Loop iteration

At this point in the inside-out method, the revised column profiles of temperature and phase compositions are used in the outer loop with the complex SRK thermodynamic models to compute updates of the approximate K and H constants. Then only one inner-loop iteration is required to obtain satisfactory convergence of the energy equations. The K and H constants are again updated in the outer loop. After one inner-loop iteration, the approximate K and H constants are found to be sufficiently close to the SRK values that overall convergence is achieved. Thus, a total of only 3 outer-loop iterations and 4 inner-loop iterations are required. 

To illustrate the efficiency of the inside-out method to converge this example, the results from each of the three outer-loop iterations are summarized in the following tables ... 

Outer-loop iteration Total liquid flows, lb moles/hr ... 

From these tables, it is seen that the stage temperatures and total liquid flows are already close to the converged solution after only one outer-loop iteration. However, the composition of the bottoms product, specifically with respect to the lightest component, C, is not close to the converged solution until after two iterations. The inside-out method does not always converge so dramatically, but is usually quite efficient,... 

A more general case than (1) is that in which fgut is specified but N is not. This amounts to a two-dimensional search in which the procedure and criteria in case (1) constitute an inner loop in an outer-loop search for the appropriate value of N. Since N is a small integer, this usually entails only a small number of outer-loop iterations. 

Clearly this double-loop-iteration procedure can be very slow. However, for some simple problems it is quite effective. 

Solutions of four cases of three- and four-component systems are presented by Tsuboka and Katayama (1976) the number of outer loop iterations ranged from 7 to 41. The four component case worked out by Henley and Seader (1981) is summarized in Example 14.9 they solved two cases with different water contents of the solvent, dimethylformamide. 

These simple model parameters become the main (or "outer loop") iteration variables, the role played by the primitive variables temperature, pressure, vapor and liquid composition and phase rates in Class I and Class II methods. 

The new outer loop iteration variables are relatively free of interaction with each other, and are relatively independent of the primitive variables, hence precise initialization is not critical to good algorithm performance. 

The inner loop iteration variable for each stage is a unique combination of temperature and phase ratio which eliminates the need to make a distinction between wide-and narrow-boiling systems. In certain cases there are additional inner loop variables. 

Example type Components K-value model Enthalpy models No. stages. No.outside loop iterations Avg. no. inside loop iterations CPU. time (sec.)... 

The difficulties associated with highly nonideal multi-stage systems have been overcome by introducing a simple model for the composition dependence of K-values. In keeping with the spirit of the inside-out concept, the parameters of the simple model become outside-loop iteration variables, and are determined by applying the actual models only in the outer loop. Further, they are as independent as possible of the primitive variables. 

Application to Simultaneous Phase and Chemical Equilibrium. The single-stage process with simultaneous phase and chemical equilibrium is another application of the inside-out concept where the Newton-Raphson method has been employed in a judicious way in the inside loop. There would appear to be no reaction parameter having characteristics that make it suitable as an outside loop iteration variable in the spirit of the inside-out concept. On the other hand, the chemical equilibrium relationships are simple in form, and do not introduce new thermophysical properties that depend in a complicated way on other variables. Thus it makes sense to include them in the inside loop, and to introduce the reaction extents as a new set of inside loop variables. 

Note Sanderson and Chien (22) reported 10 outside loop iterations using their conventional algorithm for the same system at the same pressure, but with T specified as 358 K. The number of inside loop iterations was not given. It is notable that each inside loop iteration of their algorithm requires that properties be calculated using the actual thermo-physical property models. 

Simultaneous Convergence Methods One drawback of some tearing methods is their relatively limited range of application. For example, the BP methods are more successful for distillation, and the SR-type methods are considered better for mixtures that exhibit a wide range of (pure-component) boiling points (see, however, our remarks above on modified BP and SR methods). Other possible drawbacks (at least in some cases) include the number of times physical properties must be evaluated (several times per outer loop iteration) if temperature- and composition-dependent physical properties are used. It is the physical properties calculations that generally dominate the computational cost of chemical process simulation problems. Other problems can arise if any of the iteration loops are hard to converge. 

Use for construct when the number of loop iterations is known beforehand. Likewise, use while construct when the number of loop iterations is not known beforehand. Cases include reading data from a file or from user input line by line until the end is encountered. Though this can be achieved by using a for loop with a conditional break statement, the while statement conveys the logic clearly. Some special cases require using do-while (when the first statement has to be executed before the conditional). 

This method starts off by fixing the temperature and pressure and iterating around the vapor fraction to calculate the equilibrium phase separation and compositions. The first step is an isothermal Hash calculation. If T and P are in fact the independent variables, the solution obtained in the first step is the desired solution. If either Tori and one more variable are specified, then another, outer iterative loop is required. The outer loop iterates around P or T (whichever is not fixed) until the other specified variable is satisfied. 

These are methods for calculating A -values and enthalpies in the inner loop. They are simple functions that use parameters derived from the rigorous property models. The parameters are updated at each outer loop iteration. 

The ordered directive causes the block following it to be processed sequentially in the same order as the loop iterations. Note that only the part of the... 

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