# 13.2: Objective Function and Newton-Raphson Procedure

We have seen that from a molar material balance applied to a two-phase system in equilibrium, and the definition of K_{i}, we can derive the Rachford and Rice Objective Function:

(13.3)

Equation (13.3) is a non-linear equation in one variable, and the Newton Raphson procedure is usually implemented to solve it. In general, Newton Raphson is an iterative procedure with a fast rate of convergence. The method calculates a new estimate, α_{g}^{new}, which is closer to the real answer than the previous guess, α_{g}^{old}, as follows:

(13.6)

Substituting (13.3) and (13.4) into (13.6),

(13.7)

In this iterative scheme, convergence is achieved when

(13.8)

where ε is a small number (ε = 1.0 x 10^{– 9} is usually adequate). After solving for , the liquid molar fraction and composition of each of the phases can be calculated as follows:

Liquid Molar Fraction: (13.9a)

Percentage of Liquid: (13.9b)

Percentage of Vapor: (13.9c)

Vapor Phase Composition: (12.7)

Liquid Phase Composition: (12.11)

### Contributors

Prof. Michael Adewumi (The Pennsylvania State University). Some or all of the content of this module was taken from Penn State's College of Earth and Mineral Sciences' OER Initiative.