The Entrance Limitation of Supersonic Branch

Situations where the conditions at the tube exit have not arrived at the critical conditions are discussed here. It is very useful to obtain the relationship between the entrance and the exit condition for this case. Denote 1 and 2 as the conditions at the inlet and exit respectably. From equation (8.24)

$$\frac{4fL}{D} = \left. \frac{4fL}{D}\right\vert _{{max}_1} - \lef... ... k{M_{2}}^{2} \over k {M_{2}}^{2}} + \ln \left( {M_{1} \over M_{2}} \right)^{2}$$ (8.36)

For the case that $M_1 > > M_2$ and $M_1 \rightarrow 1$ equation (8.36) is reduced into the following approximation

$$\frac{4fL}{D} = 2 \ln M_{1} -1 - \overbrace{ 1 - k{M_{2}}^{2} \over k {M_{2}}^{2}}^{\sim 0}$$ (8.37)

Solving for $M_1$ results in

$$M_1 \sim \hbox{\huge e}^{{1\over 2}\left(\frac{4fL}{D} +1\right)}$$ (8.38)

This relationship shows the maximum limit that Mach number can approach when the heat transfer is extraordinarily fast. In reality, even small $\frac{4fL}{D} > 2$ results in a Mach number which is larger than 4.5. This velocity requires a large entrance length to achieve good heat transfer. With this conflicting mechanism obviously the flow is closer to the Fanno flow model. Yet this model provides the directions of the heat transfer effects on the flow.