T C said:
Suppose the velocity would be in the subsonic level throughout the process.
This is literally always the case in a converging nozzle. The flow cannot be anything other than subsonic.
T C said:
Efficiency means the velocity at the exit would be as close to the theoretical value as possible. Suppose we have a nozzle having 5:1 ratio i.e. the inlet to throat area ratio is 5:1. That means the velocity at the exit would be five times that of inlet. I want to know with wihich geometry, the real velocity at the exit/throat would be as close to 5 times that of the inlet as possible. That I will consider to be the most efficient.
russ_watters said:
If that's the case, then every nozzle will satisfy your requirements and provide 100% efficiency by that definition of efficiency. Conservation of mass demands it.
@russ_watters is correct in pointing out that your 5:1 comment is only true for an incompressible flow. However, he is not correct in stating that every nozzle will satisfy your requirements. Conservation of mass demands that the stated efficiency metric be satisfied
on average when you compare the inlet and the exit. In the real world, viscosity causes the actual contraction ratio to be different from the one set by the nozzle. Conservation of mass states that the mass flow into the nozzle equals the mass flow out. Mathematically, this is
0 = \int_{A_{\mathrm{out}}}\rho \vec{u} \cdot\;d\vec{A} - \int_{A_{\mathrm{in}}}\rho \vec{u}\cdot \;d\vec{A}
If ##\rho## and ##u## are constant at the inlet and exit, then your statements both old.
In a real situation, even for an incompressible flow where ##\rho## is constant, ##u## is not constant. A boundary layer exists that is going to make both the inlet and the exit effectively smaller than the physical geometry dictates. This effect will be more pronounced at the exit, so the real contraction ratio will tend to be larger than what you would measure from the dimensions of your nozzle. Your outlet flow would be a bit faster. Viscosity also dissipates energy.
russ_watters said:
Usually, though, efficiency in such a case is measured by energy loss, which in that case come from static pressure loss.
That aside, the basic answer to your question is that there is no single answer to your question. That's why there is more than one nozzle design.
Honestly, for his criterion, the basic answer is simply "the biggest nozzle you can find" because that would mean the constrictions from the boundary layer are minimized relative to the physical dimension of the nozzle. Otherwise, the answer usually revolves around whichever nozzle minimizes separation in the boundary layer, which is a much more complicated question to answer.
EDIT: Fixed my equation to include vector notation since I left it off before and it therefore wasn't technically correct.