A rigorous partial NR method for multicomponent, multistage vapor-liquid separations can be devised that is midway between the complexity of the BP/SR methods on the one other and the NR methods on the other hand. The first major step in the procedure is to solve the modified M equations for the liquid-phase mole fractions by the usual tridiagonal-matrix algorithm. Then, in the second major step, new sets of stage temperatures and total vapor flow rates leaving a stage arc computed simultaneously by a Newton—Raphson method. These two major steps are repeated until a sum-of-squares criterion is satisfied. For this partial NR method:

(a) Write the two indexed equations you would use to simultaneously solve for a new set of 7; and Y,.

(b) Write the truncated Taylor series expansions for the two indexed equations in the 7’ and V, inknowns, and derive complete expressions for all partial derivatives, except that derivatives of physical properties with respect to temperature can be left as such. These derivatives are subject to the choice of physical property correlations.

(c) Order the resulting linear equations and the new variables 7’ and mto a Jacobian matrix that will permit a rapid and efficient solution.