# 13.2: Objective Function and Newton-Raphson Procedure

- Page ID
- 503

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 and Attributions

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.