Compressible Flow Free Textbook

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A Note on the Entrance Mach number

The value of Mach number, is a function of the resistance, $\frac{4fL}{D}$ and the ratio of pressure in the tank to the back pressure, . The exit pressure, is different from in some situations. As it was shown before, once the flow became choked the Mach number, is only a function of the resistance, $\frac{4fL}{D}$ . These statements are correct for both Fanno flow and the Isothermal flow models. The method outlined in Chapters 8 and 9 is appropriate for solving for entrance Mach number, .

Two equations must be solved for the Mach numbers at the duct entrance and exit when the flow is in a chokeless condition. These equations are combinations of the momentum and energy equations in terms of the Mach numbers. The characteristic equations for Fanno flow (9.50), are

$\displaystyle \frac{4fL}{D} = \left[ {\left.\frac{4fL}{D}\right\vert _{max} } \right]_{1} - \left[ {\left.\frac{4fL}{D}\right\vert _{max} } \right]_{2}$

(11.23)

and

$\displaystyle {P_2 \over P_{0}(t)} = \left[ 1+ {k-1 \over 2} {M_{2}}^{2} \right... ...k-1}{2} {M_{2}}^{2}} {1+ \frac{k-1}{2} {M_1}^{2}} \right] ^ {\frac{k+1}{k-1}} }$

(11.24)

where $\frac{4fL}{D}$ is defined by equation (9.49).

The solution of equations (11.18) and (11.19) for given $\frac{4fL}{D}$ and ${P_{exit}} \over {P_{0}(t)}$ yields the entrance and exit Mach numbers. See advance topic about approximate solution for large resistance, $\frac{4fL}{D}$ or small entrance Mach number, $M_{1}$ .

Next: Rigid Tank with Nozzle Up: General Model and Non-dimensioned Previous: Isothermal Process in The Index

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