The statement that propulsive efficiency will be 100% when the exhaust velocity is equal to the aircraft velocity is theoretically correct. But is not possible to achieve this in reality.
To understand what this is all about we need to get behind the maths and understand what is really going on.
Let’s suppose that we are in our jet on the runway with the brakes on and the engines running at max rpm. We are burning fuel to release chemical energy, which is then converted into heat and then into mechanical energy in the form of pressure within our jet pipes. This pressure exerts a rearward force on the exhaust gas, and the exhaust gas in turn exerts an equal and opposite reactionary force as predicted by Newton’s Third Law. This reaction force is our thrust.
But the rearward force which we exerted on the exhaust gas causes it to be accelerated rearwards, thereby giving the exhaust gas an enormous amount of kinetic energy. In effect we have taken all of the useful energy which we extracted from the fuel and thrown it away in the exhaust gas. None of this energy is being used to propel our aircraft forward, so our propulsive efficiency is zero.
The important thing to note in all of this is the fact that the thrust was not actually produced by the acceleration of the exhaust gas. It was produced by the rearward force which we exerted on the exhaust gas. The acceleration of the gas and the resulting loss of energy was just a side effect of the process. It happens because the air is not sufficiently stiff to resist the rearward force which we exerted upon it.
If you have a problem with the idea that the acceleration of the gas does not actually produce the thrust let’s look at a slightly different scenario. You are driving down a motorway at 70 mph. The thrust required to maintain this speed is produced by your wheels exerting a rearward force on the surface of the road. This causes the road to exert an equal and opposite reaction on your wheels, thereby pushing the car forward.
But because the road is stiff enough to resist the forces involved, it does not experience a rearward acceleration. The road surface meets your wheels at 70 mph and leaves your wheels at 70 mph. In this case all of the energy that your engine is providing to the wheels is being used to propel you forward, so your propulsive efficiency is 100%. But this is only possible because the material on which you are exerting your rearward force (the road) is stiff enough to prevent it form being accelerated rearwards.
Now it might be argued that the entire Earth is actually being accelerated backwards relative to your car, and because the mass of the Earth is enormous, the resulting acceleration is to small to detect. To test this argument let’s imagine that we have 2 identical cars standing back to back and tied together with a strong rope. The drivers start their engines and put the cars into first gear then slowly let out the clutches. The cars move forward until the rope becomes stretched and are then brought to a standstill. The cars are pointing in opposite directions so they cannot both be accelerating the road rearwards. But both cars are producing thrust and it is this which is exerting tension on the rope.
For another example we could look at an inflated baloon. The gas in the boloon exerts an outward force which is balanced by an equal and opposite inward force exerted by the baloon envelope. There is no acceleration but there are equal and opposite forces.
Now getting back to our aircraft sitting on the runway, if we release the brakes the aircraft will start to accelerate forward. As it accelerates forward its velocity becomes closer and closer to the exhaust speed. This means that the amount of acceleration that we giving to the exhaust gas is becoming less and less. This in turn means that the amount of kinetic energy that are we throwing away in the exhaust gas is becoming less and less, so more of the energy derived from the fuel is being used to propel us forward. So our propulsive efficiency is gradually increasing.
This process will continue after lift-off. If we climb at constant IAS the increasing TAS will produce an increasing propulsive efficiency. But we will never get to 100% because the exhaust gas is not sufficiently stiff to resist being accelerated rearwards.
Last edited by keith williams; 17th Dec 2016 at 19:22.