After further reading, I believe it is a simple current source paired to a PWM circuit. The PWM is used for temperature control. (Longer duty cycle = more cooling) That's why Dave said it's like a diode driver. It runs at a set current (actually max current - I don't know if it's adjustable) when "ON", and the duty cycle controls the temp.
Watts law ALWAYS applies. True, if the current is pulsed then you need to take duty cycle into consideration, but you can't get away from Watts law (or Ohm's law, for that matter) ever. When that transistor is on, the heat (in watts) produced across the FET will be equal to the voltage drop across it times the current flowing through it. Period.So linear power transistor V x A = W calculation is not really applicable
However, It may be that the Die4chill doesn't try to maintain a specific current at all. If it's pulsing the output, it might just throw whatever it can at the TEC (and to hell with whatever the max current rating is for the TEC) for a brief period and then switch off. In that case, the voltage drop across the output FET might be quite low (since it's in full-conduction mode and not trying to limit the current). So even though the current through the FET would be high, the total heat produced would be manageable for an SO-8 package with no heat sink.
The question is: what is the voltage drop across the FET? Depending on the duty cycle, that might be OK. Then too, even if we assume 100% duty cycle, if the FET is really just switching on full (and not trying to limit the current) then the voltage drop across it will be minimal. If we assume the forward voltage drop across the FET is .2 volts, that would be 1.8 watts. I think the data sheet says 2 watts max.Board indicates 12vdc and max I = 8amps ....
I suggest you bench-test it on a 6 volt PSU with both TEC's in series. That way you know you won't exceed the max current and you can measure the voltage drop across the FET (assuming you have an oscilloscope, that is). If it is indeed functioning as surmised above, then you can safely run it off the 12 volt supply once you install it in the projector.
Adam