It might be a bad thermostat, or something worse, such as a blown head gasket/cracked head. Either of those scenarios will cause the cooling system to pressurized beyond it's specified capacity, and the pressure will force the coolant out of the reservoir.
Sounds like your water level is running low. Check the level in the radiator reservoir You may also have a leak.
Check your heater hoses, that's where the water ciruclates into the heater from the water pump.
If the smoke is coming from the electric fan itself, then it needs replacing. It could be the smoke is steam that the fan is drawing from the radiator, and you have a coolant leak at the radiator.
If the reservoir keeps losing fluid.
Did you get the hose clamps tight? Did you puncture a hole in the bottom of the radiator? Did you get the petcock closed all the way? You didn't overtighten the petcock at the bottom of the rad did you? Is your water pump good? Did you put thermostat in the right way?????????????
Sounds like your water level is running low. Check the level in the radiator reservoir You may also have a leak.
It depends on what you mean by "run out" but there is probably a leak in the tank.
possibly a leak in cooling system
Where is it leaking from? Could leak from the radiator, hoses, or water pump.
blown head gasket
No. The only times the radiator would leak is if it was damaged, defective, or corroded.
Could be a hose leak, a radiator leak or a head gakket leak. The water in the reservoir will get "sucked" into the cooling system if the level get low. Look for water on the ground or signs of spray under the hood. If it's a head gasket, look for water dripping from the exhaust pipe on start-up and/or steam. That would indicate that water is getting into the combustion chamber.
It probably has a leak. Here's something to check - the radiator has a tube coming out to the reservoir, so when it gets hot and expands the coolant goes into the reservoir and should go back into the radiator when it cools, but if your reservoir has a crack and the coolant leaks out, well it's gone and can't go back into the radiator - this would cause a lot of loss of coolant.
Find leak, repair/replace cause.
It sounds like there's a leak somewhere. If it's not in the cap, it could be in the hose, or the connection, or the radiator itself. If you can't figure this out on your own, take it to a mechanic.
A radiator cap would not cause a leak! However, if the radiator cap is holding pressure, other weak parts could signal issues in need of attention.
You have a leak in the cooling system. Could be water pump, hoses, radiator, headgasket, etc.