No, it won't smoke when it's that thirsty, it'll want to drink though. Give it some good beer.
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.
That smoke is steam. You have a leak in the radiator or a hose. Take it to a mechanic.
Smoke from the radiator would be do to a leak evaporating off of it. Find the leak repair or replace necessary parts.
It could be a damaged or loose radiator hose, a damaged radiator, a faulty radiator cap, blown head gasket. Have a look around all the hoses and see where the water is coming from and then use common sense.
It is very doubtful that it is smoke. It is most likely steam. Either there is a pinhole leak in the radiator allowing hot coolant to escape that then vaporizes into steam when it hits the atmosphere... or there is something spilled on the radiator that steams when the radiator gets hot. Have the radiator pressure tested for leaks.
there is supposed to be water in the radiator. your radiator is supposed to be evenly mix 50/50 with radiator fluid and water, preferably distilled water.
There is a water reservoir tank that goes to the radiator that you can add the water to.
Radiator coolant.
You can add water to a radiator any time it is low on water.
remove your radiator form your car then poor some hot water in radiator till water comes out nice and smooth.
white smoke is a indication that water is entering the combustion chamber might be a head gasket check oil for water and or your radiator is low on water
Your water pump might be broken or your thermostat might need to be replaced. You obviously don't have a leak so somehow the fluid isn't moving through the engine. If the thermostat is broken it will not let fluid into the engine and then the engine will overheat and the radiator will release smoke.