Newton-Raphson method Almost Band Algorithm

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

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... 

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. 

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. 

In the analysis of continuous distillation columns at total reflux, three different calculational procedures are presented namely, the 6 method of convergence and two formulations of the Newton-Raphson method, one analogous to the 2N Newton-Raphson method and the other to the Almost Band Algorithm. The formulation of each of these methods for continuous distillation columns at total reflux is presented below. 

Although the same enclosures are used for the component-material balances in the formulation of the Almost Band Algorithm as were used in the formulation of the 2N Newton-Raphson method, it is convenient in this case (because of the form of the phase equilibrium relationships) to include Tr+l, Ts, vr+lt,, and usl in the set of independent variables. Thus, in the formulation of the Almost Band Algorithm, the following choice of independent variables is made. 

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