Now we have derived all the essential equations concerning thrust, power and efficiency in the previous video, we can start to tackle our original problem. As explained in the previous video, it is my intention to express thrust as a function of airspeed in order to simplify the equations of motion. I will do this for two distinct types of propulsion systems.
First the pure jet engine, and second variable pitch propellers. Now let us start with the jet engine. The basic operating principle of this engine is as follows. Air flows into the engine and is compressed, fuel is injected, combustion starts and the air is heated up to a large temperature.
Since it has a large temperature and pressure it wants to expand. During expansion it flows past the turbine which drives the compressor. After the turbine however, It still has a lot of energy and expands through the nozzle up to the jet velocity. This jet velocity mainly depends on the energy that is put into the gas by means of fuel and the pressure of the gas inside the engine.
Even though there is some compression of air in front of the engine, the majority of the compression is achieved inside the compressor. And the amount of energy that is put into the gas is limited by the maximum temperature that the material of which the combustion chamber is made can withstand. So, how does thrust of a jet engine vary with airspeed if we have a specific throttle setting?
The thrust equation is presented here. If the airspeed is increased, mass flow will increase as well. The jet velocity is not really affected to a great extent, because this is what we are looking for. This mainly depends on the compression ratio and the throttle setting. The compression in front of the engine will change a bit, but this is just a minor effect compared to the other effects.
So mass flow increases, but the velocity differences decreases. These two effects more or less cancel each other out. As a result, jet engine thrust is more or less constant as a function of airspeed. when all other variables such as throttle setting and altitude are kept constant. In this picture you can see the variation of thrust with flight speed for a real jet engine.
There are three lines indicated in the picture for three different throttle settings set by the pilot. Now in essence the pilot can select any thrust level below the maximum by decreasing the fuel flow via the throttle. All lines indicated here are fairly flat and show that thrust indeed remains more or less constant for varying airspeed like I just explained.
And because of this specific behavior I will assume for our performance calculations that thrust can be assumed constant with airspeed for a given flight altitude and a given throttle setting. Consequently the power available for a jet engine, which is thrust multiplied with airspeed becomes a straight line from the origin. So now I have expressed the propulsive force of a jet engine as a function of speed in order to simplify the equations of motion. Let's do exactly the same exercise for a propeller. A propeller blade is essentially a rotating wing which creates lift and drag.
The part of the lift and drag which act forward or perpendicular to the plane of rotation is thrust and the part of lift and drag in the plane of rotation creates a torque on the shaft which drives the propeller. This torque needs to be overcome by the engine. The velocity of a blade section depends on the rotational speed and the forward speed of the aircraft. These velocities also determine the speed of the plane.
determine the local angle of attack of a blade section together with the geometric angle attack of the blade, indicated by beta in the figure. An increase in flight speed will result in a decrease in angle of attack and thereby in a change of lift and drag. Most modern propellers have a constant rotational speed and a variable geometric pitch.
By varying the pitch of the propeller, the angle of attack and thereby the thrust can be controlled. In each flight condition the optimum setting can be selected. If one would make a diagram with propulsive efficiency as a function of airspeed for a fixed setting, it would have the following shape.
At one specific flight speed the propulsive efficiency will be maximal. However, if you would draw this same graph for various propeller pitch settings, then you can see that the optimum shifts. Because of this, it is possible to select the maximum efficiency at each flight speed, by changing the propeller pitch. If we assume that the engine delivers a constant shaft power, for example by means of a conventional piston engine, then the propulsive efficiency used for evaluation of jet engines can be replaced by the propeller efficiency, which we define as the power available divided by the shaft power. If the shaft power is constant and efficiency is kept constant by varying blade pitch, then power available will be constant as well.
This is quite an important result, because this means that power available is independent of flight speed for a given flight altitude and a given throttle setting. Consequently the thrust decreases with a 1 over x shape with increasing flight speed. This of course is theoretical behavior, because it also implies that at zero velocity Thrust will be infinite. In reality however, thrust and power available of a variable pitch propeller have the following shape.
The range of flight speeds of interest for performance calculations in the current lecture series is however in the range where power available is indeed constant. So from now on I will assume that the power available for a propeller aircraft is independent of airspeed for a given flight altitude and throttle setting. Concluding, I have expressed the propulsive force for both pure jet engines and variable pitch propellers as a function of airspeed in order to simplify the equations of motion.
Because of the simple form of the thrust and power curves, this allows us to analytically calculate aircraft performance as you will see in the next lectures. You might wonder. But what if an aircraft is equipped with a propulsion system that does not have this simplified behavior?
For example if it is equipped with a turbofan engine, which can be seen on most commercial aircraft. Well, if the engine manufacturer provides us with thrust or power variation with airspeed, then we can still do our calculations. However, in that particular situation the calculations will then have to be performed either graphically or numerically and cannot be solved analytically anymore. This concludes the treatment of the behavior of the propulsion system for aircraft performance calculations.