Contribution of the air in the jet aircraft engine

[duarthiago]

I've read that the total forward force on an engine is given by a change of momentum which is written as $P = \mu_{fuel}v_{0} + \mu_{air}(v_{0}-v)$, where $\mu_{fuel}$, $\mu_{air}$, $v_{0}$ and $v$ are respectively the rate of decrease of mass of the fuel being burnt, the rate which the air is being carried through the engine, the velocity of the material ejected by the engine and finally the velocity of the aircraft.

The book where I read it says that "The main function of the fuel that is carried with the plane is to give the ejected gases a high speed with respect to the plane, and most of the moving mass is supplied by the air" but I really don't understood it, I mean, I can't visualize it. I would to know what that $\mu_{air}(v_{0}-v)$ physically means. I thought the change of momentum was due only by the fuel burnt.

 

[jbriggs444]

$v_{0}$ would be the velocity of the air as it enters the engine, not the velocity of the air as it passes through the engine. The difference between $v_{0}$ and $v$ would be the change in velocity of air resulting from its passing through the engine. Multiply by $\mu_{air}$ and you have the rate at which [rearward] momentum is being imparted to the air.

 

[A.T.]

I thought the change of momentum was due only by the fuel burnt.

Energy comes from the reaction. Momentum goes mostly to the air.

 

[sophiecentaur]

Energy comes from the reaction. Momentum goes mostly to the air.

Unlike a rocket engine that has to provide the chemical power and all the propellant mass.

 

[duarthiago]

$v_{0}$ would be the velocity of the air as it enters the engine, not the velocity of the air as it passes through the engine. The difference between $v_{0}$ and $v$ would be the change in velocity of air resulting from its passing through the engine. Multiply by $\mu_{air}$ and you have the rate at which [rearward] momentum is being imparted to the air.

I don't know about $v_{0}$, the book seems quite clear about it be the velocity of the ejected material, although what you said fits with the diagram that I have here, with $v$ being the velocity of the air entering the engine. So what is happening is that the change of velocity of air which passes through the engine (and that change is possible thanks to the energy provided by the burnt) and the loss of fuel mass escaping with the same velocity of the air are the factors that produce the change of momentum of the aircraft?

 

[jbriggs444]

I don't know about $v_{0}$, the book seems quite clear about it be the velocity of the ejected material, although what you said fits with the diagram that I have here, with $v$ being the velocity of the air entering the engine. So what is happening is that the change of velocity of air which passes through the engine (and that change is possible thanks to the energy provided by the burnt) and the loss of fuel mass escaping with the same velocity of the air are the factors that produce the change of momentum of the aircraft?

You are correct. I had reversed $v_{0}$ and $v$.

 

[duarthiago]

Very well! Thank you everyone.

 

[A.T.]

Unlike a rocket engine that has to provide the chemical power and all the propellant mass.

Yes, the other extreme is a piston engine with a propeller, where basically all the momentum is going to the air. Jet engines are in between those two extremes.