If battery technology is already good enough to run cars on, why wouldn't APUs and emergency power cells also be batteries? Why use turbines at all, since they would suffer the same carnot losses as piston engines?

Here is an example of a small turbine:

At full power it runs for 5.5 minutes on 968 mL of aviation jet fuel (176 mL/min).

Jet fuel contains 33 MJ energy per 1000 mL.

So fuel energy is input at a rate of 95 kW (127 hp)

But it only generates 44 Newtons (10 Lbs) of thrust.

It is very inefficient, because the exhaust temperature is almost as high as the combustion temperature.

Combustion temperature = 700 deg.C
Exhaust temperature = 500 deg.C
Carnot efficiency = 20.55 percent (same as a car)
Mechanical power = 19.5 kW (26 hp)

I'm not sure if this is correct, given the specified 10 Lbs thrust, but if true it could keep a car going at highway speed, but it certainly could not get it up to highway speed in a reasonable time. A motor 3-4 times bigger would be needed for reasonable accereration. I suppose it could be used to charge a car battery, but it guzzles fuel. It would need at least 2 liters (1/2 gallon) of fuel to run for 11 minutes.

Plus, these exhaust gas temperature are extremely dangerous! Co-generation is not possible with small turbines, so the heat is guaranteed wasted. They are expensive, $3200 US dollars for the smallest model, and their warranties only cover 1-year, 200 starts, or 20 hours of operation, whichever comes first. Not very good, IMO, unless you are an insanely mad scientist, or a typical hobbiest with a little extra spending money.

I think turbines are far better for stationary powerplants, where they can hook up co-generation units to recover exhaust heat. Not very easy with portable turbines.