# Air temperature that would cause same rate of heating to human body as a 40°C hot tub

My son was wondering why a 104°F (40°C) hot tub feels so much hotter than a 104° day. I explained the difference between conduction and convection but we started wondering how hot the air would have to be to heat your body by convection at the same rate the water does by conduction.

I know that water conducts heat roughly 24X faster than air so the obvious but I assume completely incorrect answer is to multiply the difference in temperatures between human skin (~ 90°F/32°C) and the water by 24, but I’m not sure what the correct means of calculating it would be.

I’m aware an exact answer isn’t possible due to differences in air temperature/humidity, the fact that heating would slow down as your body heated up, etc, but am hoping there is some rough accepted guidance for estimating something like this.

Bonus points if it applies to sous vide vs. oven cooking, just because I’m curious about that too. :-)

Thanks!

• Not only does the conductivity play a role, but also the fact that water simply carries more thermal energy per unit than air does when having the same temperature. This is called heat capacity. – Steeven Dec 28 '17 at 21:29
• Are you assuming hot 100% humidity air? Otherwise evaporation must be included... – DJohnM Dec 28 '17 at 22:14

When you are exposed in the hot ambiance, such as 104°F, your body starts sweating and if the ambient air is dry enough to evaporate sweat (water) from your skin, your skin cools down and so you don't feel much heated. But the same thing doesn't happen with the hot tub.

There are 2 possible reasons as to why this happens.

1. Air is a very poor conductor of heat, poorer than water.

This is because air molecules are less concentrated then water molecules, hence, there would be less interaction between the molecules. This would mean that it conducts heat from the environment slower to your body than water. It should be noted at gases are in general are very poor conductors of heat, while solids are generally the best conductors of heat, with fluids having a general thermal conductivity between that of solids and gases.

1. Air possesses very little thermal energy as compared to water.

Here, I am talking about the specific heat, and not their temperatures. This is because air is about 900 times less dense than water, meaning that for a given volume, there are more water molecules than air molecules. Hence, given the same amount of time you are exposed to both mediums, you would definitely have gained more heat from water than air per unit time.

This phenomenon can be seen in daily lives when you take something out from the oven. The oven is about 210°C, but as long as you do not come into contact with the objects in the oven, you will not burn. On the other hand, dipping your hand in 100°C water would already cause you to burn. Hence, it is definitely observed that a 40°C hot tub would feel much hotter than a 40°C day. Hope this helps.