The firing temperature is believed by some to be the highest temperature attained in the system. Some however believe that it is the average temperature in the hot-section of the gas turbine.

Please help me with an illustration.
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No illustration necessary, really. For GE-design heavy duty gas turbines, "firing temperature" is something of a misnomer, and depending on the turbine, can refer to either of two locations, though both are very similar to each other. (NOTE: This discussion does *NOT* refer to the combustion reference temperature of GE-design heavy duty gas turbines equipped with DLN combustors. TTRF1 is something related but different from this discussion.)

For units without air-cooled first-stage turbine nozzles, firing temperature refers to the temperature of the combustion gases entering the first-stage turbine nozzles, sometimes called the "turbine inlet temperature". Units without DLN combustion systems and single nozzle combustors will experience flame temperatures (the temperature of the burning fuel "ball") of as much as 3000 deg F, but cooling and dilution slots and holes in the combustion liner allow compressor discharge air to mix with the hot gas and reduce the temperature to levels which are acceptable to the turbine nozzles and buckets, which for many machines is somewhere around 2000 deg F.

For units with air-cooled first-stage turbine nozzles, firing temperature refers to the temperature of the combustion gases at the trailing edge of the first-stage turbine nozzles, just before they enter the first-stage turbine buckets. This is because the air used for cooling the nozzles "exhausts" into the hot gas flowing through the nozzles, and cools the gases very slightly, by just a few deg F, if that. This added cooling allows a very slight extra amount of fuel to be burned to elevate the hot gas temperature entering the first-stage turbine nozzles which is then cooled very slightly by the nozzle cooling air before it enters the first-stage turbine buckets.

So, for GE-design heavy duty gas turbines, firing temperature does not refer to the highest temperature attained in the system (or, the "flame temperature" in the combustor). It refers to the temperature of the gases entering the turbine section, which generally must be significantly cooler than the flame temperature in the combustors.