Hard to say how long until dry...depends on ambient humidity..usually a few hours or so.
You could put brooder where it will be 'solar heated'.....you'll just have to make sure it doesn't over heat by balancing some shade with good ventilation. Chicks need to be kept pretty warm the first day 85-90F, but they also need to be able to move away from the heat to cool off too. Might be hard to do with a passive solar set.
They should get all the vitamins they need from a good chick starter ration, but maybe that' not available to you.
Here's my notes on a chick heat, the electrical parts won't apply to your situation but the behavioral and thermometer parts should help...and yo might already know this from having chicks before, but...
Here's my notes on chick heat, hope something in there might help:
They need to be pretty warm(~85-90F on the brooder floor right under the lamp and 10-20 degrees cooler at the other end of brooder) for the first day or two, especially if they have been shipped, until they get to eating, drinking and moving around well. But after that it's best to keep them as cool as possible for optimal feather growth and quicker acclimation to outside temps. A lot of chick illnesses are attributed to too warm of a brooder. I do think it's a good idea to use a thermometer on the floor of the brooder to check the temps, especially when new at brooding, later I still use it but more out of curiosity than need.
The best indicator of heat levels is to watch their behavior:
If they are huddled/piled up right under the lamp and cheeping very loudly, they are too cold.
If they are spread out on the absolute edges of the brooder as far from the lamp as possible, panting and/or cheeping very loudly, they are too hot.
If they sleep around the edge of the lamp calmly just next to each other and spend time running all around the brooder they are juuuust right!
The lamp is best at one end of the brooder with food/water at the other cooler end of the brooder, so they can get away from the heat or be under it as needed. Wattage of 'heat' bulb depends on size of brooder and ambient temperature of room brooder is in. Regular incandescent bulbs can be used, you might not need a 'heat bulb'. You can get red colored incandescent bulbs at a reptile supply source. A dimmer extension cord is an excellent way to adjust the output of the bulb to change the heat without changing the height of the lamp.