# Effect of humidity and $CO_2$ on heat transfer coefficient for a small object (breath on a thermistor)

I am writing a paper on monitoring breathing using small thermistors (in my case a narrow cylinder of 1.5 mm x 3 mm) placed just beyond the nose and mouth (e.g. 1–2 cm) and I need to explain the difference in heat transfer coefficient during expiration versus inspiration (when ambient air passes over the sensor), independent of the effect of gas velocity (the effect of which is later used to estimate breath volumes).

There are three main differences between exhaled breath and air:

• Exhaled air is composed of about 4% carbon dioxide (and only 17% oxygen)
• The humidity is very high for breath leaving the mouth, and fairly high for breath leaving the nose
• The temperature is higher than ambient (e.g. 30–36°C instead of 20–25°C)

I am aware that these changes can affect density, thermal conductivity, dynamic viscosity and specific heat capacity. I expect the conversion of some oxygen to carbon dioxide to have a small effect, but I'm not sure.

I have found it difficult to find information on the effect of humidity on heat transfer for small objects in the 20–37°C range.

The only useful link I've found is this one: https://www.electronics-cooling.com/2003/11/the-thermal-conductivity-of-moist-air/ but the reliability is unclear and it would be criticisable as a reference for a publication. The graph is also not entirely clear in the region I need. It also completely contradicts this graph in a paper related to desalination: https://www.researchgate.net/figure/The-thermal-conductivity-of-moist-air-relative-humidity-between-0-and-100_fig5_271937026

Any references would be greatly appreciated.

Many thanks.

PS: I am aware that there are many variables that affect a real-life application and things will ultimately be measured empirically, but I need to explain the physics of what is to be expected when exhaled breath is compared to ambient air. Other situations will later be considered such as mounting them in a mask or in a tube, but for now I'm just trying to explain a simplified idealised situation and the expected difference in heat transfer between breath and ambient air.

• The biggest effect is the air velocity. How do the blowing and sucking velocities compare? Can you please provide diagrams of the two cases? Commented Jul 26 at 10:26
• @ChetMiller, I have updated the question: I need to explain the differences independent of the effect of gas velocity (the effect of which is later used to estimate breath volumes). In terms of a diagram, you can imagine a thermistor in mid air 2 cm in front of the nostrils and one 2 cm in front of the lips. Other situations will later be considered such as mounting them in a mask or in a tube, but for now I'm just trying to explain a simplified idealised situation and the expected difference in heat transfer between breath and ambient air. Thanks Commented Jul 26 at 12:45

First, I don't find the paper by Boukhriss et al. (corresponding to your ResearchGate link) to be credible. The paper states only that "The first step is to calculate the thermophysical properties of moist air..." No calculation method, computer program, reference, or comparison to previous analytical or experimental values is given. The thermal conductivity of 100°C at 100% relative humidity is reported to exceed 0.06 W/m–K:

This is not supported by experimental measurements (see below). It looks to me like Boukhriss et al. correctly present humidity-dependent density and viscosity but that something went wrong in their thermal conductivity calculation or presentation, and no information is available to diagnose the problem.

Second, I agree that essentially a blog post (corresponding to your electronics-cooling.com link) is not suitable as a reference, although at least the software used—Techware's @Air—is cited. In any case, that graph, namely,

essentially matches the one given in Tsilingiris's Thermophysical and transport properties of humid air at temperature range between 0 and 100°C,

Tsilingiris describes the models employed to construct these curves and cites their origins. He also compares experimental reports from several sources from temperatures between 0 and 100°C and 0 and 100% relative humidity. None of these values exceeds 0.032 W/m–K. This is one of the reasons why I find Boukhriss et al.'s chart to be noncredible.

Of course, just because two graphs match and another doesn't doesn't mean that the former must be correct. Looking deeper into the trends of thermal conductivity with changing humidity and temperature, Emperhaff et al. in "On the influence of humidity on a thermal conductivity sensor for the detection of hydrogen" compare several binary gas-mixing models in the context of air and water vapor near room temperature. Tsilingiris's model predicts a monotonic decrease in thermal conductivity with increasing humidity, but a model by Melling et al. allows a nonmonotonic change:

Emperhoff et al.'s measurements with three humidity sensors better match the Melling et al. model:

Unfortunately, the measurements don't include your specific range of interest of and 30–36°C air at near-100% relative humidity, so we don't know if one mixing model has experimental support over the others for these sensors in that range. (Nor have I calculated the difference between the model predictions, which might be insignificant.)

Now, let's look at the bigger picture. The thermal conductivity dependence on humidity and temperature we're talking about here is on the order of 1%. That's smaller than the errors in the empirical equations you're generally going to be applying to evaluate the convection coefficient and the associated heat flux. With the aberrant report by Boukhriss et al. apparently resolved, I would suggest that a constant estimate for the thermal conductivity of 0.026 W/m–K should be suitable for all of your working conditions.

• Thanks a lot for the Tsiligiris link and for taking the time to plot comparisons. I realise I actually saw the Tsilingiris paper a long time ago but I couldn't find it again. I agree there is little effect on thermal conductivity but I needed a good reference and Tsiligiris will do nicely. Interesting that other models predict a non-monotonic change. Thanks for finding those. Commented 21 hours ago
• To calculate the heat transfer coefficient, I also have to take into account the density and dynamic viscosity (which affect the Reynolds number and thus the Nusselt number), so the differences can add up. Based on the Tsilingiris graphs, it seems that thermal conductivity increases ~2.3% between 25°C 50% (25.8 mW/m·K) and 37°C 100% (26.4 mW/m·K). Commented 21 hours ago