How do we compute the mole fraction of an ideal gas in a stream blended by mass flow controllers? For a non-ideal gas?

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?

• 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. – Martin - マーチン Feb 9 '16 at 7:53
• 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. – John Chris Feb 9 '16 at 17:42
• What have you done in your analysis so far that says this applies only to ideal gases, and not to mixtures in general. – Chet Miller Feb 22 '16 at 20:50
• Can anyone make any suggestions as to further analysis? I would be happy to add detail to the question. – John Chris Jul 28 '18 at 12:07