I think water pans help, but not everyone agrees with that thought. Another nice thing, it serves a purpose as a drip pan to help keep the smoker clean.
I agree with
@RayClem that using a shallow pan to increase surface area and would help in having better evaporation. But like mentioned, that will entail adding water and opening the smoker more, which will release any built up humidity you have in there. I always add hot or boiling water to the water pans. Whether that makes a big difference or jot, i don't know. I just have always done it with preheated water.
Because of smoking sausages, I have been experimenting with different methods of increasing humidity in the smoke chamber. With sausages, you need a higher humidity to ensure your casing has that snap and be tender when you bite it and to help keep the moisture level up in your meat mixture.
I have been using a humidifier, and recently experimented with adding a cloth to the water pan to increase the surface area and increase the conductive evaporation. You can increase the surface area by extending the towel over the sides more, but I didn't find that necessary

On a 12 hour smoke at low temperatures (starting at 150 and maxing out at 180 in incremental increases), I added 16 cups of hot water throughout the cook. I had just over 1 cup left at the end of the cook.
My cook this last weekend was 6 hours with the same incremental temperature increases and used just under 8 cups of hot water.
Higher temperatures in the pit and that would increase the water use. So water evaporation is taking place even with the lower ttemperatures that I had tested with
Thread 'Adding humidity'
https://www.traegerforum.com/threads/adding-humidity.4800/
I guess when you ask how much water you will need, it is a more complicated calculation based on temperature, relative humidity and surface area. But like Ray said, more is better.