Band algorithm

Most problems involving the separation of nonideal solutions may be solved by use of the 6 method of convergence. When used to solve such problems, the 6 method does become slower and it may be necessary to place certain restraints on the calculational procedure. [The Almost Band Algorithm which is presented in Chap. 5 may be used to solve any problem for which the 9 method fails.]... 

ALMOST BAND ALGORITHMS OF THE NEWTON-RAPHSON METHOD... 

Although the Almost Band Algorithms use a large number of independent variables, far less computer time is required to obtain a solution to a given distillation problem than might be expected. The computational speed results from the use of selected techniques of sparse matrices and the characteristics of homogeneous functions. 

Highly nonideal solutions are characterized by the fact that the activity coefficients and the partial molar enthalpies are strongly dependent upon composition. In order to compute the partial derivatives of these quantities which are needed in the application of the Newton-Raphson method, it is convenient to choose compositions or component-flow rates as members of the set of independent variables. Numerous choices of the independent variables have been made.6, lf 8 13,15 17 19-20 To demonstrate the formulation of the Newton-Raphson method, the choice of independent variables proposed by Naphtali and Sandholm17 is used. The Almost Band Algorithm may be formulated for other choices of independent variables as shown by Gallun and Holland.7,8 9... 

ALMOST BAND ALGORITHMS FOR ABSORBERS AND STRIPPERS, INDEPENDENT VARIABLES ... 

ALMOST BAND ALGORITHMS FOR CONVENTIONAL AND COMPLEX DISTILLATION COLUMNS... 

The [N(2c + 1) + 3] Formulation of the Almost Band Algorithm for a Complex Column with One Sidestream... 

In the application of the Almost Band Algorithm to problems involving sharp separations, it was required that all of the corrected flow rates be positive and that the temperatures lie within the range of the curve fits. In this procedure, the vector correction Ax was reduced by an appropriate scalar a until all of the corrected flow rates were positive and the corrected temperatures were within the range of the curve fits, that is,... 

Comparison of the 2N Newton-Raphson Method with the Almost Band Algorithm for Mixtures which Form Ideal Solutions... 

After the Broyden correction for the independent variables has been computed, Broyden proposed that the inverse of the jacobian matrix of the Newton-Raphson equations be updated by use of Householder s formula. Herein lies the difficulty with Broyden s method. For Newton-Raphson formulations such as the Almost Band Algorithm for problems involving highly nonideal solutions, the corresponding jacobian matrices are exceedingly sparse, and the inverse of a sparse matrix is not necessarily sparse. The sparse characteristic of these jacobian matrices makes the application of Broyden s method (wherein the inverse of the jacobian matrix is updated by use of Householder s formula) impractical. 

The Almost Band Algorithm presented in Chap. 5 may be used to describe a single column in which either an azeotropic or extractive distillation is carried out provided that the accumulator contains only one liquid phase and one vapor phase. In many azeotropic distillation columns, the accumulator contains two liquid phases and one vapor phase. In order to describe a column whose accumulator contains three phases, the Almost Band Algorithms presented in Chap. 5 must be modified. To illustrate the modifications of the Almost Band Algorithm which are required in order to describe a column having three phases in the accumulator, the [N(2c + 1) -f 2] formulation of the Almost Band Algorithm for a column with a two-phase partial condenser is selected as the base case. Then the modifications required to describe a three-phase partial condenser are presented. 

Alternatively, a column modular approach may be used wherein the Almost Band Algorithm is applied successively to each column of the system. After one or more trials have been made on each column of the system, the capital 0 method of convergence is applied in a manner similar to that described for systems of interconnected columns. 

To demonstrate the application of the column modular method and system modular method (the Almost Band Algorithm for Systems), Examples 6-1 and 6-2 are presented. The statements of Examples 6-1 and 6-2 are presented in Table 6-3, and the solutions are given in Tables 6-4 through 6-6. Example 6-1 is a relatively easy problem to solve while Example 6-2 is more difficult. A flow diagram of the system involved in Examples 6-1 and 6-2 is shown in Fig. 6-11. 

The following procedure was used in the solution of Example 6-1 by the column modular method. After one complete trial had been made on column 1 by use of the Almost Band Algorithm, the component flow rates bit J so obtained were used as the feecLto the second column. Then one trial on column 2 was made by use of the Almgst Band Algorithm. Next the capital 0 method was applied to the most recentfets of terminal flow rates di%l, bitl 9 dit 2, and bu 2, and corrected sets were obtained which satisfied the component-material balances enclosing each column and the specified values of Bj and B2. To... 

Column modular method Almost Band Algorithm ... 

Example 6-1 was also solved by the Almost Band Algorithm for Systems (the system modular method) as well as the column modular method. Results obtained by each of these methods are presented in Tables 6-3 through 6-6. 

Before concluding this chapter, a useful algorithm proposed by Kubifcek22 for solving the equations for the Almost Band Algorithm for Systems is presented. 

Many separations which would be difficult to achieve by conventional distillation processes may be effected by a distillation process in which a solvent is introduced which reacts chemically with one or more of the components to be separated. Three methods are presented for solving problems of this type. In Sec. 8-1, the 0 method of convergence is applied to conventional and complex distillation columns. In Sec. 8-2, the 2N Newton-Raphson method is applied to absorbers and distillation columns in which one or more chemical reactions occur per stage. The first two methods are recommended for mixtures which do not deviate too widely from ideal solutions. For mixtures which form highly nonideal solutions and one or more chemical reactions occur per stage, a formulation of the Almost Band Algorithm such as the one presented in Sec. 8-3 is recommended. 

For the case where one or more reactions occur on each stage of an absorber or distillation column and the vapor and liquid phases form highly nonideal mixtures, a formulation of the Almost Band Algorithm is recommended. In the present formulation for the case where one or more chemical reactions occur on each stage of an absorber, the following choice of N(2c + 1 + r) independent variables and N(2c 4-1 + r) independent functions are made. In particular, for the case of one chemical reaction per stage, the independent variables and functions are taken to be... 

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