How do I check the alternator?

[petroleum bone-apart]

Apologies for this basic question, but I fitted a cooling fan to reduce warming up time and this seems to use up the battery faster than the alternator is able to charge it up. It works well in reducing warm up time but the second or third time it is used during the day there is insufficient power to start the engine (SR3 24v).
The batteries are on trickle charge whilst in my garage at home so they read 13v at the outset. Three warm ups with the fan running the Aim is telling me just 10.5v left.
What revs are needed for the alternator to recharge the battery? I have tried testing by connecting a voltmeter to the engine compartment battery and running the engine with the fan on, and it shows a slow fall in voltage at tickover (about 1250rpm) and no appreciable difference at 4000rpm.
Is there a better way of testing it? Any ideas?

Thanks

David

[petroleum bone-apart]

That makes sense Josh, I will try and find the charging loom. Any clues? It is a 2011 SR3 1500.
Bert, the water heats up faster than the oil to the point where the max water temperature is reached before the oil reaches the minimum temperature, so I have to keep switching the engine off to let the water cool down then start again, repeating until both are in range. The fan keeps the water cool enough to allow the oil to rise, halving the warm up time. It also keeps the water temp down whilst sitting in a queue.
The battery reads about 12.5v whilst on tickover, shows a drop when the fan is switched on and falling to 11.5 after about ten minutes. Switch the fan off and it rises to 12.1 / 12.3v.

[petroleum bone-apart]

Thanks Josh