Rachford-Rice function

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

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. 

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. 

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). 

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... 

Consider a binary mixture of components 1 and 2 in LLE at fixed T and P. Show that the Rachford-Rice function P (11.1.20) is linear in R and can be written in terms of overall mole fractions Z and distribution coefficients C,- as... 

Equation f2-46) gives a good estimate for the next trial. Once (V/F)k+i is calculated the value of the Rachford-Rice function can be determined. If it is close enough to zero, the calculation is finished otherwise repeat the Newtonian convergence for the next trial. 

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 equation gives us the best next guess for the fraction vaporized. To use it, however, we need equations for both the function and the derivative. For 4, use the Rachford-Rice equation, (2=42). Then the derivative is... 

The fourth question is How should we do the individual convergence steps For the Rachford-Rice equation, linear interpolation or Newtonian convergence will be satisfactory. Several methods can be used to estimate the next flash drum temperature. One of the fastest and easiest to use is a Newtonian convergence procedure. To do this we rearrange the energy balance (Eq. 2-71 into the functional form,... 

Implementation of conditions (1) to (3) is done following a similar procedure as that of a two-phase system (Rachford and Rice, 1952). For a system with C components and n possible phases, satisfying a simple mass balance for each component in each phase results in the following objective function ... 

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