I've just heard that N2 is the limit when starting jet engine, But I have no idea why it is.
I know that N1 is Low compressor speed, and N2 is high compressor speed. So what makes high compressor speed more important when starting jet engine?
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If it's a turbofan, N2 represents the core engine compressor/turbine spool that is creating the compress/burn/expand energy that drives the fan, and is exposed to the most energy, that is, velocity and heat.
N1, as the fan/free turbine RPM, is effectively just a windmill tacked on to the end of the core, driving the fan at the other end, same as a free-turbine turboprop, except it's a fan instead of a propeller. You could jam a piece of wood in the fan to stop it from turing, and you could probably still start the core engine and run it at idle. This is done with routinely with free-turbine turboprops, although not with turbofans.
In a turboprop or turboshaft, the engine core is called the "gas generator" and you have the same thing. You are concerned with the gas generator RPM while starting, where all the energy is being produced, not the free-turbine/propeller, which is just going along for the ride.
If it's a turbojet with no fan but two compressor/turbine spools, it's still the same thing in the end. The interior spool that has the highest pressure compressor stages and turbine blades closest to the burner has the most critical parameters, so its speed will be part of the starting limitations, while the low pressure spool is helping but just doesn't have to deal with the same energy level and you can be sure that if the N2 is in limits the N1 will be also.
The N2 shaft is the actual jet engine proper, which has one job: turn fuel into thrust directly. It is akin to the crankshaft on a gas/diesel engine.
The N1 shaft is an extra turbine put in the thrust stream so it can steal thrust and turn it into rotating shaft power. It is akin to the transmission shaft on a car, or the Power Takeoff on a tractor.
So when you are starting a gas engine you care about crankshaft RPM not transmission RPM. Same here.
And then, once the engine is running, other equipment somewhere takes N1 shaft rotation and turns it into something valuable:
All correct answers here, N1 is the large fan at the front and suffers the same effects as turbo lag in a car. N2 does the main compression of the air before it enters the multistage combustion chambers and mixed with fuel. This then drives a series of turbines where one or two actually drives the N2 compressors and one drives the N1 fan. With this setup there is inherent lag between N2 stage and the N1 stage.