Easily compute a good approximation of heat conductivity of cooled semi-vacuum (water vapor + air)

Question

I am preparing a classroom demonstration about heat, including a demo of how a satellite loses heat only radiatively, since it's flying through a vacuum. I'd like to make the vacuum-production apparatus very simple and cheap.

The kind of pump used to semi-evacuate wine bottles to help preserve the flavor (supposedly) is the right kind of item in terms of cost (as cheap as $6); a slight modification of one of these improved the seal on its plunger dramatically, though I still doubt the pressure in the vacuum chamber gets much below 1/4th bar.

To enhance the vacuum, I'm venting steam from a kettle into the vacuum chamber (a cheap plastic jar with the wine-pump check valve attached through a hole in the lid), screwing the lid back on, pumping out as much of the hot water vapor as possible, then cooling the jar in the freezer.

I'm wondering if this really reduces the heat conductivity of the remaining gases (mainly, water vapor) very significantly. Reducing the density of a gas doesn't really reduce the heat conductivity very dramatically, and without such a reduction, the point of the exercise (simulating vacuum) is lost. I can measure the temperature easily enough, and I will get a vacuum gauge to measure the pressure, but it would be good to work from an approximate formula based on those two measures to determine the heat conductivity. I assume there will be little but water vapor in the jar when I put on the lid. I can find tables for water vapor heat conductivity above freezing, but at the temperatures and pressures where H2O only sublimates, I've found nothing so far.

What's a good approximate formula to use? Bear in mind that I'd also like to use not only regular water ice but also dry ice (packed around the jar) as the background for the satellite-simulating heat sensor to radiate against, so it would be best if the formula is a good approximation at dry-ice temperatures too. (Cosmic background is obviously not achievable cheaply, but the effect of even greater radiative cooling can reasonably be extrapolated from various other cooling approaches, such as burying the jar with water ice and dry ice.)

Answer 1

In the molecular flow region of pressure, the thermal conductivity of an ideal, monatomic gas is obtained by this equation.

$$ k = \frac{1}{\pi^{3/2} d^2}\sqrt{k_B^3T/m} $$

where $d$ is the collision or molecular diameter and m is the molar mass divided by Avogadro's number. It is independent of pressure.

Approximations have been derived for non-monatomic gases based on collision cross sections and collision integrals. In a nutshell, thermal conductivity is still independent of pressure.

Once the mean free path of the gas reaches the diameter of the container, the gas crosses to Knudsen flow. The mean free path is inversely proportional to pressure as

$$ \lambda = \frac{k_B T}{\sqrt{2} \pi \sigma^2 p} $$

where $\sigma^2$ is the collision cross sectional area (taken to first order as roughly the same as $d^2$ in some cases).

Once we reach the point where $\lambda > d_{container}$, we can assume the thermal conductivity of a gas in the container is negligible.

Use a water molecule at 0.275 nm diameter in a container with 6 cm diameter at 25 $^\mathrm{o}$C. The pressure needed to cross to Knudsen flow is 0.20 Pa. At or below this pressure should do what you need.

When the walls of the container are below the sublimation temperature of water vapor at the given partial pressure, the walls will be coated with ice. The sublimation temperature of pure water vapor at 0.20 Pa can be obtained from the picture in this post.

Consider also a tube with radius $r_t$ at $T_t$ containing a probe (with zero thickness) at its center-line at $T_p$. Assuming black-body radiation for the tube and probe, the ratio of radiation to conduction in the system to first order is

$$ R_{rc} = \frac{\sigma\left(T_p^4 - T_t^4\right)}{k \left(T_p - T_t\right) / r_t} $$

A computer application (python) could be used to plot this ratio as a function of the input parameters $(T_t, T_p, r_t)$ to appreciate the behavior when the tube is not evacuated (to have $k \rightarrow 0$).