Why is bleed air taken from some stage of the compressor used for de-icing? Why not e.g. exhaust gas?
Using air from the exhaust section either for anti-ice directly, or using a heat exchanger, would create many problems.


The air must first be used to burn fuel before reaching a turbine bleed location. Air taken from the compressor section hasn't required any fuel to be burned (other than to compress that air of course).
The air will be much, much hotter than air from the compressor. Turbines, especially in the high pressure section, are made of specialized alloys and can be additionally actively cooled to prevent them from melting. This means more specialized, and therefore more expensive and/or heavy material must be used for the whole anti-ice system.
Components must also be designed for the case of this bleed air leaking. Hotter turbine air would be a huge problem for this.
The air will have the combustion byproducts in it. This can coat the insides of the anti-ice system, reducing its effectiveness, and possibly requiring it to be disassembled and cleaned periodically.
If there is some type of engine fire, the fire typically goes downstream, which would send it into this anti-ice system as well.
In the case of using a heat exchanger, this would be yet another component to fit into spaces that are already very full, and yet more weight on the airplane.


The air from the compressor is already plenty hot. Take a look at the diagram in this answer. Notice that all the compressor bleed air goes through the precooler before going to cowl or wing anti-ice. This means that the compressor air is already hotter than is required. So taking air from the turbine provides no benefit for creating the problems listed above.

Also, taking air from the compressor does reduce engine efficiency, but you are going to have to get that energy from somewhere.



To address your points about the thermodynamics:

Assuming your numbers are correct, what is wrong with that scenario? Releasing air at -97°C that was initially at -50°C means that you extracted 47°C of energy from it to use for anti-ice, using the systems already available on the aircraft (bleed air, precooler). Kind of like a heat pump, but just using air? Wouldn't it be less efficient to burn fuel to heat this air, which would then leave the aircraft at a higher temperature, and therefore throwing away more energy?


  In principle, one can play with the numbers, increase/decrease altitude/temperatures and discuss how adiabatically this (de)compression processes are.


(Emphasis added) I'm not convinced that your assumption that these processes are adiabatic is a good basis for your model, since that is an idealization (though your initial bleed numbers do seem to match here and here). An important point is the way the air is used. A typical installation uses piccolo tubes. Small holes along the tubes release the air onto the leading edge. This whole leading edge cavity then vents to the environment. (This doesn't seem to match your assumption that the air is cooled at constant pressure) In order for this air to be useful any further, you would have to somehow return this air to for reuse. This adds more complication, when there is already an abundant supply of compressed bleed air available.

Another option would be to somehow maintain the pressure behind the surfaces with ice protection. This may be feasible for the engine inlet, but not for wing leading edges that include moving slats.

Even assuming you can somehow recover this air for use, that would be another system and more complication added to the aircraft. And you can't rely on this system, because anti-ice is not always in use. You have to provide a pretty large gain in efficiency to justify adding weight and complexity to the airplane.

Maybe you would like to see proof that most the work done on the air is used in the anti-ice process, and adding heat from combustion is not necessary. We may need to get someone like Peter Kämpf in here to run some numbers.