and Iterative Methods
In the previous section, you learned that, given proper specifications, PRO/II can solve the distillation column equations to determine the val-ues of the dependent variables. In practice, actually solving the column equations is an inherently difficult task because they are nonlinear. Since all of PRO/II's distillation column solving algorithms are iterative meth-ods, a quick review of iterative methods is in order.
Iterative methods seek to solve equations of the general form:
f(x) = 0
The goal is to find a value of x (a root) that satisfies the above equation.
We will call this value x*. All iterative methods start with an initial guess, x0, of the solution and generate a sequence of guesses: {x1,x2, ...,xn,...} that hopefully approach x*.
Newton's Method
To illustrate a typical iterative technique, consider how the Newton method solves the nonlinear equation, f(x) = 0. Figure 42 illustrates the basic steps.
Figure 42:
Newton's Method
1. Linearize the function about the current guess, xn. The linearization about the guesses x0 and x1 are the tangent lines shown in Figure 42.
2. Solve for the root of the linearized equation and call it xn+1. Since the linearized equation approximates f(x), we hope that xn+1, the root of the linearized equation, approximates the root of the nonlin-ear equation.
3. Test for convergence. For example:
If , then xn ¬ xn+1, go to step (1).
4. Done: , so xn+1 is accepted as a root of f(x).
Nonlinear Pitfalls
There are few guarantees when it comes to solving nonlinear equations.
If you asked an equation solver to find the roots of the functions plotted in Figure 43, it would fail for cases (a) and (c). Case (a) fails because the function simply has no roots. Case (c) fails because of the continuum of roots. Case (b) would solve; however the root it finds will depend on your initial guess and might not be the one you desire.
Figure 43:
Fortunately, mathematical models of physical processes will usually have a steady-state solution if the process has a steady-state; this is the least you would expect from a model. In steady-state distillation, multi-ple solutions are uncommon except in problems involving chemical reaction or the formation of heterogeneous azeotropes.
The reason you should be concerned with these solvability issues is that you provide some of the equations (in the form of performance specifi-cations and choice of thermodynamic methods) that PRO/II must solve.
If your specifications cause the distillation equations to have no solu-tions, or infinitely many solusolu-tions, PRO/II will not be able to solve your flowsheet. Such specifications usually constitute physically invalid oper-ating conditions. For example, if you request an overhead product rate of 200 lb/hr from a column whose total feed rate is only 100 lb/hr, the mass balance equations will have no solutions. Although the defect in this par-ticular example is obvious, there are many examples where seemingly valid specifications result in a column that has no solutions.
Even when your equations are well posed and do have a solution, there is no guarantee that the iterative method will be able to locate the solution.
The success is somewhat dependent on the initial guess that you provide.
Figure 44 illustrates some interesting behavior of Newton's method.
Figure 44:
Initial Guesses for Newton's Method
■ Close to x*, the function looks very similar to the one in Figure 42, and an initial guess that is sufficiently close to x* will lead to conver-gence.
■ If, however, your initial guess is close to a local extremum, the value at the next iteration will be either very large or very small, depending on which side of the extremum your initial guess lies. An initial guess at point A, for example, will yield a very large value of x for the next guess. The algorithm will probably not be able to recover from this.
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■ If you were lucky enough to guess point B, the next guess would be very close to the solution as indicated in Figure 44. Points A and B illustrate a paradox in iterative methods. Although point A is closer to the correct solution (x*) than point B, an initial guess at point A will not converge but an initial guess at point B will.
■ Certain guesses close to the local minimum will lead to bounded oscillations that will never settle down on a steady state. In Figure 44, guess C leads to guess D, and guess D leads to guess C. This CDCD... sequence will repeat forever
The set of initial guesses that can be iterated to the solution is called the basin of attraction for that solution. A large basin is desirable; it means that the algorithm is forgiving of poor initial guesses. One of the great strengths of PRO/II's I/O distillation and Enhanced I/O algorithms is that they tend to have a large basin of attraction, and properly posed prob-lems usually converge. The Chemdist and Sure algorithms, on the other hand, tend to have smaller basins of attraction, therefore the quality of the initial guess can make the difference between convergence and diver-gence for these methods. The basin of attraction is also affected by the shape of the function you are trying to solve. Figure 45 shows two func-tions, f(x) and g(x), that have the same root but very different basins of attraction for Newton's method. As indicated in the graph, f(x) can be converged from a larger range of initial guesses.
Figure 45:
Basins of Attraction for Newton's Method
By now you should be convinced that it is important to provide good ini-tial guesses for the column algorithms. To assist you in generating these initial guesses, PRO/II provides several tools, called initial estimate gen-erators, which will be discussed shortly.
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Choosing Feasible Specifications
It is very easy to make mistakes in providing specifications, in fact, most column simulation errors are caused by improper specifications. Col-umns with physically infeasible specifications will fail to converge. You must provide the simulator with meaningful specifications that uniquely define the problem.
As you gain experience simulating distillation columns, you will find that certain types of specifications are more likely to converge than oth-ers. Consider the difference between imposing a purity and a recovery specification on a column.
Generally speaking, purity specifications are more likely to be infeasible than recovery specifications. For example, thermodynamics dictates that you will never be able to obtain 80 mole percent ethanol in a two stage column. This separation simply requires more than two stages, and the simulation will reflect this infeasibility by failing to converge. You have asked PRO/II to solve a system of equations that has no solution.
Recovery specifications, on the other hand, are almost always feasible and hence are safer constraints to impose on columns. Your column could recover 80% of the ethanol fed to it by simply maintaining a high overhead flowrate.
A flowrate specification is an example of a potentially dangerous specifi-cation in hydrocarbon systems that contain hydrogen and other non-con-densable gases. Since only part of the overhead product can be condensed, an overhead liquid flowrate specification can easily be infea-sible.
Key Components
Understanding the concept of light and heavy keys will help you choose appropriate specifications for your distillation problems. Distributed components are the components that appear, in significant quantities, in both the bottom and overhead products. The light key is the lightest of the distributed components and the heavy key is the heaviest. The light and heavy keys appear in small nonzero quantities in the bottom and overhead streams respectively. You can identify the key components by determining the percent recovery of each component in the overhead and making a table such as the one in Table 22.
Components should be listed in volatility order for this analysis. For the example in Table 22, note that components A and B do not appear in the bottom product and components H, I, and J do not appear in the over-head product. In other words, these components are non-distributed. You should be careful in formulating specifications based on non-distributed components. Such specifications are usually meaningless because the tendency of the non-distributed components to exit from the top or bot-tom of the column is so strong that operating adjustments will not affect their split. It is almost impossible to operate a crude column to recover 2% of the methane in the bottom product; methane is simply so volatile at the operating conditions of a crude column, that all of it will exit in the overhead stream. This is an example of a specification that leads to a dis-tillation operation with no solutions, like the function in Figure 43(a).
At the opposite extreme, if you specify that 100% of the methane exits in the overhead stream, your crude column would have an infinite number of solutions, like the function in Figure 43(c). Although this specifica-tion is feasible, it is not a meaningful constraint on the column's opera-tion because 100% of the methane would have exited in the overhead whether you asked for it or not. Although this specification may be met, it will not affect the operations of the crude column. It is vital that your specifications contain quantities that the column variables have some control over. This is why it is safe to formulate distillation specifications around key component recovery.
Table 22: Separation Keys COMP NBP %OVHD Most Volatile A 50 100
B 65 100
Light Key C 79 98
Distributed Components D 102 50
E 120 12 F 125 15 Heavy Key G 135 2
H 150 0 I 270 0 Least Volatile J 290 0
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Tip: For modeling purposes, non-distributed components can be combined to simplify the modeling effort. This will help to speed up the simulation.The separation key table is also useful for determining the ideality of the separation. In Table 22, components E and F do not distribute as you would expect from their normal boiling points. You would expect that proportionally more of component E would appear in the column over-head product than component F. Therefore, there is some non-ideal behavior with one or both of these components. Hydrocarbon systems usually distribute according to their volatilities, making simplified analy-ses possible. Unfortunately, systems with strong chemical interactions are not as easy to analyze. For these systems, the presence of one chemi-cal could radichemi-cally alter the behavior for another. In fact, this is the prin-ciple used in extractive and azeotropic distillation.