Rachford-Rice equation

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. 

Equation (2=42), which is known as the Rachford-Rice equation, has excellent convergence properties. It can also be modified for three-phase (liquid-liquid-vapor) flash systems fChien. 19941. 

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... 

Newtonian convergence procedures do not always converge. One advantage of using the Rachford-Rice equation with the Newtonian convergence procedure is that there is always rapid convergence. This is illustrated in Example 2-2. 

If V/F and p j-um specified, then T trum n st be determined. This can be done by picking a value for Tdrum, calculating Kp and checking with the Rachford-Rice equation, f2-421. A plot of f(V/F) vs. 

B. Explore. Since and Pdrum given, a sequential solution can be used. We can use the Rachford-Rice equation to solve for V/F and then find Xj, y, L, and V. 

C. Plan. Calculate K from DePriester charts or from Eq. (2-301. Use Newtonian convergence with the Rachford-Rice equation, Eq. (2=46), to converge on the correct V/F value. Once the correct V/F has been found, calculate X from Eq. (2=38) and y from Eq. (2=39). Calculate V from V/F and L from overall mass balance, Eq. (2=5). 

Eq. (2-38) or (2-39) to solve for V/F. The Rachford-Rice equation is then the check equation on Tj j-unj. If the Rachford-Rice equation is not satisfied, we select a new tenperature—using Eq. (2=49)—and repeat the procedure. In the second approach, we guess V/F and calculate from Eq. (2=38) or (2=32). We then determine the drum tenperature from this. The Rachford-Rice equation is again the check. If it 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,... 

C. Plan. Since this is a double trial and error, all calculations will be done on the conputer and summarized here. Newtonian convergence will be used for both the Rachford-Rice equation and the energy balance estimate of new drum tenperature. e = 0.02 is used for energy convergence (Eq. 2z56b). The Rachford-Rice equation is considered converged when... 

We can now follow exacdy the same steps as were used to derive the Rachford-Rice equation [Eqs. (2z 37 to (2-42) to derive two equations for the three-phase flash. 

If temperature and pressure are specified and correlations for the equilibrium parameters are available, these two forms of the Rachford-Rice equation can be solved simultaneously for /F and V/F. Then... 

Derive the Rachford-Rice equation for multiconponent flash distillation, and use it with Newtonian convergence to determine V/F... 

C2. Solve the Rachford-Rice equation for V/F for a binary system,... 

C5. Choosing to use V/F to develop the Rachford-Rice equation is conventional but arbitrary. We could also use L/F, the fraction remaining liquid, as the trial variable. Develop the Rachford-Rice equation as f(L/F). 

C7. Plot Eq. (2-40) vs. V/F for Example 2-2 to illustrate that convergence is not as linear as the Rachford-Rice equation. 

D22. A flash drum is separating a feed that is 50 wt % n-propanol and 50 wt % isopropanol with F=100 kmol/h. T um 0°C and Pdrum 101.3 kPa. Use the Rachford-Rice equation to find V/F. Note This does not have to be trial and error. Then find y and x. Determine K values from Raoult s law using Antoine equation for vapor pressure. Watch your units. 

Here, (f)] (4>g) is the fugacity coefficient of the liquid (gas) phase. The constants Kj. should be substituted into the Rachford-Rice equation instead of K . The value of inJ(P ) is taken from a previous iteration. Then, the new value of P is found from the known liquid molar fraction P... 

At each value of s, the system of equations (58) can be solved by Newton-Raphson iterations with regard to z,(fi) and P s). Another method of solution is that of subsequent substitutions based on Eq. (59), which plays the role of the Rachford-Rice equation for the problem considered [108]. At each iteration of the last method, Eq. (59) is solved at fixed values of z (fi) with regard to the pressure P s). Then, new values of Zj(e) are found fi om Eq. (58), on whose right-hand side, old Zj 8) are used. These iterations are repeated until convergence. 

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