Relative Humidity Formula

Search for psychrometric chart, hope this helps.

It is not that complicated if you take the time, and it may fuel more questions we all will try to answer.

"formula for relative hummidity in terms of distance? i.e. the hummidity of the air in a room where the cooler is on the window of the house."

Buy an inexpensive hygrometer and move it around the room. Leave it in one place at a time for 15 to 30 minutes, take a reading and move it to a new location.

Due to air circulation of the air in the room you are not likely to find much if any variation due to location in the room.

Most likely what you are feeling is the extra cooling effect of the air movement from the cooler when near to it.

This is a slight oversimplification, but I believe what you are looking for doesn't exist. The humidity in a room is the partial pressure of water vapor. Just like a barometer reading, it is the same everywhere in the room. When water condenses on a cool surface inside the room, it is because the temperature of the surface is below the dew point of the surrounding air. As moisture condenses on the surface, the partial pressure of water vapor is reduced, and if it happens in large quantities, we call it dehumidification or air conditioning. Although it's true that condensation lowers the water pressure adjacent to the cooled surface, nature abhors a vacuum, and moist (higher water pressure) air immediatly takes the place of the dehumidified air.

The difference in hummidity at different locations in the room will not last very long after you start the cooler probably a few seconds or minutes. You may be able to calculate a time to reach uniform hummidity by evaluating the diffusion rates of the two gases (air and water).

There may be differences around the room but they are below the threshold of what you can measure. Air flow will equilibrate the relative humidity throughout the room. Of course, as the air temperature decreases in the room, the relative humidity will increase because colder air holds less water vapor and thus the percentage increases. However, the air passing over the cooling coils will remove water vapor and you have a net gain on relative humidity if everything is working correctly.