17.3: Solution Algorithms for VLE Problems

Last updated
Save as PDF

Page ID: 574

Michael Adewumi
Pennsylvania State University via John A. Dutton: e-Education Institute

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

We have essentially reduced the typical VLE problem to that of solving a system of non-linear algebraic equations. The objective function is the nucleus of the VLE calculation. Rather than solve the material balance equations in their elemental form, the use of an objective function will allow for a more stable iteration scheme. We have seen that three objective functions are available for this calculation:

\[F_{y}\left(\alpha_{g}\right)=\sum_{i=1}^{n} \frac{z_{i} K_{i}}{1+\alpha_{g}\left(K_{i}-1\right)}-1=0 \label{17.7a}\]

\[F_{x}\left(\alpha_{g}\right)=\sum_{i=1}^{n} \frac{z_{i}}{1+\alpha_{g}\left(K_{i}-1\right)}-1=0 \label{17.7b}\]

\[F\left(\alpha_{g}\right)=\sum_{i=1}^{n} \frac{z_{i}\left(K_{i}-1\right)}{1+\alpha_{g}\left(K_{i}-1\right)}=0 \label{17.7c}\]

Expression \ref{17.7c} is the Rachford-Rice Objective Function; we have proven that its numerical solution is cleaner than that of equations (17.7a) and (17.7b). We also discussed that the key problem in solving for “α_g” is the need for values for the equilibrium ratios (K_i’s). With the aid of equations (17.5) and (17.6) these problems can be solved simultaneously by iteration techniques.

In the problem we have posed, we have (3n+1) unknowns:

Contact your instructor if you are unable to see or interpret this graphic.

i=1,2, … n
y_i i=1, 2, … n
x_i i=1, 2, … n

and α_g.

And we also have (3n+1) equations:

\[K_{i}=\frac{\phi_{i}^{L}\left(P, T, x_{i}\right)}{\phi_{i}^{V}\left(P, T, y_{i}\right)} i=1,2, \ldots n \label{17.8a}\]

\[y_{i}=K_{i} x_{i} \quad i=1,2, \dots n \label{17.8b}\]

\[(x_{i}=\frac{z_{i}}{1+\alpha_{g}\left(K_{i}-1\right)} \quad i=1,2, \ldots n \label{17.8d}\]

\[\sum_{i=1}^{n} \frac{z_{i}\left(K_{i}-1\right)}{1+\alpha_{g}\left(K_{i}-1\right)}=0 \label{17.8d}\]

where P, T, and z_i are the values that we know. See how this is only a restatement of the original set of equations that contain the equilibrium conditions, material balances, and molar fraction constraints:

Contact your instructor if you are unable to see or interpret this graphic. i=1,2, … n(17.9a)
i=1,2, … n (17.9b)
i=1,2, … n (17.9c)
or (17.9d)

Since we have the same number of equations as unknowns, this system is determined and has a unique solution. To solve this system of simultaneous non-linear equations, there are two categories of solution techniques:

Newton-type methods,
Substitution-type methods.

Newton-type methods for more than one unknown require finding (n x n) elements of a Jacobian matrix at every iteration step. This could be computationally expensive. Additionally, the Newton Raphson method requires a very good initial guess (i.e., close to the actual solution) to guarantee convergence to the true values. This is not always possible, especially at the start of the procedure.

The most popular method, and the easiest to implement, is the substitution type. However, substitution type methods can be quite slow for some conditions of interest. In these sort of cases, we either switch to a Newton-Raphson solver (that has a much better rate of convergence) or implement an acceleration procedure to the substitution method itself.

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.