Prompted by this thread, I did some instrumented testing today, on a car which is not a VW TDI, but for which there are all sorts of aftermarket blingy bits available including a number of aftermarket cold-air intakes.
Now, the advantage of this not being a VW TDI and being a non-turbo gas engine (in fact it is a Fiat 1.4 MultiAir), is that the intake air temperature is actually the intake air temperature ... not the post-turbo-compressor post-intercooler temperature, which is what you get with a TDI if you have a scangauge connected to it.
As with every VW, the air intake is coming through a duct which has its inlet in front of the radiator (before the heat source) and high up (to keep it away from potential water ingestion).
To make matters worse, on this particular car, the air filter is inside the plastic engine cover which is spread out across the top of the hot engine itself. So you'd think there would be heat going from the engine and soaking into the engine cover and then into the air stream. And, true, there is some evidence of this.
Outside air temp -5 C at time of test. Scangauge reported within one degree of that on cold engine start. Close enough.
After driving for about 15 km through the countryside, outside temp was still around -5 C, intake air temp was a whopping 3 degree C. So 8 degree C temperature rise from ambient, measured at the throttle body.
After sitting in a coffee shop for a wee bit, outside temp had risen to -2 C, and the heat-soaked intake air was now reported at 19 C. Over 4 or 5 km of driving, this dropped down pretty quickly, and after 10 or so km. it was now reporting 6 C intake air temperature. So aside from heat-soak conditions (which sorts itself out in a few minutes) it's safe to say that intake temperature is 8 degrees above ambient.
The difference in air density between 0 C (273 K) and 8 C (281 K) is ... 2.8 percent. If one were to construct a perfectly insulated intake system with absolutely zero heat transfer from the engine compartment, you could get air 2.8 percent more dense into the engine and maaaaaybe get 2.8 percent more power (probably less). In reality that (typically) metal air intake duct on your typical average "cold-air-intake" system conducts heat through it much faster than the plastic that the OEM cold-air-intake is made from. On the other hand, you wouldn't be stuck with having the air filter spread over the top of the engine. I think it's safe to say that the potential real world power improvement from the real world change in intake air temperature is pretty close to nil. Certainly not worth spending 300 bucks on.
Then there's the claim that stock air intakes are "restrictive".
Key-on engine-off the intake pressure sensor reports 98 bar. We'll take that as barometric pressure.
Going up a hill with the accelerator pedal located on the floor
the intake pressure sensor reports 97 - 98 bar depending on RPM. I didn't test beyond 4000 rpm as there was no point. There is so little difference between barometric pressure and intake manifold pressure that the "restriction" of the stock air intake duct and generously-sized stock air filter element can safely be called "diddly-squat".
An interesting demonstration of the MultiAir system (variable intake valve timing and lift) is that even at part throttle, the intake manifold pressure remains high (it seems to like keeping it around 87 bar) under light load cruising conditions. It's using the variable intake system to manage how much air the engine is getting, not so much the throttle. Only at idle or shut-throttle coasting does the intake manifold pressure actually drop way down.
Experiment concluded. Not worth buying a cold-air-intake for that car, either. (I didn't expect anything different.)