Formulation of the N r 2 Newton-Raphson Method

Consider first the case where the reaction given by Eq. (8-3) occurs in the liquid phase on each stage. The rate expression for this reaction is given by Eq. (8-12) and the moles of A reacted per unit time per unit volume on stage j is denoted by r]j. In this case r — 1 and N(r + 2) = 3N. The 3N independent variables are the Lj/Vj, Tj, and. The variable Lj/Vj is normalized by introduction of the new variable Qj which is defined as follows 

By taking (Lj/Vj)a to be equal to the most recent value of Lj/Vj found by the calculational procedure, it is evident that Oj approaches unity as the solution set of values of the variables is approached. 

Corresponding to these 3N independent variables, 3N independent functions must be selected. These are taken to be the dew-ooint functions Fj, the enthalpy balance functions G,-, and the reaction functions Rj. The dew-point functions are given by Eq. (4-3), and the reaction functions are given by Eq. (8-16). The enthalpy balance functions are formulated by stating Eq. (8-22) in functional form to give 

As discussed below, Gj = 0 when the solution set of values of the variables has been found. 

For any choice of values for the independent variables, the corresponding values of the dependent variables, the y, Vjxh and Lj may be found by use of the component-material balances, the equilibrium relationships, and the total-material balances. 

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Formulation of the N(r + 2) Newton-Raphson Method for Distillation Columns... 

The equations presented f r the formulation of the N(r + 2) Newton-Raphson method for absorbers may e applied to distillation columns in a manner similar to that demonstrated in (Thap. 4 for the 2N Newton-Raphson method for absorbers and distillation columns. 

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