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11.7.6.2: Fanno Flow Supersonic Branch

Fig. 11.25 The Mach numbers at entrance and exit of tube and mass flow rate for Fanno Flow as a function of the \(ld\).

There are several transitional points that change the pattern of the flow. Point \(a\) is the choking point (for the supersonic branch) in which the exit Mach number reaches to one. Point \(b\) is the maximum possible flow for supersonic flow and is not dependent on the nozzle. The next point, referred here as the critical point \(c\), 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 \(d\), 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 11.25): The 0-\(a\) range, the mass flow rate is constant because the flow is choked at the nozzle. The entrance Mach number, \(M_1\) is constant because it is a function of the nozzle design only. The exit Mach number, \(M_2\) decreases (remember this flow is on the supersonic branch) and starts (\(\dfrac{4\,f\,L}{D}=0\)) as \(M_2= M_1\). At the end of the range \(\mathbf{a}\), \(M_2=1\). 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 \(\mathbf{b}-\mathbf{b'}\), \(\left(\dfrac{4\,f\,L}{D}\right)_{max}\), 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, \(M_2\) is still one. However, the entrance Mach number, \(M_1\), reduces with the increase of \(\dfrac{4\,f\,L}{D}\). It is worth noticing that in the \(\mathbf{a} - \mathbf{c}\) the mass flow rate nozzle entrance velocity and the exit velocity remains constant! In the last range \(c−\infty\) 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 (\(\dot{m} \propto M_1\)). To summarize the above discussion, Figures 11.25 exhibits the development of \(M_1\), \(M_2\) mass flow rate as a function of \(\dfrac{4\,f\,L}{D}\). 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 \(M_2\) is a continuous monotonic function that decreases with \(\dfrac{4\,f\,L}{D}\).  The entrance Mach \(M_1\) is a non continuous function with a jump at the point when shock occurs at the entrance "moves'' into the nozzle.

Fig. 11.26 \(M_1\) as a function \(M_2\) for various \(ld\).

Figure ?? exhibits the \(M_1\) as a function of \(M_2\). The Figure was calculated by utilizing the data from Figure 11.20 by obtaining the \(\left.\dfrac{4\,f\,L}{D}\right|_{max}\) for \(M_2\) and subtracting the given \(\dfrac{4\,f\,L}{D}\) and finding the corresponding \(M_1\).

Fig. 11.27 \(M_1\) as a function \(M_2\) for different \(ld\) for

The Figure (??) exhibits the entrance Mach number as a function of the \(M_2\). 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 ?? only for \(\dfrac{4\,f\,L}{D}=0.1\) and \(\dfrac{4\,f\,L}{D}=0.4\) two extremes are shown. For \(\dfrac{4\,f\,L}{D}= 0.2\) shown with only shock at the exit only. Obviously, and as can be observed, the larger \(\dfrac{4\,f\,L}{D}\) creates larger differences between exit Mach number for the different shock locations. The larger \(\dfrac{4\,f\,L}{D}\) larger \(M_1\) must occurs even for shock at the entrance. For a given \(\dfrac{4\,f\,L}{D}\), 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
\begin{align*}
    \dfrac{4\,f\,L}{D} > - \dfrac{1 }{k} + \dfrac{1+k }{ 2\,k} \ln\left(\dfrac{ k+1 }{ k-1}\right)
\end{align*}
For cases of \(\dfrac{4\,f\,L}{D}\) 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.

The Pressure Ratio, \(\left.{P_2 }\right/{ P_1}\), effects

In this section the studied parameter is the variation of the back pressure and thus, the pressure ratio \(\left(\left.P_2 \right/ P_1\right)\) variations. For very low pressure ratio the flow can be assumed as incompressible with exit Mach number smaller than \(<0.3\). As the pressure ratio increases (smaller back pressure, \(P_2\)), the exit and entrance Mach numbers increase. According to Fanno model the value of \(\dfrac{4\,f\,L}{D}\) is constant (friction factor, \(f\), is independent of the parameters such as, Mach number, Reynolds number et cetera) thus the flow remains on the same Fanno line. For cases where the supply come from a reservoir with a constant pressure, the entrance pressure decreases as well because of the increase in the entrance Mach number (velocity). Again a differentiation of the feeding is important to point out. If the feeding nozzle is converging than the flow will be only subsonic. If the nozzle is "converging-diverging'' than in some part supersonic flow is possible. At first the converging nozzle is presented and later the converging-diverging nozzle is explained.

Fig. 11.28 The pressure distribution as a function of \(\dfrac{4\,f\,L}{D}\) for a short \(\dfrac{4\,f\,L}{D}\).

Contributors

  • Dr. Genick Bar-Meir. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or later or Potto license.