Section 3.7.1.4: Flash Calculations with Fixed Relative Volatility

If the phase equilibrium properties of a mixture can be represented adequately by fixed relative volatilities, see notes here, then the equations describing a vapour-liquid `flash' system become particularly convenient to handle.

The situation is as illustrated in the diagram.

Specified Component Recovery

The simplest type of problem is when the recovery r_j of one species j (usually in the vapour product) is given in terms of the ratio of:

flow of component in vapour : flow of same component in feed

If we write:

r_j = v_j / f_j

Then we can calculate the ratio of vapour to liquid flows of the component:

s_j = v_j / l_j

Using the component material balance equation:

f_j = v_j + l_j

which with a bit of algebra gives:

s_j = r_j / (1 - r_j)

From the definition of relative volatility and `K value':

$\begin{displaymath}\alpha_{i,r} = \frac{K_i(T,P)}{ K_r(T,P)} = \frac{y_i/x_i}{y_r/x_r} \end{displaymath}$

If the total flow of e.g. the vapour stream is V then:

v_i = V y_i

So:

$\begin{displaymath}\frac{v_i}{v_r} = \frac{V y_i}{V y_r} = \frac{v_i}{v_r} \end{displaymath}$

Hence, using this and the corresponding relationship for the liquid stream we can write:

$\begin{displaymath}\alpha_{i,r} = \frac{v_i/l_i}{v_r/l_r} \end{displaymath}$

Problem 1

Show that for a two component flash this problem can be defined as follows:

Given component 1 as reference component, hence r₁and $\alpha_{2,1}$
Given the feed f₁ and f₂
The problem then involves as unknowns: s₁, s₂, v₁, l₁, v₂, l₂, r₂
There are thus 7 equations to be solved.
These can be written in a form suitable for direct solution.

In order to do this:

Write down all the relevant equations
Draw up an ordered incidence table
Rewrite the equations suitably rearranged as formulas and reordered for solution
Using the modelling language, create a spreadsheet to solve this version of the flash problem
Test for an equimolar feed of two species having relative volatility of 5 with a less volatile component recovery in the vapour of 5%.

Specified Vapour Fraction

Let the total fraction of the feed which is vaporised be specified, i.e.:

V_s = V / F

In this case it is not possible to arrange the equations for direct solution. However, if the reference component recovery is assumed (or `torn') then the above procedure, and program, can be used to determine all component flows, and hence to calculate V/F, for the assumed r_j

r_j is then adjusted iteratively until the equations are satisfied, i.e. until:

$\begin{displaymath}\phi = V_s - V/F = 0 \end{displaymath}$

Problem 2

For the same two component mixture as above:

Draw up a new ordered incidence table showing a single tear variable, when a value of V_s is specified. There is now at least one more unknown, r₁, and the above additional equation. It may be convenient to provide V as a variable and an equation to calculate it.
Write the ordered and rearranged set of equations. Note that the new equation is not rearranged but its l.h.s. is evaluated as written until it becomes zero.
For the same feed as above solve for a ratio of V/F of 0.5

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