# 17.4: Successive Substitution Method (SSM)

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- 575

In a substitution-type method, we start with initial guesses for all of the unknowns and loop around the equations to obtain “better” approximations for each of them. We test the goodness of the solution at every time step by comparing the new, better approximation to the previous guess. If the correction is small under certain convergence criteria, the procedure is stopped and we use the results from the last iteration as the final answer.

As we discussed in the previous section, reliable values for the equilibrium constants (K_{i}’s) must be obtained before we can solve the Rachford-Rice Objective Function. Generally, first estimates for equilibrium constants are calculated by using Wilson’s empirical equation (equation 13.15). However, Wilson’s correlation yields only approximate values for equilibrium ratios. Here, we apply thermodynamic equilibrium considerations in order to obtain more reliable predictions for K_{i}’s.

Let us recall that for a system to be in equilibrium, any net transfer (of heat, momentum, or mass) must be zero. For this to be true, all potentials (temperature, pressure, and chemical potential) must be the same in all of the phases. Provided that the temperature and pressure of both phases are the same, a zero net transfer for all components in the mixture results when all chemical potentials are the same. A restatement of this: all *fugacities* for *all* components in each phase are equal. Since fugacity is a measure of the potential for transfer of a component between two phases, equal fugacities of a component in both phases results in zero net transfer. SSM (Successive Substitution Method) takes advantage of this fact. Equation (17.5) showed that the fugacity criterion for equilibrium allows writing the equilibrium ratios K_{i} as a function of fugacity coefficients as follows:

\[K_{i}=\frac{\phi_{i}^{L}}{\phi_{i}^{V}} \label{17.10}\]

Equation (17.10) presupposes equality of fugacity (\(f_{i}^{V}=f_{i}^{L}\)), but during an iterative procedure, fugacities may not be equal (\(f_{i}^{V} \neq f_{i}^{L}\)) until convergence conditions have been attained. Therefore, if fugacities are not equal (convergence has not been achieved), Equation \ref{17.10} becomes:

\[K_{i}=\frac{\phi_{i}^{L}}{\phi_{i}^{V}}=\frac{f_{i}^{L} /\left(x_{i} P\right)}{f_{i}^{V} /\left(y_{i} P\right)}=\frac{y_{i}}{x_{i}}\left(\frac{f_{i}^{L}}{f_{i}^{V}}\right) \label{17.11}\]

Using the above equation, a correction step can be formulated to improve the current values of K_{i}. The SSM-updating step is written as:

\[ \begin{align} K_{i}^{n+l} &=\left(\frac{y_{i}}{x_{i}}\right)^{n}\left(\frac{f_{l i}}{f_{g i}}\right)^{n} \\[4pt] &=K_{i}^{n}\left(\frac{f_{l i}}{f_{g i}}\right)^{n} \label{17.12a} \end{align}\]

SSM updates all previous equilibrium ratios (K_{i}) using the fugacities predicted by the equation of state. This iteration method requires an initial estimation of K_{i}-values — for which Wilson’s correlation is used. It can easily be concluded that the convergence criteria will be satisfied whenever the fugacity ratios of all the components in the system are close to unity. Such condition is achieved when the following inequality is satisfied:

\[\sum_{i}^{n}\left(\frac{f_{l i}}{f_{g i}}-1\right)^{2}<10^{-14} \label{17.12b}\]

## 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.