necessary for the *b* term, however the *a* term does require a combining rule. The customary combining rule is

where *k _{ij}* is referred to as a

In the absence of experimental data or literature values for *k _{ij}*, we may make a first-order approximation by letting

Review of the concepts from Section 10.8 may help put the approaches in context. Keep in mind that the objective is still to perform bubble, dew and flash calculations, but after relaxing the ideal solution assumption.

We begin with a reminder that for phase equilibria calculations, that the fugacities of components are needed. The tool that we need for VLE calculations is the *K*-ratio and an expression for the component fugacity. In Section 10.9 we demonstrated that the component fugacity for an ideal gas component is equal to the partial pressure. In this chapter we develop a method of “correcting” the partial pressure to provide the fugacity. As a variation of the Venn diagram presented in Fig. 11.8, we present the schematic shown in Fig. 15.1. Because the equation of state is capable of representing liquid phases by using the smaller root, we show both vapor and liquid phases.

The method of deriving the fugacity is an extension of Eqn. 10.39. If we compare the chemical potential in the real mixture to the chemical potential for an ideal gas, we see that the difference is given by the component derivative of the Gibbs departure.

We have seen the Gibbs departure in Eqns. 9.23 and 9.31. For the virial equation, we have

where we recognize that the virial coefficient depends on composition via Eqn. 15.2. By differentiation of this expression, we obtain the chemical potential. We can calculate the component fugacity if we use Eqn. 11.22 and replace the standard state with the ideal gas mixture state. Since the component fugacity in the ideal gas state is the partial pressure, the fugacity coefficient becomes

We define the ratio of the component fugacity to the partial pressure (ideal gas component fugacity) as the component fugacity coefficient.

Differentiation of the Gibbs departure leads to the component fugacity coefficients for a binary,

which will be shown in more detail later. The fugacity coefficient of a component in a mixture may be directly determined at a given *T* and *P* by evaluating the virial coefficients at the temperature, then using this equation to calculate the fugacity coefficient.

Differentiation of the Gibbs departure function is difficult for a pressure-explicit equation of state like the Peng-Robinson equation of state. The difficulty arises because the Gibbs departure function is given in terms of volume and temperature rather than pressure (Eqns. 8.36 and 9.33), and differentiation at constant pressure as required by Eqn. 15.12 is difficult. As in the case of pure fluids, classical thermodynamics provides the means to solve this problem. Instead of differentiating the Gibbs departure function, we differentiate the Helmholtz departure function. Recalling,

and noting,

we also use *A* = *G – PV*, or *dA* = *dG – d*(*PV*):

Equating coefficients of *dn _{i}* we see an alternative method to find the chemical potential,

Note that *T, P*, and *V* identify the same conditions for the real fluid. Therefore, when we evaluate the departure, the ideal gas state must be corrected from *V ^{ig}* to

where the notation (*A* – *A ^{ig}*)

General form for fugacity coefficient for a pressure explicit equation of state such as the Peng-Robinson, that is, *Z*(*T*,ρ).

Therefore, the fugacity coefficient is calculated using

To apply this, consider the Peng-Robinson equation as an example.

By extending the method of reducing the equation of state parameters developed in Eqns. 7.21 and 7.22, and , where . Then, differentiation as we will show in Example 15.5 on page 592, yields for a binary system

As we saw in the case of equations of state for pure fluids, there is no fundamental reason to distinguish between the vapor and liquid phases except by the magnitude of *Z*. The equation of state approach encompasses both liquids and vapors very simply. We replace the vapor phase mole fractions with liquid phase mole fractions in all formulas including those for *A* and *B*, resulting in

Recalling that at equilibrium, we write the equality and rearrange to find the expression for the *K*-ratio used to solve VLE problems.

Eqn. 15.20 provides the primary equations for VLE via equations of state. Different equations of state provide different formulas for .

Given *K _{i}* for all

*Note: Eqns. 15.20 provide the primary equations for VLE via equations of state. These equations are implemented by iteration procedures summarized in Appendix C. Only the bubble method will be presented in the chapter in detail. Although cubic equations can represent both vapor and liquid phases, note that the virial equation cannot be used for liquid phases.*

For a bubble-pressure calculation, the *T* and all *x _{i}* are known as shown in Table 10.1 on page 373. Like the simple calculation performed in the preceding chapter, the criterion for convergence is which needs to be expressed in terms of variables for the current method. Rearranging Eqn. 15.20, this sum becomes . Unlike the activity model calculations, we cannot explicitly solve for pressure because all and depend on pressure. Additionally, all depend on composition of the vapor phase, which is not exactly known until the problem is solved. Typically, we use Raoult’s law with the shortcut vapor pressure equation for the first guesses of

As we observed for pure fluids, it is important to select the proper root when applying an equation of state. Considering the Workbook Prfug.xlsx, for the one-root region, we should select that row for the fugacity coefficients. For the three-root region, we should choose the root with the lowest mixture fugacity. At low pressure and near room temperature, systems are usually in the three-root region for both liquid and vapor compositions, but that may change as we approach the critical region. The number of roots depends on composition as well as *T* and *P.* For example, it often occurs that one root occurs using the vapor composition when one component is supercritical in equilibrium with a liquid phase. This means we need to select among at least four possibilities for each phase when computing the *K*-values: largest *Z* root for vapor composition, smallest *Z* root with liquid composition, single root with vapor composition, single root with liquid composition. If we compute *K*-values with all four ratios, only one of the possibilities provides meaningful results and these are the ones to apply in the next iteration.

Example 15.2. *K*-values from the Peng-Robinson equation

The bubble-point pressure of an equimolar nitrogen (1) + methane (2) system is to be calculated by the Peng-Robinson equation and compared to the shortcut *K*-ratio estimate at 100 K. The shortcut *K*-ratio estimate will be used as an initial guess: *P* = 0.4119 MPa, *y*_{N2} = 0.958. Apply the formulas for the fugacity coefficients to obtain an estimate of the *K*-values for nitrogen and methane and evaluate the sum of the vapor mole fractions based on this initial guess.

Solution

The spreadsheet Prfug.xlsx may be used to follow the calculations. The *K*-values using the vapor root with vapor composition and liquid root with liquid composition are valid throughout the iterations of this example. From the shortcut calculation, *P* = 0.4119 MPa at 100 K. Applying Eqns. 7.21 and 7.22 for the pure component parameters:

For N_{2}: *A*_{11} = 0.09686; *B*_{1} = 0.011906;

For CH_{4}: *A _{22}* = 0.18242;

By the square-root combining rule Eqn. 15.9:

Based on the vapor composition of the shortcut estimate at *y*_{1} = 0.958, the mixing rule gives *A ^{V}* = 0.099913;

Then

Many of the terms are the same for the methane in the mixture:

To save some tedious calculations, the liquid formulas have already been applied to obtain: ; . Determining the *K* values,

*y _{1}* = 0.5 · 1.955 = 0.978;

A higher guess for *P* would be appropriate for the next iteration in order to make the *K*-values smaller. and would need to be evaluated at the new pressure. The calculations are obviously tedious. *K _{i}* calculations are possible in Excel by first copying the “Fugacities” sheet on Prfug.xlsx, using one sheet for liquid and the other for vapor, and then referencing cells on one of the sheets to calculate the

Since a compositional derivative is necessary to obtain the partial molar quantities, and the compositions are present in summation terms, we must understand the procedures for differentiation of the sums. Since *all of the compositional dependence* is embedded in these terms, if we understand how these terms are handled, we can then apply the results to *any* equation of state. Only three types of sums appear in most forms of equations of state, which have been introduced above. The first type of derivative we will encounter is of the form

Since the compositional dependence is within the mixing rule, if we understand how to differentiate the general mixing rules, then we can easily apply them to the models that use them.

where . For a binary *nb* = *n*_{1}b_{1} + *n*_{2}b_{2}, and *k* will be encountered once in the sum, whether *k* = 1 or *k* = 2, thus:

and the general result is

The second type of derivative which we will encounter is of the form

*n*^{2}*a* may be written as . For a binary mixture, . Taking the appropriate derivative,

The general result is

For the virial equation, we need to differentiate a function that will look like:

Differentiation by the product rule gives

The double sum in the derivative is *n ^{2}B* which we have evaluated in equivalent form in Eqn. 15.25. The second term is just

The general result is

Example 15.3. Fugacity coefficient from the virial equation

For moderate deviations from the ideal-gas law, a common method is to use the virial equation given by:

*Z* = 1 + *BP/RT*

where . Develop an expression for the fugacity coefficient.

Solution

For the virial equation, we have the result of Eqn. 9.30,

Applying Eqn. 15.12

the argument we need to differentiate looks like .

Differentiation has been performed in Eqn. 15.29, which we can generalize as

which has been shown earlier for a binary in Eqn. 15.14.

Example 15.4. Fugacity coefficient from the van der Waals equation

Van der Waals’ equation of state provides a simple but fairly accurate representation of key equation of state concepts for mixtures. The main manipulations developed for this equation are the same for other equations of state but the algebra is a little simpler. Recalling van der Waals’ equation from Chapter 6,

where and . Develop an expression for the fugacity coefficient.

Solution

We need to apply Eqn. 15.17. For the departure, we apply Eqn. 8.27 because the differentiation indicated above is performed at constant volume, not constant pressure.

Apply Eqn. 15.17, but instead of differentiating directly, use the chain rule, Eqn. 6.16.

Example 15.5. Fugacity coefficient from the Peng-Robinson equation

The Peng-Robinson equation is given by

where and . Develop an expression for the fugacity coefficient.

Solution

We need to apply Eqn. 15.17. From integration for the pure fluid,

The next steps look intimidating. Basically, they apply the same procedure for differentiation as the last example.

Note a simplification that is not obvious:

Substituting the following definitions,

which has been shown in Eqns. 15.18–15.19 for a binary.

At the end of Section 15.2, the bubble-pressure method was briefly introduced to show how the fugacity coefficients are incorporated into a VLE calculation, without concentrating on the details of the iterations. Section 15.3 offered derivations of formulas for the fugacity coefficients that were presented without proof at the beginning of the chapter. Now, it is time to turn to the applied engineering objective: calculation of phase equilibria. Refer again to Table 10.1 on page 373, that lists the types of routines that are needed and the convergence criteria. Note that Table 10.1 is independent of the model used for calculating VLE. As an example of the iteration procedure for cubic equations of state, the bubble-pressure flow sheet is presented in Fig. 15.2. The flow sheet puts detail to the procedure discussed superficially in Example 15.2 and immediately preceding the example. Flow sheets for bubble temperature, dew, and flash routines are available in Appendix C. As with ideal solutions, the bubble-pressure routine is the easiest to apply, so we cover it in detail in the following examples. Iterative phase equilibrium calculations can be tedious and difficult to automate. We can facilitate the calculations to some extent by combining two copies of the PrFug spreadsheet into a single workbook, which we call Prmix.xlsx. The four possible *K*-value representations are included for convenient selection, as described in Example 15.2. This workbook forms only a starting basis with an emphasis on clearly showing the fundamental steps.

Example 15.6. Bubble-point pressure from the Peng-Robinson equation

Use the Peng-Robinson equation (*k _{ij}* = 0) to determine the bubble-point pressure of an equimolar solution of nitrogen (1) + methane (2) at 100 K.

Solution

The calculations proceed by first calculating the short-cut *K*-ratio as in Example 15.2 on page 587. The ideal-solution (* ^{is}*) bubble pressure was ;

Noting that these sum to a number greater than unity, we must choose a greater value of pressure for the next iteration. Before we can start the next iteration, however, we must develop new estimates of the vapor mole fractions; the ones we have do not make sense because they sum to more than unity. These new estimates can be obtained simply by dividing the given vapor mole fractions by the number to which they sum. This process is known as **normalization** of the mole fractions. For example, to start the second iteration, *y*_{1} = 0.978/1.033 = 0.947. After repeating the process for the other component, the mole fractions will sum to unity. Since the result for the second iteration is less than one, the pressure guess is too high.

The third iteration consists of applying the interpolation rule to obtain the estimate of pressure and use of the normalization procedure to obtain the estimates of vapor mole fractions. *P* = 0.4119 + (1 – 1.033)/(0.956 – 1.033) · (0.45 – 0.4119) = 0.428 MPa. Since the estimated vapor mole fractions after the third iteration sum very nearly to unity, we may conclude the calculations here. This is the bubble pressure. Note how quickly the estimate for *y*_{1} converges to the final estimate of 0.945.

Example 15.7. Isothermal flash using the Peng-Robinson equation

A distillation column is to produce overhead products having the following compositions:

Suppose a partial condenser is operating at 320 K and 8 bars. What fraction of liquid would be condensed according to the Peng-Robinson equation, assuming all binary interaction parameters are zero (*k _{ij}* = 0)?

Solution

This is an isothermal flash calculation. Refer back to the same problem (Example 10.1 on page 382) for an initial guess based on the shortcut *K*-ratio equation. *V*/*F* = 0.25 ⇒ {*x _{i}*} = {0.1829, 0.7053, 0.1117} and {

The computations for the flash calculation are basically analogous to those in Example 10.1, except that *K _{i}* values are calculated from Eqn. 15.20. A detailed flow sheet is presented in Appendix C. For this example, the

Using these *x*’s and *y*’s for guesses we find *K* = 1.7276, 0.8318, and 0.6407, respectively. These *K-*values are similar to those estimated at the compositions derived from the ideal-solution approximation, and will yield a similar *V/F*. Therefore, we conclude that this iteration has converged (a general criterion is that the average % change in the *K*-values from one iteration to the next is less than 10^{-4}). Comparison to the shortcut *K*-ratio approximation shows small but significant deviations—*V/F* = 0.13 for Peng-Robinson versus 0.25 for the shortcut *K*-ratio method.

Based on this example, we may conclude that the shortcut *K*-Ratio approximation provides a reasonable first approximation at these conditions. Note, however, that none of the components is supercritical and all the components are saturated hydrocarbons.

It is tempting to expand further on Prmix.xlsx to facilitate greater automation and simple-minded application. An online supplement provides a very preliminary step in this direction through the use of macro’s. Ultimately, however, this literature comprises specialized research that is beyond our introductory scope. In general, the analysis requires detailed consideration of phase stability and criticality. References cited in Chapter 16 describe works by Michelsen and Mollerup, Eubank, and Tang that can help to create more reliable algorithms. It is a useful exercise to customize your workbooks to increase your confidence in achieving reliable solutions, but do not spend excessive time trying to program every possibility. Chapter 16 and the references cited there are recommended for advanced programming.

Example 15.8. Phase diagram for azeotropic methanol + benzene

Methanol and benzene form an azeotrope. For methanol + benzene the azeotrope occurs at 61.4 mole% methanol and 58°C at atmospheric pressure (1.01325 bars). Additional data for this system are available in the *Chemical Engineers’ Handbook*. Use the Peng-Robinson equation with *k _{ij}* = 0 (see Eqn. 15.9) to estimate the phase diagram for this system and compare it to the experimental data on a

The experimental data for this system are as follows:

Solution

Solving this problem is computationally intensive, but still approachable with Prmix.xlsx. The strategy is to manually set a guessed *k _{ij}* and then perform a bubble pressure calculation at the azeotrope temperature (331.15 K) and composition,

The resultant *k _{ij}* is used to perform bubble-temperature calculations across the composition range, resulting in Fig. 15.3. Note that we might find a way to fit the data more accurately than the method given here, but any improvements would be small relative to estimating

Example 15.9. Phase diagram for nitrogen + methane

Use the Peng-Robinson equation (*k _{ij}* = 0) to determine the phase diagram of nitrogen + methane at 150 K. Plot

Solution

First, the shortcut *K*-ratio method gives the dotted phase diagram in Fig. 15.4. Applying the bubble-pressure procedure with the program Prmix.xlsx, we calculate the solid line in Fig. 15.4. For the Peng-Robinson method we assume *K*-values from the previous solution as the initial guess to get the solutions near *x*_{N2} = 0.685. The program Prmix.xlsx assumes this automatically, but we must also be careful to make small changes in the liquid composition as we approach the critical region.

Fig. 15.4 was generated by entering liquid nitrogen compositions of 0.10, 0.20, 0.40, 0.60, 0.61, 0.62..., 0.68, and 0.685. This procedure of starting in a region where a simple approximation is reliable and systematically moving to more difficult regions using previous results is often necessary and should become a familiar trick in your accumulated expertise on phase equilibria in mixtures. We apply a similar approach in estimating liquid-liquid equilibria.

Comparing the two approximations numerically and graphically, it is clear that the shortcut approximation is significantly less accurate than the Peng-Robinson equation at high concentrations of the supercritical component. This happens because the mixture possesses a critical point, above which separate liquid and vapor roots are impossible, analogous to the situation for pure fluids. Since the mixing rules are in terms of *a* and *b* instead of *T _{c}* and

Instead of depending simply on *T* and *P* as they did for pure fluids, however, *A _{c}* and

The mixture critical point also leads to computational difficulties. If the composition is excessively rich in the supercritical component, the equation of state calculations may obtain the same solution for the vapor root as for the liquid root and, since the fugacities are equal, the program will terminate. The program may indicate accurate convergence in this case due to some slight inaccuracies that are unavoidable in the critical region. Or the program may diverge. It is often up to the competent engineer to recognize the difference between accurate convergence and a spurious answer. Plotting the phase envelope is an excellent way to stay out of trouble.

The shortcut *K*-ratio method provides an initial estimate when a supercritical component is at low liquid-phase compositions, but incorrectly predicts VLE at high liquid-phase concentrations of the supercritical component.

Example 15.10. Ethane + heptane phase envelopes

Use the Peng-Robinson equation (*k _{ij}* = 0) to determine the phase envelope of ethane +

Solution

Note that these phase envelopes are similar to the one from the previous problem, except that we are changing the temperature instead of the composition along each curve. They are more tedious in that both dew and bubble calculations must be performed to generate each curve. The lines of constant composition are sometimes called **isopleths.** The results of the calculations are illustrated in Fig. 15.5. The results at mole fractions of 0 and 1.0 are indicated by dash-dot curves to distinguish them as the vapor pressure curves. Phase equilibria on the *P-T* plot occurs at the conditions where a bubble line of one composition intersects a dew line of a different composition.

Some practical considerations for high pressure processing can be inferred from the diagram. Consider what happens when starting at 90 bars and ~445 K and dropping the pressure on a 30 mole% C7 mixture at constant temperature. Similar situations could arise with flow of natural gas through a small pipe during natural gas recovery. As the pressure drops, the dew-point curve is crossed and liquid begins to condense. Based on intuition developed from experiences at lower pressure, one might expect that dropping the pressure should result in more vapor-like behavior, not condensation. On the other hand, dropping the pressure reduces the density and solvent power of the ethane-rich mixture. This phenomenon is known as **retrograde condensation.** It occurs near the critical locus when the operating temperature is less than the maximum temperature of the phase envelope. Since this maximum temperature is different from the mixture critical temperature, it needs a distinctive name. The name applied is the “critical condensation temperature” or **cricondentherm.** Similarly, the maximum pressure on the phase envelope is known as the **cricondenbar.** Note that an analogous type of phase transition can occur near the critical locus when the pressure is just above the critical locus and the temperature is changed.

To extend the analysis, imagine what happens in a natural gas stream composed primarily of methane but also containing small amounts of components as heavy as C80. A retrograde condensation region exists where the heavy components begin to precipitate, as discussed in Example 14.12 on page 567. But a different possibility also exists because the melting temperature of the heavy components may often exceed the operating temperature, and the precipitate that forms might be a solid that could stick to the walls of the pipe. This in turn generates a larger constriction which generates a larger pressure drop during flow, right in the vicinity of the deposit. In other words, this deposition process may tend to promote itself until the flow is substantially inhibited. Wax deposition is a significant problem in the oil and natural gas industry and requires considerable engineering expertise because it often occurs away from critical points, as well as in the near-critical regions of this discussion. A wide variety of solubility behavior can occur, as we show in Chapter 16.

For some problems, such as generation of a phase diagram, the examples in this chapter can be followed directly. Often, however, it takes some thought to decide which type of VLE routine is appropriate to apply in a given situation. Do not discount this step in the problem solution strategy. Part of the objective of the homework problems is to increase understanding of phase behavior by encouraging thought about which routine to use. Since the equation of state routines are complicated enough to require a computer, they are also solved relatively rapidly, *once the VLE routine has been identified*. Use Table 10.1 on page 373 and the information in Section 10.1 to identify the correct procedure to apply before turning to computational techniques. Use the problem statement to identify whether the liquid, vapor, or overall mole fractions are known. Study the problem statement to see whether the *P* and *T* are known. Often this information together with Table 10.1 will determine the routines to apply. Sometimes more than one approach is satisfactory. Sometimes the ideal-solution approximation may be applied, and a review of Section 10.8 may be helpful. Before using software that accompanies the text, be sure to read the appropriate section of Appendix A, and the instructions or readme.txt files that accompany the program.

The essence of the equation of state approach to mixtures is that the equation of state for mixtures is the same as the equation of state for pure fluids. The expressions for *Z, A*, and *U* are exactly the same. The only difference is that the parameters (e.g., *a* and *b* of the Peng-Robinson equation) are dependent on the composition. That should come as no large surprise when you consider that these parameters must transform from pure component to pure component in some continuous fashion as the composition changes. What may be surprising is the wealth of behaviors that can be inferred from some fairly simple rules for modeling this transformation. Everything from azeotropes to retrograde condensation, and even liquid-liquid separation can be represented with qualitative accuracy based on this simple extension of the equation of state.

So, are we done with phase behavior modeling? Unfortunately, the keyword in the preceding paragraph is “qualitative.” Equations of state are sufficiently accurate for most applications involving hydrocarbons, gases, and to some extent, ethers, esters, and ketones. For many oil and gas wells, it may suffice to treat the hydrocarbon-rich phases in this way and treat water separately. But any applications involving strongly hydrogen-bonding species tend to require greater accuracy than currently attainable from equations like the Peng-Robinson EOS. For example, if methanol is used as a hydrate inhibitor in a gas well, its partitioning may require a more sophisticated treatment. One idea is to adapt multiparameter activity models like the UNIQUAC model as the basis for mixing rules. This is the approach of the Wong-Sandler model.^{3} Another approach is to analyze hydrogen bonding as simultaneous reaction and phase equilibria, as discussed in Chapter 19.

Once again the mixing rules play an important role in defining the thermodynamics. Since the Peng-Robinson mixing rules have the same form as the van der Waals mixing rules, including a single binary interaction parameter, *k _{ij}*, the Peng-Robinson model cannot match the skewness of the Gibbs excess curve, only the magnitude. Outside the critical region, you might as well use an activity model. The advantage of the Peng-Robinson model is that it provides a holistic framework that applies seamlessly to vapor, liquid, and critical region. Noting how activity models artificially designate different methods for different phases, it is gratifying to see that such conceptual simplicity is feasible. The key equation for establishing this feasibility is deriving the fugacity coefficient for a pressure-explicit equation of state:

Given this equation, it is straightforward to derive fugacity coefficients, and *K*-ratios, for any equation of state or mixing rule. Two related equations that often appear are Eqns. 15.23 and 15.26.

Look for ways to rearrange the equations before differentiating such that these terms appear and then differentiation becomes much simpler, often reducing to simple substitution. Finally, the EOS method melds with every other phase equilibrium computational procedure when the expression is derived for the partition coefficient, as given by a slight variation on Eqn. 15.20.

Here we have generalized Eqn. 15.20 slightly by recognizing that the upper phase could be vapor or it could be the upper phase of LLE. The beauty of the EOS perspective is that the fluid phase model is the same for liquid or vapor; only the proper root must be selected.

**P15.1.** Repeat all the practice problems from Chapter 10, this time applying the Peng-Robinson equation.

**P15.2.** Acrolein (C_{3}H_{4}O) + water exhibits an atmospheric (1 bar) azeotrope at 97.4 wt% acrolein and 52.4°C. For acrolein: *T _{c}* = 506 K;

**a.** Determine the value of *k _{ij}* for the Peng-Robinson equation that matches this bubble pressure at the same liquid composition and temperature. (ANS. 0.015)

**b.** Tabulate *P, y* at 326.55 K and *x* = {0.57, 0.9, 0.95, 0.974} via the Peng-Robinson equation using the *k _{ij}* determined above. (ANS. (1.33, 0.575), (1.16, 0.736), (1.06, 0.841), (1.0, 0.860))

**P15.3.** Laugier and Richon (*J. Chem. Eng. Data*, 40:153, 1995) report the following data for the H_{2}S + benzene system at 323 K and 2.010 MPa: *x*_{1} = 0.626; *y*_{1} = 0.986.

**a.** Quickly estimate the vapor-liquid *K*-value of H_{2}S at 298 K and 100 bar. (ANS. 0.21)

**b.** Use the data to estimate the *k _{ij}* value, then estimate the error in the vapor phase mole fraction of H

**P15.4.** The system ethyl acetate + methanol forms an azeotrope at 27.8 mol% EA and 62.1°C. For ethyl acetate, *T _{c}* = 523.2 K;

**a.** What is the estimate of the bubble-point pressure from the Peng-Robinson equation of state at this composition and temperature when it is assumed that *k _{ij}* = 0? (ANS. 0.98 bars)

**b.** What value of *k _{ij}* gives a bubble-point pressure of 1 bar at this temperature and composition? (ANS. 0.0054)

**c.** What is the composition of the azeotrope and value of the bubble-point pressure at the azeotrope estimated by the Peng-Robinson equation when the value of *k _{ij}* from part (b) is used to describe the mixture? (ANS.

**a.** Assuming zero for the binary interaction parameter (*k _{ij}* = 0) of the Peng-Robinson equation, predict whether an azeotrope should be expected in the system CO

**b.** Assuming a value for the binary interaction parameter (*k _{ij}* = 0.11) of the Peng-Robinson equation, predict whether an azeotrope should be expected in the system CO

**a.** Assuming zero for the binary interaction parameter (*k _{ij}* = 0) of the Peng-Robinson equation, estimate the bubble pressure and vapor composition of the pentane + acetone system at

**b.** Use the experimental liquid composition and bubble condition of the pentane + acetone system at *x _{P}* = 0.728,

**P15.7.** Calculate the dew-point pressure and corresponding liquid composition of a mixture of 30 mol% carbon dioxide, 30% methane, 20% propane, and 20% ethane at 298 K using

**a.** The shortcut *K*-ratios (ANS. 32 bar)

**b.** The Peng-Robinson equation with *k _{ij}* = 0 (ANS. 44 bar)

**P15.8.** The equation of state below has been suggested for a new equation of state. Derive the expression for the fugacity coefficient of a component.

*Z* = 1 + 4*cb*ρ/(1 – *b*ρ)

**15.1.** Using Fig. 15.5 on page 602, without performing additional calculations, sketch the *P-x-y* diagram at 400 K showing the two-phase region. Make the sketch semi-quantitative to show the values where the phase envelope touches the axes of your diagram. Label the bubble and dew lines. Also indicate the approximate value of the maximum pressure.

**15.2.** Consider two gases that follow the virial equation. Show that an ideal mixture of the two gases follows the relation *B* = *y*_{1}*B*_{11} + *y*_{2}*B*_{22}.

**15.3.** Consider phase equilibria modeled with . When might ϕ* _{i}* be replaced by ϕ

**15.4.** Calculate the molar volume of a binary mixture containing 30 mol% nitrogen(1) and 70 mol% *n*-butane(2) at 188°C and 6.9 MPa by the following methods.

**a.** Assume the mixture to be an ideal gas.

**b.** Assume the mixture to be an ideal solution with the volumes of the pure gases given by

and the virial coefficients given below.

**c.** Use second virial coefficients predicted by the generalized correlation for *B*.

**d.** Use the following values for the second virial coefficients.

Data:

*B*_{11} = 14 *B*_{22} = –265 *B*_{12} = –9.5 (Units are cm^{3}/gmole)

**e.** Use the Peng-Robinson equation.

**15.5.** For the same mixture and experimental conditions as problem 15.4, calculate the fugacity of each component in the mixture, . Use methods (a) – (e).

**15.6.** A vapor mixture of CO_{2} (1) and *i*-butane (2) exists at 120°C and 2.5 MPa. Calculate the fugacity of CO_{2} in this mixture across the composition range using

**a.** The virial equation for mixtures

**b.** The Peng-Robinson equation

**c.** The virial equation for the pure components and an ideal mixture model.

**15.7.** Use the virial equation to consider a mixture of propane and *n*-butane at 515 K at pressures between 0.1 and 4.5 MPa. Verify that the virial coefficient method is valid by using Eqn. 7.10.

**a.** Prepare a plot of fugacity coefficient for each component as a function of composition at pressures of 0.1 MPa, 2 MPa, and 4.5 MPa.

**b.** How would the fugacity coefficient for each component depend on composition if the mixture were assumed to be ideal, and what value(s) would it have for each of the pressures in part (a)? How valid might the ideal-solution model be for each of these conditions?

**c.** The excess volume is defined as , where *V* is the molar volume of the mixture, and *V _{i}* is the pure component molar volume at the same

**d.** Under which of the pressures in part (a) might the ideal gas law be valid?

**15.8.** Consider a mixture of nitrogen(1) + *n*-butane(2) for each of the options: (*i*) 395 K and 2 MPa; (*ii*) 460 K and 3.4 MPa; (*iii*) 360 K and 1 MPa.

**a.** Calculate the fugacity coefficients for each of the components in the mixture using the virial coefficient correlation. Make a table for your results at *y*_{1} = 0.0, 0.2, 0.4, 0.6, 0.8, 1.0. Plot the results on a graph. On the same graph, plot the curves that would be used for the mixture fugacity coefficients if an ideal mixture model were assumed. Label the curves.

**b.** Calculate the fugacity of each component in the mixture as predicted by the virial equation, an ideal-mixture model, and the ideal-gas model. Prepare a table for each component, and list the three predicted fugacities in three columns for easy comparison. Calculate the values at *y*_{1} = 0.0, 0.2, 0.4, 0.6, 0.8, 1.0.

**15.9.** The virial equation *Z* = 1 + *BP*/*RT* may be used to calculate fugacities of components in mixtures. Suppose *B* = *y*_{1}*B*_{11} + *y*_{2}*B*_{22}. (This simple form makes calculations easier. Eqn. 15.1 gives the correct form.) Use this simplified expression and the correct form to calculate the respective fugacity coefficient formulas for component 1 in a binary mixture.

**15.10.** The Lewis-Randall rule is usually valid for components of high concentration in gas mixtures. Consider a mixture of 90% ethane and 10% propane at 125°C and 170 bar. Estimate for ethane.

**15.11.** One of the easiest ways to begin to explore fugacities in nonideal solutions is to model solubilities of crystalline solids dissolved in high pressure gases. In this case, the crystalline solids remain as a pure phase in equilibrium with a vapor mixture, and the fugacity of the “solid” component must be the same in the crystalline phase as in the vapor phase. Consider biphenyl dissolved in carbon dioxide, using *k _{ij}* = 0.100. The molar volume of crystalline biphenyl is 156 cm

**a.** Calculate the fugacity (in MPa) of pure crystalline biphenyl at 310 K and 330 K and 0.1, 1, 10, 15, and 20 MPa.

**b.** Calculate and plot the biphenyl solubility for the isotherm over the pressure range. Compare the solubility to the ideal gas solubility of biphenyl where the Poynting correction is included, but the gas phase nonidealities are ignored.

**15.12.** Repeat problem 15.11, except consider naphthalene dissolved in carbon dioxide, using *k _{ij}* = 0.109. The molar volume of crystalline naphthalene is 123 cm

**15.13.** A vessel initially containing propane at 30°C is connected to a nitrogen cylinder, and the pressure is isothermally increased to 2.07 MPa. What is the mole fraction of propane in the vapor phase? You may assume that the solubility of N_{2} in propane is small enough that the liquid phase may be considered pure propane. Calculate using the following data at 30°C.

**15.14.** A 50-mol% mixture of propane(1) + *n*-butane(2) enters a flash drum at 37°C. If the flash drum is maintained at 0.6 MPa, what fraction of the feed exits as a liquid? What are the compositions of the phases exiting the flash drum? Work the problem the following two ways.

**a.** Use Raoult’s law.

**b.** Assume ideal mixtures of vapor and liquid (*K _{i}* is independent of composition).

**15.15.** A mixture containing 5 mol% ethane, 57 mol% propane, and 38 mol% *n*-butane is to be processed in a natural gas plant. Estimate the bubble-point pressure, the liquid composition, and *K*-ratios of the coexisting vapor for this mixture at all pressures above 1 bar at which two phases exist. Set *k _{ij}* = 0. Use the shortcut

**15.16.** Vapor-liquid equilibria are usually expressed in terms of *K* factors in petroleum technology. Use the Peng-Robinson equation to estimate the values for methane and benzene in the benzene + methane system with equimolar feed at 300 K and a total pressure of 30 bar and compare to the estimates based on the shortcut *K*-ratio method.

**15.17.** Benzene and ethanol form azeotropic mixtures. Prepare a *y-x* and a *P-x-y* diagram for the benzene + ethanol system at 45°C assuming the Peng-Robinson model and using the experimental pressure at *x _{E}* = 0.415 to estimate

**15.18.** A storage tank is known to contain the following mixture at 45°C and 15 bar on a mole basis: 31% ethane, 34% propane, 21% *n*-butane, 14% *i*-butane. What is the composition of the coexisting vapor and liquid phases, and what fraction of the molar contents of the tank is liquid?

**15.19.** The *CRC Handbook* lists the atmospheric pressure azeotrope for ethanol + methylethylketone at 74.8°C and 34 wt% ethanol. Estimate the value of the Peng-Robinson *k*_{12} for this system.

**15.20.** The *CRC Handbook* lists the atmospheric pressure azeotrope for methanol + toluene at 63.7°C and 72 wt% methanol. Estimate the value of the Peng-Robinson *k*_{12} for this system.

**15.21.** Use the Peng-Robinson equation for the ethane/heptane system.

**a.** Calculate the *P-x-y* diagram at 283 K and 373 K. Use *k*_{12} = 0. Plot the results.

**b.** Based on a comparison of your diagrams with what would be predicted by Raoult’s law at 283 K, does this system have positive or negative deviations from Raoult’s law?

**15.22.** One mol of *n*-butane and one mol of *n*-pentane are charged into a container. The container is heated to 90°C where the pressure reads 7 bar. Determine the quantities and compositions of the phases in the container.

**15.23.** Consider a mixture of 50 mol% *n*-pentane and 50 mol% *n*-butane at 15 bar.

**a.** What is the dew temperature? What is the composition of the first drop of liquid?

**b.** At what temperature is the vapor completely condensed if the pressure is maintained at 15 bar? What is the composition of the last drop of vapor?

**15.24.** LPG gas is a fuel source used in areas without natural gas lines. Assume that LPG may be modeled as a mixture of propane and *n*-butane. Since the pressure of the LPG tank varies with temperature, there are safety and practical operating conditions that must be met. Suppose the desired maximum pressure is 0.7 MPa, and the lower limit on desired operation is 0.2 MPa. Assume that the maximum summertime tank temperature is 50°C, and that the minimum wintertime temperature is –10°C. [Hint: On a mass basis, the mass of vapor within the tank is negligible relative to the mass of liquid after the tank is filled.]

**a.** What is the upper limit (mole fraction) of propane for summertime propane content?

**b.** What is the lowest wintertime pressure for this composition from part (a)?

**c.** What is the lower limit (mole fraction) of propane for wintertime propane content?

**d.** What is the highest summertime pressure for this composition from part (b)?

**15.25.** The *k _{ij}* for the pentane + acetone system has been fitted to a single point in problem P15.6. Generate a

**15.26.** The synthesis of methylamine, dimethylamine, and trimethylamine from methanol and ammonia results in a separation train involving excess ammonia and converted amines. Use the Peng-Robinson equation with *k _{ij}* = 0 to predict whether methylamine + dimethylamine, methylamine + trimethylamine, or dimethylamine + trimethylamine would form an azeotrope at 2 bar. Would the azeotropic behavior identified above be altered by raising the pressure to 20 bar? Locate experimental data relating to these systems in the library. How do your predictions compare to the experimental data?

**15.27.** For the gas/solvent systems below, we refer to the “gas” as the low molecular weight component. Experimental solubilities of light gases in liquid hydrocarbons are tabulated below. The partial pressure of the light gas is 1.013 bar partial pressure. Do the following for each assigned system.

**a.** Estimate the partial pressure of the liquid hydrocarbon by calculating the pure component vapor pressure via the Peng-Robinson equation, and by subsequently applying Raoult’s law for that component.

**b.** Estimate the total pressure and vapor composition using the results of step (a).

**c.** Use the Peng-Robinson equation with *k _{ij}* = 0 to calculate the vapor and liquid compositions that result in 1.013 bar partial pressure of the light gas and compare the pressure and gas phase composition with steps (a) and (b).

**d.** Henry’s law asserts that , when *x _{i}* is near zero, and

**e.** Calculate the solubility expected at 2 bar partial pressure of light gas by using Henry’s law as well as by the Peng-Robinson equation and comment on the results.

**15.28.** Estimate the solubility of carbon dioxide in toluene at 25°C and 1 bar of CO_{2} partial pressure using the Peng-Robinson equation with a zero binary interaction parameter. The techniques of problem 15.27 may be helpful.

**15.29.** Oxygen dissolved in liquid solvents may present problems during use of the solvents.

**a.** Using the Peng-Robinson equation and the techniques introduced in problem 15.27, estimate the solubility of oxygen in *n*-hexane at an oxygen partial pressure of 0.21 bar.

**b.** From the above results, estimate the Henry’s law constant.

**15.30.** Estimate the solubility of ethylene in *n*-octane at 1 bar partial pressure of ethylene and 25°C. The techniques of problem 15.27 may be helpful. Does the system follow Henry’s law up to an ethylene partial pressure of 3 bar at this temperature? Provide the vapor compositions and total pressures for the above states.

**15.31.** Henry’s law asserts that , when *x _{i}* is near zero, and

**15.32.** A gas mixture follows the equation of state

where *b* is the size parameter, , and *a* is the energetic parameter, . Derive the formula for the partial molar enthalpy for component 1 in a binary mixture, where the reference state for both components is the ideal gas state of *T _{R}, P_{R}*, and the pure component parameters are temperature-independent.

**15.33.** The procedure for calculation of the residual enthalpy for a pure gas is shown in Example 8.5 on page 316. Now consider the residual enthalpy for a binary gas mixture. For this calculation, it is necessary to determine *da/dT* for the mixture.

**a.** Write the form of this derivative for a binary mixture in terms of *da*_{1}*/dT* and *da*_{2}*/dT* based on the conventional quadratic mixing rule and geometric mean combining rule with a nonzero *k _{ij}*.

**b.** Provide the expression for the residual enthalpy for a binary mixture that follows the Peng-Robinson equation.

**c.** A mixture of 50 mol% CO_{2} and 50 mol% N_{2} enters a valve at 7 MPa and 40°C. It exits the valve at 0.1013 MPa. Explain how you would determine whether CO_{2} precipitates, and if so, whether it would be a liquid or solid.

**15.34.** A gaseous mixture of 30 mol% CO_{2} and 70 mol% CH_{4} enters a valve at 70 bar and 40°C and exits at 5.3 bar. Does any CO_{2} condense? Assume that the mixture follows the virial equation. Assume that any liquid that forms is pure CO_{2}. The vapor pressure of CO_{2} may be estimated by the shortcut vapor pressure equation. CO_{2} sublimes at 0.1013 MPa and –78.8°C, although freezing is less likely.

**15.35.** The vapor-liquid equilibria for the system acetic acid(1) + acetone(2) needs to be characterized in order to simulate an acetic anhydride production process. Experimental data for this system at 760 mmHg have been reported by Othmer (1943)^{4} as summarized below. Use the data at the equimolar composition to determine a value for the binary interaction parameter of the Peng-Robinson equation. Based on the value you determine for the binary interaction parameter, determine the percent errors in the Peng-Robinson prediction for this system at a mole fraction of *x*_{(1)} = 0.3.

**15.36.** A mixture of methane and ethylene exists as a single gas phase in a spherical tank (10 m^{3}) on the grounds of a refinery. The mixture is at 298 K and 1 MPa. It is a spring day, and the atmospheric temperature is also 298 K. The mole fraction of ethylene is 20 mol%. Your supervisor wants to draw off gas quickly from the bottom of the tank until the pressure is 0.5 MPa. However, being astute, you suggest that depressurization will cause the temperature to fall, and might cause condensation.

**a.** Provide a method to calculate the change in temperature with respect to moles removed or tank pressure valid up until condensation starts. Assume the depressurization is adiabatic and reversible. Provide relations to find answers, and ensure that enough equations are provided to calculate numerical values for all variables, but you do not need to calculate a final number.

**b.** Would the answer in part (a) provide an upper or lower limit to the expected temperature?

**c.** Outline how you could find the *P, T, n* of the tank where condensation starts. Provide relations to find answers, and ensure that enough equations are provided to calculate numerical values for all variables, but you do not need to calculate a final number.

**a.** At 298 K, butane follows the equation of state: *P(V* – *b)* = *RT* at moderate pressures, where *b* is a function of temperature. Calculate the fugacity for butane at a temperature of 298 K and a pressure of 1 MPa. At this temperature, *b* = –732 cm^{3}/mol.

**b.** Pentane follows the same equation of state with *b* = –1195 cm^{3}/mol at 298 K. In a mixture, *b* follows the mixing rule: where *b*_{12} = –928 cm^{3}/mol. Calculate the fugacity of butane in a 20 mol% concentration in pentane at 298 K and 1 MPa, assuming the mixture is an ideal solution.

**15.38.** The Soave equation of state is:

where the mixing and combining rules are given by Eqns. 15.8 and 15.9. Develop an expression for the fugacity coefficient and compare it to the expression given by Soave (1972. *Chem. Eng. Sci*. 27:1197).

**15.39.** The following equation of state has been proposed for hard-sphere mixtures:

where

Derive an expression for the fugacity coefficient.

**15.40.** The equation of state below has been suggested. Derive the expression for the fugacity coefficient.

*Z* = 1 + 4*c*ρ/(1 – *b*ρ)

**15.41.** The following free energy model has been suggested as part of a new equation of state for mixtures. Derive the expression for the fugacity coefficient of component 1.

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