Editing Flash evaporation (section)

==Equilibrium flash of a multi-component liquid==
The '''equilibrium flash''' of a multi-component liquid may be visualized as a simple [[distillation]] process using a single [[equilibrium stage]]. It is very different and more complex than the flash evaporation of single-component liquid. For a multi-component liquid, calculating the amounts of flashed vapor and residual liquid in equilibrium with each other at a given temperature and pressure requires a trial-and-error [[Iterative method|iterative]] solution. Such a calculation is commonly referred to as an equilibrium flash calculation. It involves solving the ''Rachford-Rice equation'':<ref>
{{cite book|author=Harry Kooijman and Ross Taylor|url=http://www.chemsep.com/downloads/docs/book2.pdf|title=The ChemSep Book|edition=2nd |year=2000|publisher=H.A. Kooijman and R. Taylor |isbn=3-8311-1068-9}} See page 186.</ref><ref>[https://www.e-education.psu.edu/png520/m13_p2.html Analysis of Objective Functions] (Pennsylvania State University)</ref><ref>[http://www.mathworks.com/matlabcentral/fileexchange/loadFile.do?objectId=9341&objectType=file Flash Calculations using the Soave-Redlich-Kwong equation of state] (view full-size image)</ref><ref name="whitson">Curtis H. Whitson, Michael L. Michelsen, ''The Negative Flash'', Fluid Phase Equilibria, 53 (1989) 51&ndash;71.</ref>

:<math>\sum_i\frac{z_i \, ( K_i -1)}{1 + \beta \, (K_i - 1)}=0</math>

where:

* ''z<sub>i</sub>'' is the mole fraction of component ''i'' in the feed liquid (assumed to be known);
* ''β'' is the fraction of feed that is vaporised;
* ''K<sub>i</sub>'' is the equilibrium constant of component ''i''.

The equilibrium constants ''K<sub>i</sub>'' are in general functions of many parameters, though the most important is arguably temperature; they are defined as:

:<math>y_i = K_i \, x_i</math>

where:

* ''x<sub>i</sub>'' is the mole fraction of component ''i'' in liquid phase;
* ''y<sub>i</sub>'' is the mole fraction of component ''i'' in gas phase.

Once the Rachford-Rice equation has been solved for ''β'', the compositions ''x<sub>i</sub>'' and ''y<sub>i</sub>'' can be immediately calculated as:

:<math>\begin{align}
       x_i &= \frac{z_i}{1+\beta(K_i-1)}\\
       y_i &= K_i\,x_i.
       \end{align}</math>

The Rachford-Rice equation can have multiple solutions for ''β'', at most one of which guarantees that all ''x<sub>i</sub>'' and ''y<sub>i</sub>'' will be positive. In particular, if there is only one ''β'' for which:

:<math>\frac{1}{1-K_\text{max}} = \beta_\text{min} < \beta < \beta_\text{max} = \frac{1}{1-K_\text{min}}</math>

then that ''β'' is the solution; if there are multiple  such ''β'''s, it means that either ''K''<sub>max</sub><1 or ''K''<sub>min</sub>>1, indicating respectively that no gas phase can be sustained (and therefore ''β''=0) or conversely that no liquid phase can exist (and therefore ''β''=1).

It is possible to use [[Newton's method]] for solving the above water equation, but there is a risk of converging to the wrong value of ''β''; it is important to initialise the solver to a sensible initial value, such as (''β<sub>max</sub>''+''β<sub>min</sub>'')/2 (which is however not sufficient: Newton's method makes no guarantees on stability), or, alternatively, use a bracketing solver such as the [[bisection method]] or the [[Brent method]], which are guaranteed to converge but can be slower.

The equilibrium flash of multi-component liquids is very widely utilized in [[Oil refinery|petroleum refineries]], [[petrochemical]] and [[chemical plant]]s and [[natural gas processing]] plants.