I have my 2011 MBA plugged to an external monitor (1680x1050, nothing fancy) and the fan is turns on and gets noisy after 5/10 minutes.
Does this happen with more modern MBA like the 2014 and 2015?
Bash is the Unix shell, before you quit that (is it usually running when you're not using terminal?) find out what processes are running in it...
Thing is, if I plug my MBA into my 46" Sony TV with an HDMI adapter and watch a movie, the battery level drops much faster than watching it on the built-in screen. Even if I dim the built-in screen all the way down, the battery still runs down faster when using the external screen.
So more energy is being used to drive the second screen, and that might result in a warmer computer. However, I have never seen the fan kick in while doing this.