Momentum equation application
Introduction: The propeller is a mechanical devise that is used to increase the fluid momentum. Many times it is used for propulsion purposes of airplanes, ships and there devices (thrust). The propeller can be stationary like in cooling tours, fan etc. The other common used of propeller is used mostly to move fluids as a pump.
The propeller analysis of unsteady is complicated due to the difficulty in understanding the velocity field. For a steady state the analysis is simpler and used here to provide an example of steady state. In the fluid flows from the left to the right. Either it is assumed that some of the fluid enters into the container and fluid outside is not affected by the propeller. Or there is a line (or surface) in which the fluid outside changes only the flow direction. This surface is called slip surface. Of course it is only approximation but is provided a crude tool. Improvements can be made to this analysis. Here, this analysis is used for academic purposes.
As first approximation, the pressure around around control volume is the same. Thus, pressure drops from the calculation. The one dimensional momentum equation is reduced
Combining the control volume between points 1 and 3 with (note that there are no external forces) with points 4 and 2 results
In This analysis provide way to calculate the work needed to move this propeller. Note that in this analysis it was assumed that the flow is horizontal that z1 = z2 and/or the change is insignificant.
Jet Propulsion: Jet propulsion is a mechanism in which the air planes and other devices are propelled. Essentially, the air is sucked into engine and with addition heating (burning fuel) the velocity is increased. Further increase of the exit area with the increased of the burned gases further increase the thrust. The analysis of such device in complicated and there is a whole class dedicated for such topic in many universities.
Here, a very limited discussion related to the steady state is offered
The difference between the jets propulsion and propellers is based on the energy supplied. The propellers are moved by a mechanical work which is converted to thrust. In Jet propulsion, the thermal energy is converted to thrust. Hence, this direct conversion can be, and is, in much case more efficient. Furthermore, as it will be shown in the Chapter on compressible flow it allows achieving velocity above speed of sound, a major obstacle in the past.
The inlet area and exit area are different for most jets and if the mass of the fuel is neglected then
An academic example to demonstrate how steady state calculations are done for a moving control volume. Notice that
MOMENTUM EQUATION APPLICATION: According to the statement of question the friction force is
Momentum in the x direction becomes
Toy velocity is then
Increase of the friction reduce the velocity. Additionally larger toy mass decrease the