Advance Topics

The term $\frac{4fL}{D}$ is very large for small values of the entrance Mach number which requires keeping many digits in the calculation. For small values of the Mach numbers, equation (11.18) can be approximated as

$$\frac{4fL}{D} = \frac{1}{k} \frac{{M_{exit}}^{2}-{M_{in}}^{2}} {{M_{exit}}^{2}{M_{in}}^{2}}$$ (11.58)

and equation (11.19) as

$$\frac{P_{exit}}{P_{0}(t)} = \frac{M_{in}}{M_{exit}}.$$ (11.59)

The solution of two equations (11.53) and (11.54) yields

$$M_{in} = \sqrt{ \frac{ 1- \left[ \frac{P_{exit}}{P_{0}(t)}\right]^{2} } { k{\frac{4fL}{D}} } }.$$ (11.60)

This solution should used only for $M_{in} < 0.00286$ ; otherwise equations (11.18) and (11.19) must be solved numerically.

The solution of equation (11.18) and (11.19) is described in ``Pressure die casting: a model of vacuum pumping'' Bar-Meir, G; Eckert, E R G; Goldstein, R. J. Journal of Manufacturing Science and Engineering (USA). Vol. 118, no. 2, pp. 259-265. May 1996.