Why is thrust (in a high bypass turbofan for e.g.) non-linear with respect to N1?

Question

I've been studying jet engines for a short while now and I think I have a pretty decent understanding of how thrust is generated, but one thing I'm struggling a bit with is how thrust varies across the N1 range. More specifically, I have seen a few sources that state that it's more of an exponential relationship (i.e. going from 80 to 90% N1 results in a much larger increase in thrust than 50 to 60%), whereas if I had to guess I would have expected it to have been relatively linear (or at least something close to that).

What are the root causes of this non-linear behaviour?

If context is required, let's assume we are talking about modern day high-bypass turbofans on commercial aircraft (subsonic flight only).

*Edited to add a link to a source which figuratively shows what I mean by the relationship being exponential, this suggests Thrust to be roughly equal to N1^3.5 in this example:

Answer 1

Gas turbine engines, e.g. a turbofan engine, are (is) designed for a specific operating point, for that point the engine operates most efficiently, any deviation from that point causes the air flow in the core of the engine to be of a different volume than the design point flow.

A compressor (or a fan) is designed for a certain compression ratio at a certain rotational speed. When the rotational speed changes (why? reducing the fuel flow causes less hot pressurized gas to be available to expand in the turbine, the less energy is available to compress air), air is compressed less and thus larger in volume. Knowing the passage of the compressor has a fixed cross section per compression stage, less compressed air causes air to flow faster through the stages. As a result the compressor velocity triangles change and blades have a different inflow and as such a different efficiency (lift generation, it can even trigger stall of the blades, unless countermeasures like bleed off-take are taken).

Compressors will work at other conditions than the design condition, the relation is usually displayed in a graph that we call the compressor characteristic (see e.g. What are the "beta lines" of a gas turbine engine component?), this characteristic relates (corrected) flow and pressure ratio to efficiency for given spool speed (using a fictive parameter called the beta lines to ensure a single solution for a given set of parameters is given). The further one moves from the optimum design point, the lower the efficiency (this is not a linear phenomena). Note that gas turbines are aerodynamically balanced and each compressor (or fan) is connected to a turbine (could be through a gearbox) which depends on a similar characteristic. The equilibrium that is reached determines how the engine operating point moves through the compressor characteristic.

When we sweep a fuel flow for a turbofan from high to low at a fixed flight condition (static on the ground), we can plot the results of such an experiment to show the effects of blades receiving the inflow at far less favorable conditions. E.g. the fan operating line:

We can take this further to create an image similar as posted in the question (note that the axes are swapped) for the fan spool speed N1 and the core spool speed N2:

Why the relation is not linear is that the performance depends on various aerodynamic effects (which are not linear) of various components.

Answer 2

A turbojet basically does something quite simple:

• it gets some fresh air from the outside;
• it compresses this air some 20 times so that its pressure and temperature raise;
• when this hot and thick air comes into contact with kerosene, this latter just catches fire;
• the temperature suddenly increases, as well as the volume of the mixture air/kerosene;
• part of this suddenly expanding mixture impinges on the blades of a turbine making it spinning; the rest is ejected outside and pushes (thrusts) everything on the opposite direction.

Et voilà, fresh air has just been converted in thrust.

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So, how do we extract some numbers from this indeed very simple process? Well, since we are dealing with air, we need for sure the tools of aerodynamics. Is it enough? No: this air reacts with kerosene creating new substances, so we need the tools of chemistry too. End of the story? Nope: in the process, air, kerosene and their products change pressure, temperature and density, so also the tools of thermodynamics are needed. And finally, the combustion products make the turbine rotates which, in turn, brings the compressor in rotation, so here the tools of mechanics are needed as well... Now, what started out as a very simple process, turned out, to be understood, to actually need all the knowledge from almost every known scientific branch!

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What are the root causes of this non-linear behaviour?

Let's make a simple example where we analyse what happens in the combustion chamber when the throttle is set a bit back in order to decrease thrust:

1. the quantity of fuel reaching the combustion chamber reduces;
2. the combustion fuel/air reduces as well since there's less fuel to burn;
3. lower combustion means lower temperature and expansion of the combustion products;
   • the relation among the combustion energy and the temperature is quite simple $Q=mc_p∆T$ where $Q$ is the heat supplied by the combustion, $m$ is the mass of air+fuel, $c_p$ is the isobaric specific heat of air+fuel and $∆T$ the temperature change; this looks like a linear relation, double heat $Q$ double $∆T$; unfortunately $c_p$ depends itself in a non linear way on the pressure and temperature (see for example the plots here) and therefore the relation between combustion heat and temperature's raise is also not linear;
   • ok, temperature changes with the heat $Q$, what about the volume $V$? Even if we consider an ideal gas (that the combustion products are not), the relation is $V/T=\text{constant}$; so if the temperature changes of $∆T=T_2-T_1$, the volume changes of $∆V=V_2(1-T_1/T_2)$ which is also a not linear relation;
4. finally, from Newton's second+third law (aka momentum conservation) and from thermodynamics we get that the relation between the "push" transferred from the expanding combustion gases to the turbine is proportional to the square of the rotational speed $\omega$ i.e. $c_p∆T=K\omega²$ which is also not linear.

So, even only considering what happen in the combustion chamber we see that the relation among the several phisical variables involved in the process is in general not linear.

Answer 3

Any gas turbine engine is inefficient at low rpm. Why in technical terms I don’t know, maybe it is that the compression ratio required for efficient combustion is not achieved by a piston squeezing but by a continuous flow needing several stages to achieve. But it is certainly the nature of the beast. Throttling back below about 85% N2 (and the fan rpm is related to the core rpm) in the engines I operated from the PT6 to the PW4060 and the CF6-80 resulted in fuel inefficiency. That is why if you are short of fuel in a four engine plane you can shut one down. The three remaining have to work harder and the faster you spin them the more fuel efficient they are. Lockheed Orion marine patrol aircraft used to do this as SOP when patrolling down low. Similarly a modern twin airliner that has depressurised and has to fly at 10,000 ft uses more fuel on two engines than it does on one. If you used anywhere near normal cruise thrust down low on two engines you would overspeed the aircraft, and when you throttle back enough to keep the speed below max the engines come below their efficient rpm range. Not that you would do it on purpose, but if you were down to one engine in a twin down low, that engine has to work harder and that is what they like. Even considering the bit of extra drag from asymmetric flight and APU burn you will still use less fuel on one than two.