A Note on the Entrance Mach number

The value of Mach number, $M_1$ is a function of the resistance, $\frac{4fL}{D}$ and the ratio of pressure in the tank to the back pressure, $P_B/P_1$ . The exit pressure, $P_2$ is different from $P_B$ in some situations. As it was shown before, once the flow became choked the Mach number, $M_1$ 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, $M_1$ .

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

$$\frac{4fL}{D} = \left[ {\left.\frac{4fL}{D}\right\vert _{max} } \right]_{1} - \left[ {\left.\frac{4fL}{D}\right\vert _{max} } \right]_{2}$$ (11.23)

and

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