My experience says the heater core is clogged. You may want to flush the core before you go ripping the dashboard apart. I just did this in my 2000 Taurus. I'm not saying they are all identical, but this seems to be a common problem.
Seems the heater core bypass pipe provides a path of lesser resistance for the anti-freeze. During the summer, while the heat is off, the scale and rust builds up and gradually blocks the coil.
here's a link to the pictures I took while flushing my Taurus this weekend:
Once the heater core is flushed, I would recommend an entire cooling system flush to be sure all rust and scale is no longer present in the system. It seems that the water pump impeller can rust and deteriorate becoming a major source of coil-clogging debris.
This is the second time I ve done this and the first time I ve done a fully system flush.
Be sure the engine is COLD!
Now I have more heat than I know what to do with.
Figure about 3 hours to complete this coil flush - start to finish.
Monday, November 5th, 2007 AT 10:47 AM