[calling the temperature gradients in wire an error source]
Well, the gradients are NOT an error source, but you'll
never convince him, because he can show you a counterexample.
Take a length of soft iron wire (sacrifice a coathanger, if you have to).
Connect the ends to an ammeter.
You'll see zero circuit current. Now fire up a torch and heat the wire
to make a hot spot. Still zero circuit current. Now slowly move the
hot spot.
When the hot spot in the wire goes one way, the current turns negative. Move
it the other way, the current goes positive.
The temperature gradient in the wire DOES make a thermocouple net voltage
here, because your hot spot is a different allotrope (sometimes called Austenite)
of iron crystal than the cold wire is. The Austenite-Martensite transition at
the trailing edge of your moving hot spot occurs at lower temperature than
the transition at the leading edge, because the transition has hysteresis. The
iron wire is thus two different metals, with connections at different temperatures.
So, if the thermocouple materials undergo phase transitions, the gradients
in the connecting wire do have a net effect on the circuit. That's a good argument
for knowing a LOT about the thermocouple wire materials before you measure
with them.
