How long does it take to recharge the battery after starting the car?

I imagine this varies by car, but assuming a fairly recent (12-year-old) gasoline powered sedan. We rarely drive our car more than 20 minutes at a time, and we’ve been having trouble with a 2-year-old battery that we just replaced.

So, it’s possible the alternator is going, or the battery just didn’t do so well in the cold after 2 years, but I’m more curious about the charging time after starting.