2N Newton methods

In both the BP and SR methods (Secs. 4.2.5 and 4.2.7) the temperatures and the total flow rates are calculated separate of each other. An alternative is to calculate the temperatures and total flow rates together in a Newton-Raphson technique. The name 2N Newton comes from that there are two equations per stage for a total of 2 x N functions and variables per column for the Newton-Raphson. In all three of these approaches, the component flow rates are still calculated in an intermediate step. 

The 2N Newton-Raphson method. This method of Holland (8) is similar to the Tomich method but adds two innovations. The summation equa-... 

The multiplier, by being a correction to the ratio of total flow rates, adds a measure of stability to the solution by preventing the existence of negative flow rates. The steps of the 2N Newton-Raphson method follow those of Tomich. 

Compositions or flows calculated by component material balances in the theta method and 2N Newton-Raphson method. Sees. 4.2.5 and 4.2.6. 

The 2N Newton-Raphson method may be applied to any type of distillation column or to any system of interconnected columns. Absorbers, strippers, reboiled absorbers, and distillation columns are treated in Sec. 4-1. Selected numerical methods for solving the 2N Newton-Raphson equations are presented in Sec. 4-2. In Sec. 4-3, two methods for solving problems involving systems of columns interconnected by recycle streams are presented. 

Recommended procedures for solving problems involving single columns as well as systems of columns interconnected by recycle streams are summarized in Table 4-1 for the case where mixtures to be separated form ideal or near solutions throughout the column. As shown there, the 2N Newton-Raphson method is recommended for solving problems involving absorber-type columns (any column which does not possess both an overhead condenser and a reboiler such as absorbers, strippers, and reboiled absorbers). 

Systems of absorber-type and distillation-type columns with recycle streams 0 method for the distillation-type columns, the 2N Newton-Raphson method for absorber-type columns, and the capital 0 method for the system This chapter and Chaps. 2 and 3... 

Formulation of the 2N Newton-Raphson Method for an Absorber (or Stripper) with Any Number of Equilibrium Stages... 

Next, the 2N Newton-Raphson method is applied to reboiled absorbers, conventional distillation columns, and complex distillation columns, and then a procedure which makes use of the calculus of matrices for solving these equations is presented. 

The sketch for a typical reboiled absorber is shown in Fig. 4-2. To demonstrate the formulation of the 2N Newton-Raphson method for reboiled absorbers, two different sets of specifications are considered. 

Complex columns were defined in Chap. 3 and illustrated by Figs. 3-1 and 3-4. To illustrate the application of the 2N Newton-Raphson method to the solution of problems involving complex columns, consider the simple case where the sidestream Wt is withdrawn in the liquid phase from some interior plate p. The withdrawal of the sidestream gives rise to one specification in addition to those stated for conventional columns, in items 1 through 4 of Table 4-2. When this additional specification is taken to be either the total-flow rate Wx or the ratio Wl/Lp, the sets of specifications, independent variables, and functions for this complex column are the same as those stated in Table 4-2 except that either Wt or Wx /Lp should be added to each set of specifications. 

In all of the applications of the 2N Newton-Raphson method which follow, both the functions and the variables were normalized for the purpose of reducing roundoff error. The functions Fj and Gj are stated in a normalized form. The definition of the 0/s contains a normalizing factor, namely, (Lj/Vj)a. Temperatures were normalized by dividing each temperature by some base temperature. Although other more precise methods of normalization may be used such as the... 

To demonstrate the application of the 2N Newton-Raphson method to reboiled absorbers, Example 4-4 was solved. The statement and solution of this example appears in Table 4-8. The application of the 2N Newton-Raphson method to conventional and complex distillation columns is illustrated by Examples 4-5 and 4-6. Example 4-5 which involves a conventional distillation column, is a restatement of Example 2-7 (see Table 2-2). The... 

---