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The iterative method I have learned to make flash calculations for a binary system where I know the inlet feed (F [mol/s]), total composition of the two components (z1 and z2), the amount of the mixture which is evaporated (q) and the total pressure (P [mmHg]) is as follows:

  1. We seek T, xi (composition of liquid) and yi (composition of gas). We know F, zi, q, P.
  2. Guess T
  3. Calculate P_1° and P_2°. In our course we are using Antoine's equation to do this. Antoine's equation can be written in many ways but lets for simplicity stick with this version. That is: P_i°(T)=10^{A_i-\frac{B_i}{C_i+T}} where T is in °C and P in mmHg. A_i,B_i,C_i are the Antoine constants and they are known.
  4. Calculate x_i and y_i. For simplicity we say that \gamma_i=1. This gives us that: x_i=\frac{Fz_i}{V\frac{P_i°(T)}{P}+L} and y_i=\frac{P_i°(T)x_i}{P} where V is the gas outlet stream (V=q\cdot F) and L is the liquid outlet stream (L=(1-q)\cdot F).
  5. If \sum x_i=1 \underline{and} \sum y_i=1 you have found the correct T, x_i and y_i. Else go back to 2.

So my question is. Do we really need to calculate \underline{both} y_i and x_i? If I find the temperature which satisfies \sum x_i=1 doesn't that guarantee that also \sum y_i=1? I plotted this and played around with different values for the Antoine constants and values for F, q and P. Here is a link to my plot: https://www.desmos.com/calculator/m37slkdebb. For all of my cases it was true that if for a temperature T \sum x_i=1 then also \sum y_i=1 for that same temperature T. I asked my professor about this and he insisted that I would have as a requirement to stop my iteration when \sum x_i=1 \underline{and} \sum y_i=1.

Am I correct that \sum x_i=1 guarantees \sum y_i=1?

Are there cases I have missed where you can have several temperatures where \sum x_i=1 but only one of these temperaturs satisfies both \sum x_i=1 and \sum y_i=1?

What happens when \gamma_i\neq1, is this simplification too much? If i use another model for P_i° what will happen then?

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    \begingroup In my judgment your professor is correct. The sum of the mole fractions of vapor is a constraint on this problem. There is only one T and set of x's that satisfies this constraint. \endgroup Commented Feb 25, 2021 at 12:53

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