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I've recently done an experiment in which I was studying the variation of the cooling rate of hot water in a draft (wind). The air was blown directly over the surface of the water (kept in a cylindrical container). After plotting the graph, I saw that the cooling rate saturates for values of wind velocity greater than $4\; \text{m} \, \text{s}^{-1}$. This is intuitive, but I would like to prove it mathematically.

Assuming that evaporation was the dominant factor of cooling, I believe that if one is going develop a model for the situation described above, then one must understand models developed for evaporation. I found two main models, which might be of interest here:

  1. The Penman Equation: This models evaporation of water in lakes.

  2. Langmuir's Evaporation Equation: I am still trying to understand its derivation, but in my opinion this has the most relevance to my question.

The reason why I am interested in understanding these models is because I want to first learn how one can even mathematically think about particles and the whole process of evaporation. And more importantly, if there are areas where one can tweak the steps (adding the velocity of air molecules blowing over the surface of the water) to arrive at a new equation from the same idea.

In conclusion, I would like to know two things:

  1. Can these models be extended to explain the phenomenon that I've observed?

  2. Any alternative approach to solving this problem (instead of understanding evaporation models first and then tweaking them to account for velocity of moving air molecules)?

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  • $\begingroup$ This is a situation in which there is forced convection. I do not think the Penman Equation is useful here. Rate of cooling can be modelled with Newton's Law of Cooling (rate of temperature decrease proportional to difference in temperature with air) but this does not include speed of air flow. $\endgroup$ Jun 14, 2016 at 2:03
  • $\begingroup$ Exactly! My teacher thinks that Forced Convection is not involved. $\endgroup$
    – Student
    Jun 14, 2016 at 5:29
  • $\begingroup$ Just found the following, which may be useful, but does not give a complete answer. I am still working on an Answer. www98.griffith.edu.au/dspace/bitstream/handle/10072/41089/… $\endgroup$ Jun 14, 2016 at 16:58
  • $\begingroup$ This question is clearly the one you want answered rather than the one about kinetic energy. Title says it all, but I recommend that you revise to make it clear that you are looking for help in finding or developing a theoretical model of your experiment to compare with your results, rather than research recommendations. Mention any ideas you have already. Any edit to your question will cause the 'on hold' decision to be reviewed. If this decision is reversed I have spotted a couple of members who have answered similar questions, and will try to get them interested. $\endgroup$ Jun 14, 2016 at 17:10
  • $\begingroup$ I found one link which was quite useful :bado-shanai.net/Map%20of%20Physics/mopLangmuirEvaporation.htm Maybe this will give you some ideas... $\endgroup$
    – Student
    Jun 14, 2016 at 19:05

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As someone has already mentioned, this is a problem of forced convection. As the dry air moves over the water, the latent heat of evaporation will cool the water. Once the air takes on 100% humidity, no further cooling will occur, which explains the threshold you observed. For the energy budget, the latent heat probably matters more than the transfer of sensible heat.

Solving this problem theoretically is much harder than it may seem. The flow may be turbulent and the water surface may be rough. There is a velocity gradient as well as a humidity gradient. Often an empirical approach is taken to this problem (e.g., wind over ocean is an important application). The much-cited paper by S. D. Smith, J. Geophys. Res. 93, 15467 (1988) doi: 10.1029/JC093iC12p15467 provides an entry point to the theory as well as empirical formulas. So, I am not providing an answer, only an entry point to the literature. I have in fact also wanted to see a derivation of this, perhaps for a laminar flow in an isothermal environment.

The Langmuir equation you mention is about evaporation into vacuum, and is not relevant here.

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    $\begingroup$ "Once the air takes on 100% humidity, no further cooling will occur, which explains the threshold you observed." I don't see how this would explain the 4 m s$^{-1}$ threshold. I'd have thought that the faster the flow of (initially) dry air, the less likely it will be to saturate with evaporated water molecules (though I agree with you about the possible role of turbulence). $\endgroup$ Dec 12, 2021 at 10:11
  • $\begingroup$ @PhilipWood You may well be right. It's possible the heat transfer plateaus for another reason than saturation. $\endgroup$
    – Norbert S
    Dec 12, 2021 at 18:07

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