Compressible Flow Free Textbook

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Fanno Flow Supersonic Branch

There are several transitional points that change the pattern of the flow. Point $\mathbf{a}$ is the choking point (for the supersonic branch) in which the exit Mach number reaches to one. Point $\mathbf{b}$ is the maximum possible flow for supersonic flow and is not dependent on the nozzle. The next point, referred here as the critical point $\mathbf{c}$ , is the point in which no supersonic flow is possible in the tube i.e. the shock reaches to the nozzle. There is another point $\mathbf{d}$ , in which no supersonic flow is possible in the entire nozzle-tube system. Between these transitional points the effect parameters such as mass flow rate, entrance and exit Mach number are discussed.

At the starting point the flow is choked in the nozzle, to achieve supersonic flow. The following ranges that has to be discussed includes (see Figure (9.8)):

0	$\frac{4fL}{D}$	$\left(\frac{4fL}{D}\right)_{choking}$	$0 \rightarrow \mathbf{a}$
$\left(\frac{4fL}{D}\right)_{choking}$	$\frac{4fL}{D}$	$\left(\frac{4fL}{D}\right)_{shockless}$	$\mathbf{a} \rightarrow \mathbf{b}$
$\left(\frac{4fL}{D}\right)_{shockless}$	$\frac{4fL}{D}$	$\left(\frac{4fL}{D}\right)_{chokeless}$	$\mathbf{b} \rightarrow \mathbf{c}$
$\left(\frac{4fL}{D}\right)_{chokeless}$	$\frac{4fL}{D}$	$\infty$	$\mathbf{c} \rightarrow \infty$

**Figure 9.8:** $M_{in}$ and $\dot{m}$ as a function of the $\frac{4fL}{D}$

The 0- $\mathbf{a}$ range, the mass flow rate is constant because the flow is choked at the nozzle. The entrance Mach number,

is constant because it is a function of the nozzle design only. The exit Mach number,

decreases (remember this flow is on the supersonic branch) and starts ( $\frac{4fL}{D}=0$ ) as

. At the end of the range $\mathbf{a}$ ,

. In the range of $\mathbf{a}-\mathbf{b}$ the flow is all supersonic.

In the next range $\mathbf{a}-\mathbf{b}$ The flow is double choked and make the adjustment for the flow rate at different choking points by changing the shock location. The mass flow rate continues to be constant. The entrance Mach continues to be constant and exit Mach number is constant.

The total maximum available for supersonic flow $\mathbf{b} - \mathbf{b'}$ , $\left(\frac{4fL}{D}\right)_{max}$ , is only a theoretical length in which the supersonic flow can occur if nozzle is provided with a larger Mach number (a change to the nozzle area ratio which also reduces the mass flow rate). In the range $\mathbf{b} -\mathbf{c}$ , it is a more practical point.

In semi supersonic flow $\mathbf{b} -\mathbf{c}$ (in which no supersonic is available in the tube but only in the nozzle) the flow is still double choked and the mass flow rate is constant. Notice that exit Mach number, is still one. However, the entrance Mach number, , reduces with the increase of $\frac{4fL}{D}$ .

It is worth noticing that in the $\mathbf{a} - \mathbf{c}$ the mass flow rate nozzle entrance velocity and the exit velocity remains constant!^9.10

In the last range $\mathbf{c} - \infty$ the end is really the pressure limit or the break of the model and the isothermal model is more appropriate to describe the flow. In this range, the flow rate decreases since ( $\dot{m} \propto M_1$ )^9.11.

To summarize the above discussion, Figures (9.8) exhibits the development of , mass flow rate as a function of $\frac{4fL}{D}$ . Somewhat different then the subsonic branch the mass flow rate is constant even if the flow in the tube is completely subsonic. This situation is because of the ``double'' choked condition in the nozzle. The exit Mach is a continuous monotonic function that decreases with $\frac{4fL}{D}$ . The entrance Mach is a non continuous function with a jump at the point when shock occurs at the entrance ``moves'' into the nozzle.

**Figure:** M1 as a function M2 for various 4fL/D
$\begin{figure}\centerline{\includegraphics {calculations/fannoM1M2}}\end{figure}$

Figure (9.9) exhibits the

as a function of

. The Figure was calculated by utilizing the data from Figure (9.2) by obtaining the $\left.\frac{4fL}{D}\right\vert _{max}$ for

and subtracting the given $\frac{4fL}{D}$ and finding the corresponding

**Figure:** M1 as a function M2
$\begin{figure}\centerline{\includegraphics {calculations/fannoM1M2Supper}}\end{figure}$

The Figure (9.10) exhibits the entrance Mach number as a function of the . Obviously there can be two extreme possibilities for the subsonic exit branch. Subsonic velocity occurs for supersonic entrance velocity, one, when the shock wave occurs at the tube exit and two, at the tube entrance. In Figure (9.10) only for $\frac{4fL}{D}=0.1$ and $\frac{4fL}{D}=0.4$ two extremes are shown. For $\frac{4fL}{D}= 0.2$ shown with only shock at the exit only. Obviously, and as can be observed, the larger $\frac{4fL}{D}$ creates larger differences between exit Mach number for the different shock locations. The larger $\frac{4fL}{D}$ larger must occurs even for shock at the entrance.

For a given $\frac{4fL}{D}$ , below the maximum critical length, the supersonic entrance flow has three different regimes which depends on the back pressure. One, shockless flow, tow, shock at the entrance, and three, shock at the exit. Below, the maximum critical length is mathematically

$\displaystyle \frac{4fL}{D} > - {1 \over k} + {1+k \over 2k} \ln{ k+1 \over k-1}$

For cases of $\frac{4fL}{D}$ above the maximum critical length no supersonic flow can be over the whole tube and at some point a shock will occur and the flow becomes subsonic flow^9.12.

Next: The Pressure Ratio, P2/P1, Up: Variations of The Tube Previous: Fanno Flow Subsonic branch Index

Created by:Genick Bar-Meir, Ph.D.
On: 2007-11-21