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:

$$F\left({\alpha }_g\right)={\displaystyle \sum }_{i=1}^n\frac{z_i\left(K_i-1\right)}{1+{\alpha }_g\left(K_i-1\right)}=0$$ (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, ${\alpha }_g{}^{new}$ , which is closer to the real answer than the previous guess, ${\alpha }_g{}^{old}$ , as follows:

$${\alpha }_g^{new}={\alpha }_g^{old}-\frac{F\left({\alpha }_g^{old}\right)}{F\text{'}\left({\alpha }_g^{old}\right)}$$ (13.6)

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

$${\alpha }_g^{new}={\alpha }_g^{old}+\frac{{\displaystyle \sum }_{i=1}^n\frac{z_i\left(K_i-1\right)}{1+{\alpha }_g^{old}\left(K_i-1\right)}}{{\displaystyle \sum }_{i=1}^n\frac{z_i{\left(K_i-1\right)}^2}{{\left[1+{\alpha }_g^{old}\left(K_i-1\right)\right]}^2}}$$ (13.7)

In this iterative scheme, convergence is achieved when

$$\left|{\alpha }_g^{new}-{\alpha }_g^{old}\right|<\epsilon$$ (13.8)

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

$$\text{Liquid Molar Fraction: }{\alpha }_l=1-{\alpha }_g$$ (13.9a)

$$\text{Percentage of Liquid: }\%L=100*{\alpha }_l$$ (13.9b)

$$\text{Percentage of Vapor: }\%V=100*{\alpha }_g$$ (13.9c)

$$\text{Vapor Phase Composition: }{y}_i=\frac{z_i{K}_i}{1+{\alpha }_g\left(K_i-1\right)}$$ (12.7)

$$\text{Liquid Phase Composition: }{x}_i=\frac{z_i}{1+{\alpha }_g\left(K_i-1\right)}$$ (12.11)