I have seen an argument both ways. There is an article somewhere I read that said NO they do not work. Temperature was measured on the outside of the barrel and never took into account inner cooling.
I say they work. The heat inside the barrel after shooting sits and radiates with nowhere to go. If you introduce an air current you move the hot air out and allow more efficient cooling of the bore. If you ever blow on a hot spoon of soup to cool it down it work on the same effect (I believe, im no scientist)
Out of curiosity do you have any idea how many CFM, rather Cubic Inches per minute of air the fan will move assuming charged battery? And can I suggest caliber adapters? (I assume the motor and fans are identical despite the caliber of output tube... which may be moot anyway as the output only has to enter the chamber, not the throat so a 338 might work for say the .376" base diameter of a 222/223)
Would you care to state a ballpark figure for cost per unit?