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Some books and learning resources use steady-state distillation columns as an example problem to introduce linear systems of equations in linear algebra and numerical computing; for instance:

I am an applied mathematician, and I have difficulties understanding the modelling part of these examples. I would like to get an explanation of the general idea of the chemical process behind these examples, assuming little previous knowledge of chemical engineering. From what I understand, these columns contain a sequence of "stages" that separate two different chemicals with different boiling points; but it is unclear to me how the 'balance' associated to these stages is computed. For instance, from the first reference:

$L$ mass or molar flow rate of the liquid reflux returned to the column from the condenser (mass time$^-1$ or mole time$^-1$); also generic flow rate of the liquid phase in the rectifying section

$\overline L$ = mass or molar flow rate of the liquid leaving the bottom of the column and entering the reboiler (mass time-1 or mole time-1); also generic flow rate of the liquid phase in the stripping section

$n$ = generic stage number, stage 1 is at the top of the column

$V$ = mass or molar flow rate of vapor leaving the top of the column and entering the condenser (mass time-1 or mole time-1); also generic flow rate of the vapor phase in the rectifying section

$\overline V$ = mass or molar flow rate of the gaseous boilup returned to the column from the reboiler (mass time-1 or mole time-1); also generic flow rate of the vapor phase in the stripping section

$x_n$ = mass or mole fraction of the light key in the liquid leaving stage $n$

$y_n$ = mass or mole fraction of the light key in the vapor leaving stage $n$

[...]

Material balance on stages $1$-$n$, the rectifying section of the column

$$ y_{n+1} = \frac{L}{V} x_n + y_1 - \frac{L}{V}x_0 $$

What are L and V? How does one obtain the equation? Which quantities are known and which are unknown?

You can assume a simplified model for clarity: e.g., two chemical species, ignoring the units at the top/bottom named "reboiler/condenser" and focus on the column itself.

Federico Poloni
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The first source that you linked is very confusing, a more satisfactory description is given here in a logical order (The Wikipedia pages for VLE, distillation and McCabe-Thiele are excellent). Unfortunately it cannot be made simpler by ignoring the reboiling and reflux as these variables actually have an effect on the equilibrium over the intermediate stages: if you input more energy at the bottom by more boiling, the balance will change. Without reboil/reflux, the process is called a 'flash' distillation and does not have intermediate stages.

Now to answer the question of 'the general chemical process'. The equilibrium balance at each stage is computed using the 'Vapour-Liquid-Equilibrium' curve for the mixture. For a mixture of two chemicals, it can be represented by 2 variables, x and y, which are the liquid and vapour fraction of one component. For common mixtures the data is available from experiments and in references like Perry's Chemical Engineer's Handbook. The composition at the neighbouring stages requires you to take into account the feed, reboiler and condenser as mentioned earlier.

Example of VLE curve from Wikipedia

To answer the question about the equation you reference:

$$ y_{n+1} = \frac{L}{V} x_n + y_1 - \frac{L}{V}x_0 $$

This can be arranged as

$$ Vy_{n+1} + Lx_0 = L x_n + V y_1 $$

V is total amount of stuff coming OUT of the top and L is the total amount falling back INTO the top of the column, and $y_{n+1}$ and $x_0$ are the fractions of that stuff that is component 1. This is a mass balance on 1 half of the column for 1 component, so on the LHS is what is going 'out' and on the RHS is what is going 'in'

Geoffrey Liddell
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