The 2N Newton-Raphson Method

When the Newton-Raphson method7 is formulated in terms of two independent variables per stage, the temperature 7 and the flow ratio Lj/Vj, the resulting procedure is called the IN Newton-Raphson method. (Originally, this procedure was called the multi-0 method.16,17) When both the vapor and liquid phases form ideal solutions on each stage, this procedure is an exact application of the Newton-Raphson method. 

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). 

For solving problems involving distillation-type columns (any column possessing both an overhead condenser and a reboiler), the Chaps. 2 and 3 is recommended. For systems of coli absorber-type and distillation-type columns, it is recoi Newton-Raphson method be used for the absorber-type c 

Conventional and complex distillation columns 0 method Chaps. 2 and 3 

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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. 

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... 

Applications of approximate formulations of the Newton-Raphson method such as the ones proposed by Sujata22 and Holland17 may fail to converge for some examples. For example, the 2N Newton-Raphson method converged for the test problem called Example 3 by Boyum4 while the approximate methods of Sujata and Holland failed. 

To illustrate the application of the column modular approach, consider the particular system of columns shown in Fig. 4-5, which consists of a reboiled absorber (column 1) and a distillation column (column 2). For such a system, a combination of the 0 method and the IN Newton-Raphson method is recommended. The 2N Newton-Raphson method is used for solving the reboiled absorber and the 0 method is recommended for the distillation column. At the end of one complete trial for each of the two columns, the capital 0 method is applied to place the system in component-material balance and in agreement with the specified values of the terminal flow rates. 

In the application of the 2N Newton-Raphson method, Vx is regarded as specified and the temperature T0 of B2 entering column 1 or the intercooler duty Qc, is taken to be the new independent variable. In this case the 2N + 1 variables X and the IN + 1 functions f are as follows... 

The proposed calculational procedure consists of first making one trial calculation on the reboiled absorber by use of the 2N Newton-Raphson method and then one trial on the distillation column by use of the 0 method. Then the capital 0 method is applied one time to the system in order to place it in overall component-material balance and in agreement with the specified values of the terminal flow rates. To initiate the calculational procedure the composition of any recycle stream which is needed is assumed. After the first trial through the system, the composition of such recycle streams found by the method are used. The steps of the proposed calculational procedure follow. 

For distillation-type columns (Examples 4-5 through 4-7) the 0 method, which is presented in Chaps. 2 and 3, is seen to be from 5 to 20 times faster than the 2N Newton-Raphson method. 

Since the number of operations required to invert or to find the LU factorization of the jacobian matrix requires n3 or n2 operations, respectively, it is fortunate that most absorber-type problems are characterized by mixtures which contain a relatively large number of components and by columns which contain a relatively small number of plates. For problems of this type, the 2N Newton-Raphson method is best suited and is recommended. For separations carried out in distillation type columns involving large numbers of components and plates, the 0 method is the most efficient and is recommended. 

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