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November 6th, 2019, 12:12 PM   #1
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Thermal Conductivity

Hello All

Currently trying to get through the question below, but any formula I find is too complex for the brief detail given, with regard thermal conductivity of gases.

It is proposed to use a katharometer to measure the amount (about 5%) of oxygen in nitrogen, in the presence of a small amount (0.5%) hydrogen. How constant would the proportion of hydrogen have to be in order to limit errors in measurement of % oxygen to + 0.1%?

The thermal conductivities are:

nitrogen 0.993
oxygen 1.052
hydrogen 6.993
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November 6th, 2019, 06:15 PM   #2
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Question 1. Are you using a specific model for the thermal conductivity of a mixture of gases? If we were looking for specific heats, energy, or similar properties of a mixture of ideal gases, we would use a mass-weighted average. In this type of model, the thermal conductivity would look something like this:
$\displaystyle k_{mixture} = [1-f_{O2}-f_{H2}] k_{N2} + f_{O2} k_{O2} + f_{H2} k_{H2}$, where $f_i$ (or $mf_i$, or $c_i$) is the mass fraction of gas $i$.

However, I also know of literature that claims that kinetic theory does not predict the thermal conductivity of gas mixtures well, and slightly more complex empirical formulas have been proposed.

Either way, you are going to have some equation $k_{mixture} = k(f_{N_2},f_{O_2},f_{H_2})$.

Since no information was given about the accuracy of the katharometer or the Wheatstone bridge, I assume the question is about keeping the conductivity of the mixture sufficiently constant.
If $f_{H2} = 0.005 \pm u_{fH2}$, then barring any other sources of uncertainty,
$\displaystyle k_{mixture} = k(0.945,0.05,0.005) \pm \frac{\partial k_{mixture}}{\partial f_{H2}} u_{fH2}$.
Whatever equation you use for the thermal conductivity of the mixture, take the partial derivative with respect to the hydrogen mass fraction and evaluate at the nominal conditions. Then,
$\displaystyle u_k = \frac{\partial k_{mixture}}{\partial f_{H2}} u_{fH2}$.

But then, you are going to measure (indirectly) the thermal conductivity, and back out the mass fraction of the oxygen, and it is this number you want to keep within ±0.1 %.
$\displaystyle u_{fO2} = \frac{\partial f_{O2}}{\partial k_{mixture}} u_k$.

So, evaluate both partial derivatives at the nominal conditions, and solve,
$\displaystyle \left| \pm 0.001 \right| \geq \left| \frac{\partial f_{O2}}{\partial k_{mixture}} \frac{\partial k_{mixture}}{\partial f_{H2}} u_{fH2} \right|$
for the maximum allowable uncertainty $u_{fH2}$.

If your question was specifically about how to calculate the thermal conductivity of the mixture, then I can guide you to some literature, but I am not an expert and could not guarantee the reliability of one model over another.
Thanks from Baggy
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conductivity, thermal

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