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Consider a gas blending system consisting of two mass flow controllers, A and B, set to control flow to $f_\mathrm{A}$ and $f_\mathrm{B}$, respectively. Consider that flow controller A is metering the pure gas of interest, which has an actual mole fraction of $x_\mathrm{A}$. Flow controller B is metering a make-up gas which coincidently may contain trace amounts of gas A at a mole fraction of $x_\mathrm{B}$.

The mass flow controllers provide continuous estimates of $f_\mathrm{A}$ and $f_\mathrm{B}$; these are measured quantities. Suppose also that we did some experiments and found the values of $x_\mathrm{A}$ (very close to unity or $\pu{1E6 ppm}$) and $x_\mathrm{B}$ (approaching zero, likely ones to tens of ppm).

To first order, the following equations provide reasonable estimates of the concentration of gas A in the blended stream at near atmospheric pressure, room temperature. Experiments were carried out with $\ce{CO2}$ (gas A) in $\ce{N2}$ (gas B). A $\ce{CO2}$ gas analyzer was used to estimate the mole fraction of gas A in the blend, $x_\mathrm{T}$. We can compute the mass flow ratio of gas A by

$r = \dfrac{f_\mathrm{A}}{(f_\mathrm{A} + f_\mathrm{B})}$

or

$r = \dfrac{f_\mathrm{A}}{f_\mathrm{T}}$

where $f_\mathrm{T} = f_\mathrm{A} + f_\mathrm{B}$. Further, we compute the mole fraction of gas A in the blend as

$x_\mathrm{T} = r \cdot \left(x_\mathrm{A} - x_\mathrm{B} \right) + x_\mathrm{B}$

My question again is: How do we compute the mole fraction of an ideal gas in a stream blended by mass flow controllers?

I'd like to do this in a rigorous manner, or correctly, possibly accounting for any non-ideal behavior of $\ce{CO2}$ in a $\ce{N2}$ make-up gas.

I came across this reference and believe the key to my problem is addressed therein. The solution may involve the empirical rule of Lewis & Randall whereby the mole fraction of a component is equated to a ratio of fugacities of the component in a mixture to the pure component. Certainly being able to compute the virial coefficients for the components may prove useful. A key may also be converting a flow into a volume with an assumed time step.

Should I attempt to apply the some of the equations presented in the reference noted above?

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    $\begingroup$ Welcome to Chemistry.SE! Take the tour to get familiar with this site. Mathematical expressions and equations can be formatted using $\LaTeX$ syntax. If you want to know more, please have a look here and here. We prefer to not use MathJax in the title field, see here for details. This appears to be a homework question, please share your thoughts and attempts towards the solution. $\endgroup$ – Martin - マーチン Feb 9 '16 at 7:53
  • $\begingroup$ Thank you for the comments. I can apply latex formatting; I wasn't aware this was supported. This is not a homework question. I will happily explore the solution further if deemed necessary. Please provide feedback as to whether my most recent edits are okay, or whether I should add more content / show more work towards the solution. $\endgroup$ – John Chris Feb 9 '16 at 17:42
  • $\begingroup$ What have you done in your analysis so far that says this applies only to ideal gases, and not to mixtures in general. $\endgroup$ – Chet Miller Feb 22 '16 at 20:50
  • $\begingroup$ Can anyone make any suggestions as to further analysis? I would be happy to add detail to the question. $\endgroup$ – John Chris Jul 28 '18 at 12:07

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