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13.2: Objective Function and Newton-Raphson Procedure

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

    \[F\left(\alpha_g\right)=\sum_{i=1}^n \dfrac{z_i\left(K_i-1\right)}{1+\alpha_g\left(K_i-1\right)}=0 \nonumber \]

    Equation \(\PageIndex{1}\) 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, \(\alpha_g{ }^{\text {new }}\), which is closer to the real answer than the previous guess, \(\alpha_g{ }^{\text {old }}\), as follows:

    \[\alpha_g^{\text {new }}=\alpha_g^{\text {old }}-d\frac{F\left(\alpha_g^{\text {old }}\right)}{F^{\prime}\left(\alpha_g^{\text {old }}\right)} \nonumber \]

    Substituting \(\PageIndex{1}\) and (13.4) into \(\PageIndex{2}\),

    \[\alpha_g^{\text {new }}=\alpha_g^{\text {old }}+\dfrac{\sum_{i=1}^n \frac{z_i\left(K_i-1\right)}{1+\alpha_g^{\text {old }}\left(K_i-1\right)}}{\sum_{i=1}^n \dfrac{z_i\left(K_i-1\right)^2}{\left[1+\alpha_g^{\text {od }}\left(K_i-1\right)\right]^2}} \nonumber \]

    In this iterative scheme, convergence is achieved when

    \[\left|\alpha_g^{\text {new }}-\alpha_g^{\text {old }}\right|<\varepsilon \nonumber \]

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

    Liquid Molar Fraction: \(\alpha_l=1-\alpha_g\)

    Percentage of Liquid: \(\% L=100 * \alpha_l\)

    Percentage of Vapor: \(\% V=100 * \alpha_g\)

    Vapor Phase Composition: \(y_i=\dfrac{z_i K_i}{1+\alpha_g\left(K_i-1\right)}\)

    Liquid Phase Composition: \(x_i=\dfrac{z_i}{1+\alpha_g\left(K_i-1\right)}\)

    This page titled 13.2: Objective Function and Newton-Raphson Procedure was last modified on Mon, 13 Apr 2026 01:35:47 GMT and is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Michael Adewumi (John A. Dutton: e-Education Institute) via source content that was edited to the style and standards of the LibreTexts platform.