The concept of an efficiency for a nozzle

The physics given in this chapter gives a very useful insight to the behaviour of gases in a convergent-divergent nozzle. The actual behaviour will not be so much different but the major difference will lie in the effect of friction on the expansion. That friction will be have two components, the friction between the gas and the solid boundaries of the nozzle and the internal friction stemming from irregularities in the flow. The combined effect will be to reduce the amount of the original internal energy that is extracted and converted to kinetic energy of the flow in the emerging jet. We must decide what effect this has on the flow.

 

If we choose stagnation conditions and a diameter for the throat, according to our model, the pressure at exit for supersonic flow in the divergence will be determined by the diameter at exit. Using our model we can determine a temperature at exit. We can use

The expression  or its derivative form  can be used to find the velocity at exit. So, using the model we can find the exit area, the pressure and the temperature.

 

Now we want to know what happens when a nozzle with these proportions operates with the inevitable friction. We can go back to the origins of . This came from  which is the application of the energy equation to a gas. The sum of the terms  becomes  the enthalpy. Now there is nothing in this that requires the process involved between 1 and 2 to be reversible, the only requirement is that the process is adiabatic. It follows that it can be used for our model of reversible adiabatic flow and for a flow involving friction.

 

I do not think that there can be any doubt that the effect of friction in the nozzle will be to reduce the amount of energy transferred from the internal motion of the molecules of the gas to the kinetic energy of the mass centre of the gas. So, if the kinetic energy is reduced from that produced in a reversible adiabatic flow, the result must be an increase in the absolute temperature at the end of expansion over that for reversible adiabatic flow.

 

For a nozzle with stagnation temperature of  we can denote the temperature after a real expansion as  and after a reversible adiabatic expansion as  and then if the corresponding velocities are  and  and .

 

If the function of a nozzle is to produce kinetic energy then an efficiency  for the nozzle can be  and this is the same as .

I have pointed out that the function of a nozzle in a rocket is to produce a force and the creation of a jet of fast moving gas is the means to produce that force. For rockets one can equate force to the momentum of the jet and then a suitable efficiency would be the ratio of forces. But the only way to improve the ratio of the forces is to improve the process by which the kinetic energy is produced so one nozzle efficiency will do for both.

 

We have said in several contexts that there will be very little loss in convergent flow but significant loss in a divergence. This will be true of the convergent-divergent nozzle. There is an advantage to be gained if the flow up to the throat is treated as reversible and all the loss presumed to take place in the divergence. Then we can relate the throat diameter to stagnation conditions and mass flow and these quantities will not change during calculations concerning the divergence.