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Turbine engines are started with both electrical starters and with larger engines, air starters (to save weight). The start valve is nothing more than an air valve, which admits air to the starter; a relatively lightweight motor which is powered by air. The starter is normally attached to the engine via a sparague clutch and worm gear. The starter motor itself is a little like a small turbine engine, with pneumatic air blowing over the blades to move the starter.
The start concept for the turbine engine is to get the engine spinning as fast as possible using the starter. The engine speed is measured in a percent of it's operating RPM, rather than the actual engine speed. That is, rather than looking for 6,000 rpm, for example, we see 15%...something like that. The 100% operating speed of the engine is typically 18,000-35,000 rpm, and we see some percentage of that, to make it easier to deal with.
When the engine has reached the highest percent RPM the starter can spin it, fuel is introduced, along with ignition. At some point, the fuel lights off, and the exhaust gases from the burning fuel act on the turbine wheels downstream to turn the engine faster. At some point, typically about 50% RPM, the starter is no longer providing any useful assistance, and is shut down by closing the air valve to the starter motor. (If the starter is allowed to go to higher speeds, it may "burst" or suffer a failure which damages not only itself, but surrounding components).
Pneumatic air from a ground "huffer" cart, the auxilliary power unit, or even from other engines is used to power the starter motor. Typically the APU or ground cart can produce a greater air volume than the engines do at idle, for "bleed" air...the air which is bled off the engines to provide air conditioning, pressurization, move leading edge devices, etc. To do a cross bleed start, using the air from one engine to start another, the operating engine or engines must be operated at a higher value than idle in order to boost the airflow and pressure in the aircraft pneumatic manifold high enough to start an engine.
If the air pressure used to move the starter motor isn't high enough, it won't turn the engine up to a fast enough speed during the start process. If this happens and the engine lights off at a low speed, it may not accelerate properly; it's temperature will increase faster than the engine speed increases, and a hung start or a hot start occurs. In a jet engine, most of the airflow through the actual engine portion (the core) is used for cooling...not for burning. If not enough airflow is passing through the engine then there's not enough cooling air, and not enough air to prevent the flame in the burner cans inside the engine from contacting the burner can walls. The result can be an unsatisfactory start and even engine damage.
Most of the functions that go on in the engine during the start are automatic. Various valves open, such as acceleration bleed valves, which manage the airflow through the engine by bleeding off excess pressure at some places while it builds in others. These are self-contained within the engine, and the pilot and/or flight engineer is scarcely aware of their function. Some engines do everything automatically,and some require the pilot or FE to move certain levers at certain times and conduct each part of the start process manually. Some engines automatically regulate the temperature, others put it all on the flight crew.
The engine start isn't over once the engine lights off. Monitoring temperatures is very critical to the life of the engine, as is the speed at which the engine operates, at which it accelerates, and where it idles. Time limits are imposed which must be watched during the start; total starter time, time to light off, time to on-speed, etc. Some powerplants have events which must be noted at different points in the start process, such as starter cutout, or the automatic opening of secondary fuel valves, etc. Typically a jet engine will take about 45 seconds from start to on-speed, but may be as long as a minute and a half. Even after the engine is started and on-speed, some engines must be closely monitored as a tailwind or other factors could cause the engine temps to take of and require a rapid shutdown to save the engine.
Turbine engines are very reliable and put a lot of thrust out for their weight. A small mistake during the start process can do millions of dollars in damage, however, so it's a process to take quite seriously.