Entrance Mach number, M1, effects

Figure 9.15: Schematic of a ``long'' tube in supersonic branch

In this discussion, the effect of changing the throat area on the nozzle efficiency is neglected. In reality these effects have significance and needs to be accounted for some instances. This dissection deals only with the flow when it reaches the supersonic branch reached otherwise the flow is subsonic with regular effects. It is assumed that in this discussion that the pressure ratio $P_2 \over P_1$ is large enough to create a choked flow and $\frac{4fL}{D}$ is small enough to allow it to happen.

The entrance Mach number, $M_1$ is a function of the ratio of the nozzle's throat area to the nozzle exit area and its efficiency. This effect is the third parameter discussed here. Practically, the nozzle area ratio is changed by changing the throat area.

As was shown before, there are two different maximums for $\frac{4fL}{D}$ ; first is the total maximum $\frac{4fL}{D}$ of the supersonic which depends only on the specific heat, $k$ , and second the maximum depends on the entrance Mach number, $M_1$ . This analysis deals with the case where $\frac{4fL}{D}$ is shorter than total $\left.\frac{4fL}{D}\right\vert _{max}$ .

Obviously, in this situation, the critical point is where $\frac{4fL}{D}$ is equal to $\left.\frac{4fL}{D}\right\vert _{max}$ as a result in the entrance Mach number.

The process of decreasing the converging-diverging nozzle's throat increases the entrance^9.14Mach number. If the tube contains no supersonic flow then reducing the nozzle throat area wouldn't increase the entrance Mach number.

This part is for the case where some part of the tube is under supersonic regime and there is shock as a transition to subsonic branch. Decreasing the nozzle throat area moves the shock location downstream. The ``payment'' for increase in the supersonic length is by reducing the mass flow. Further, decrease of the throat area results in flushing the shock out of the tube. By doing so, the throat area decreases. The mass flow rate is proportionally linear to the throat area and therefore the mass flow rate reduces. The process of decreasing the throat area also results in increasing the pressure drop of the nozzle (larger resistance in the nozzle^9.15)^9.16.

In the case of large tube $\frac{4fL}{D} > \left.\frac{4fL}{D}\right\vert _{max}$ the exit Mach number increases with the decrease of the throat area. Once the exit Mach number reaches one no further increases is possible. However, the location of the shock wave approaches to the theoretical location if entrance Mach, $M_1= \infty$ .

Figure 9.16: The extra tube length as a function of the shock location

The extra tube length as a function of the shock location, $\frac{4fL}{D}$ supersonic branch