A theoretical model for evaporation occurring when wind is blowing over the surface of water

Question

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)?

Answer 1

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.