Rachford-Rice

Equation (7-8). However, for liquid-liquid equilibria, the equilibrium ratios are strong functions of both phase compositions. The system is thus far more difficult to solve than the superficially similar system of equations for the isothermal vapor-liquid flash. In fact, some of the arguments leading to the selection of the Rachford-Rice form for Equation (7-17) do not apply strictly in the case of two liquid phases. Nevertheless, this form does avoid spurious roots at a = 0 or 1 and has been shown, by extensive experience, to be marltedly superior to alternatives. 

A step-limited Newton-Raphson iteration, applied to the Rachford-Rice objective function, is used to solve for A, the vapor to feed mole ratio, for an isothermal flash. For an adiabatic flash, an enthalpy balance is included in a two-dimensional Newton-Raphson iteration to yield both A and T. Details are given in Chapter 7. 

DFA Partial derivative of the Rachford-Rice objective function (7-13) with respect to the vapor-feed ratio. 

Change in extract-feed ratio from one iteration to the next. Partial derivative of Rachford-Rice objective function with respect to extract-feed ratio. 

F Rachford-Rice objective function for liquid-liquid separa-... 

This is called the Rachford-Rice equation. Notice that, if the A -values and inlet compositions Zi are known, this is a nonlinear equation to solve for u. Thus, you can apply here the same methods used with Excel and MATLAB in Chapter 2. Once the value of f is known, you can calculate the value of the liquid compositions, x,, and vapor compositions, y,, using Eqs. (3.1) and (3.3). The mole balance is then complete. 

Step 3 You can then use Goal Seek to make the Rachford-Rice equation (cell F9) zero by changing the fraction of the feed that is vapor (cell DI), giving the result shown. Once you find the fraction vapor, the mole fractions in the two phases are easy to calculate using Eqs. (3.8) and (3.1), and these are included in columns G and H. 

Step 2 You need the Rachford-Rice equation to sum to zero, which is evident in the spreadsheet. 

Step 4 You can also check one term in the Rachford-Rice equation using detailed calculations, and then copy the equation down. This ensures that the formula is correct for all components. 

For most of these options, the activity coefficient depends upon the liquid mole fractions, and that means the A -values depend upon the liquid mole fractions, too. Thus, there is not just one -value that you can use in the calculations. Instead, one option is to choose the liquid mole fractions, determine the activity coefficients, then the A -values and see if the liquid mole fractions changed significantly. If so, you might use the new values and repeat the process. If this procedure did not work, you would have to use more sophisticated methods to solve the Rachford-Rice equation using process simulators such as Aspen Plus. 

The following stream is at 100 psia and 178°F. Calculate the fraction that is vapor by solving the Rachford-Rice equation (1) using Excel (2) using MATLAB. 

Using the Rachford-Rice equation, Eq. (3.9), prove that for two phases to co-exist, at least one component needs a f-value greater than one and another component needs a A"-value less than one. 

Now the separator must satisfy the Rachford-Rice equation, Eq. (3.9) ... 

In this example, you have two iterations - one because of the circular reference due to the recycle streams and one because of the nonlinear Rachford-Rice equation. Some computer programs cannot handle both of these complications together. Neither Goal Seek nor Solver worked for this example, and you iterated the vapor fraction by hand. 

Ideally you would have calculated the equilibrium in the reactor, too. Then you would have three interacting iterations, and it would be the rare problem that Excel could solve. The difficulty is that, during the iterations, the values may be physically unrealistic. Then, the equilibrium relation or the Rachford-Rice equation gives even more unrealistic values. Programs such as Aspen Plus can realize this and take precautionary steps to avoid it. As the flow sheet gets more and more complicated, and involves more and more thermodynamics, the power of Aspen Plus is welcome. See Chapters 6 and 7 for examples. 

What happened to the mass balances when you introduced a purge stream (You can run it without carbon dioxide, too.) What happened to the mass balances when vapor-liquid equilibrium was required Did the ratio of nitrogen to hydrogen in the recycle stream change Why or why not What if you had to solve the Rachford-Rice equation in the separator, the chemical equilibrium equation in the reactor, and set the purge fraction to maintain a maximum mole fraction of carbon dioxide in the inlet to the reactor. Could you do that all in Excel Would it converge Speculate. 

At a given temperature, pressure, and assumed composition, Equation 5 is solved for V and the fugacity coefficients of each component in the mixture in both vapor and liquid are calculated by Equation 15. The K-ratio for each component is calculated by Equation 4. The Rachford-Rice (4) form of the flash equation is used to calculate the amount of vapor per mole of the overall mixture v/F ... 

Table 7.2 Rachford-Rice procedure for isothermal flash calculations when ff-values are independent of composition...

Rachford-Rice flash algorithm, 273-274 Raffinate, 103 reflux, 415 Raoult s law, 92 K-values, 161 Raschig rings, 57 Rayleigh equation, 363 RDC coluitm, 82, 516 Reboiled absorption, 8, 12 Reboiled stripper, 9, 13, 464 Reboilers partial, 327 total, 327... 

The motivation for posing the LLE problem in this way is that it allows us to take advantage of the Rachford-Rice procedure [11], which is a robust algorithm traditionally applied to isothermal flash calculations. To develop that procedure, we introduce a distribution coefficient Q for each component this quantity is defined by... 

But the mole fractions in each phase must sum to unity, so we define the Rachford-Rice function F by... 

This Rachford-Rice approach offers two principal advantages over other formulations of the LLE problem (i) Equation (11.1.20) is one equation in the unknown R, independent of the number of components present. (The corresponding disadvantage is that we must make initial guesses for all C distribution coefficients.) (ii) Equation (11.1.20) readily lends itself to a solution by Newton s method. (See A.6 in Appendix A.) In Newton s method the value of R(k) at the end of the iteration is replaced by the next guess R k+1) by applying... 

Unfortunately, the distribution coefficients Q are not constants, and the Rachford-Rice function F(R) actually represents a family of curves, as in Figure 11.3. At each iteration of the calculation, the C, values change, moving the search from one curve to another. Nevertheless, each curve in the family is monotone in R, so the computation often converges. 

A flow diagram for the Rachford-Rice method is shown in Figure 11.4. The algorithm divides into two parts an initialization stage followed by a single search loop over R, the fraction of material in one phase. During initialization, we set values for all... 

Figure 11.3 Schematic of a few members of the family of Rachford-Rice curves F(R) (11.1.20) for a ternary mixture in LLE. All curves here are for one set of overall mole fractions 2 at the same T and P however, each curve is for a different set of values for the distribution coefficients C. ...

Figure 11.4 Rachford-Rice algorithm applied to the gamma-gamma method for solving multi-component liquid-liquid equilibrium problems...

With the compositions, we solve our selected model equations for all activity coefficients, and use (11.1.15) to obtain new values for the Q. Then we can compute the Rachford-Rice function F from (11.1.20) and test for convergence. If convergence is lacking, we apply Newton s method (11.1.21) to get a new guess for R and iterate. 

The Rachford-Rice algorithm can find nonphysical values for R. If the procedure gives R > 1 (or R < 0), then the system is a single liquid phase, rich in those components that dominate the P (or a) phase. However, we caution that solutions having R > 1 and R < 0 may be false roots that is, there may actually exist two-phase roots (0 < R < 1) that solve the problem. These can usually be found by changing the initial guesses made for the distribution coefficients Q. 

In the special case of a binary mixture, we can obtain useful information from the Rachford-Rice procedme, even if we do not know the overall mole fractions z to do so, we assume values for the z,. Then, if a two-phase solution is found (0 < R < 1), the computed values for the compositions, y and xP, will be correct regardless of the values assumed for z. However, the value for R changes with z,. So if we have a binary and if we need only the compositions of the two liquid phases, and do not need R, then the Rachford-Rice remains a viable approach. Unfortunately, for two-phase situations with C > 3, the compositions, as well as R, depend on the z,-. 

How do we use the Rachford-Rice algorithm to compute a triangular diagram for liquid-liquid equilibria in a ternary mixture ... 

To compute a point on the liquid-liquid saturation curve at the specified T and P, we choose an overall composition and apply the Rachford-Rice algorithm. For example, consider the overall composition having z = 0.3436 and Z2 = 0.3092. We set the convergence tolerance to e = 10 (or 10 near the critical point). Then we guess the fraction of material in the water-rich phase, R = 0.4, and guess the component distribution coefficients = 7, C2 = 2, and C3 = 0.5. Here we use C = jxf, where a indicates the water-rich phase and (1 indicates the organic-rich phase. 

Table 11.5 Results from Rachford-Rice algorithm for two liquid phases in equilibrium at 333 K and 1.0133 bar...

Figure 11.5 Liquid-liquid equilibria in mixtures of benzene(l), acetonitrile(2), and water(3) at 1.0133 bar and 333 K, computed from the Rachford-Rice algorithm in Figure 11.4 using the NRTL model for activity coefficients. Filled circle is an estimate of the liquid-liquid critical point the estimate lies near Xj = 0.012, X2 = 0.36, X3 = 0.628. Dashed line is the tie line used in the example to illustrate the computational procedure.

With values known for (C + 1) properties, we aim to compute values for (3C - 1) other properties (C - 1) independent mole fractions x for liquid phase a, (C - 1) fractions xP for liquid phase p, (C - 1) fractions y for the vapor, the fraction of material L in liquid a, and the fraction of material V in the vapor. To compute these quantities, we use a double implementation of the Rachford-Rice scheme. That is, we combine phase-equilibrimn and material balance equations to obtain two equations that can be solved for the fractions L and V. 

As in the traditional Rachford-Rice approach, our strategy at this point is to reduce the number of unknowns by summing over the unknown mole fractions. Then we define two functions, analogous to the Rachford-Rice function in (11.1.20),... 

Newton-Raphson method. This is a trial-and-error method for solving simultaneous, nonlinear, algebraic equations. For our VLLE problem we would guess the two unknowns, L and V, use (11.1.31) and (11.1.32) to calculate values for the Rachford-Rice functions, Fj and and then test for convergence. If our convergence criteria are... 

The central portion of the algorithm in Figure 11.6 exactly parallels the standard Rachford-Rice procedure. First, we use (11.1.27)-(11.1.29) to compute the mole fractions for all phases, then we compute all fugacity coefficients and all activity coefficients. With those quantities we can obtain new estimates for the Cs and Ks from the phase-equilibrium relations (11.1.15) and (11.1.24). Now we use (11.1.31) and (11.1.32) to calculate values for the Rachford-Rice functions, Fj and F2, and test for convergence. If our convergence criteria are not met at iteration k, then we use the Newton-Raphson method to estimate the unknown L and V at the next iteration (fc + 1). 

Figure 11.6 Double Rachford-Rice algorithm for using the combined gamma-phi and gamma-gamma methods to solve multicomponent (C > 3) vapor-Uquid-Uquid equilibrium problems. This method fails for binary mixtures (C = 2).

This VLLE algorithm is prone to the same kinds of problems discussed in 11.1.5 for the two-phase Rachford-Rice procedure the algorithm is sensitive to the initial guesses made for the Cs and Ks, and nonphysical results for L and V may be false roots, or they may indicate that three phases do not form at the given conditions. The latter interpretation may hinge on the models chosen for the equation of state and for the activity coefficients. In addition, the absence of three phases can cause the coefficient matrix in (11.1.37) to become singular. 

A principal advantage of this algorithm is that it applies to any number of components C > 3, though in every case we solve only fhe two equations (11.1.31) and (11.1.32). However, this method fails for binary mixtures. To see why, note that for binaries in three-phase equilibrium, (11.1.23) requires us to specify values for T = 3 variables. We then have five equations that can be solved for five unknowns. The five equations are four phase-equilibrium relations (11.1.15) and (11.1.24) plus the one Rachford-Rice function (11.1.31). In the Rachford-Rice approach, the five unknowns would be i/p plus the fractions L and V. However, L and V appear in only one... 

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