# 11.7.6.1: Variations of The Tube Length ($$ld$$) Effects

In the analysis of this effect, it should be assumed that back pressure is constant and/or low as possible as needed to maintain a choked flow. First, the treatment of the two branches are separated.

### Fanno Flow Subsonic branch

Fig. 11.24 The effects of increase of $$\dfrac{4\,f\,L}{D}$$ on the Fanno line.

For converging nozzle feeding, increasing the tube length results in increasing the exit Mach number (normally denoted herein as $$M_2$$). Once the Mach number reaches maximum ($$M=1$$), no further increase of the exit Mach number can be achieved with same pressure ratio mass flow rate. For increase in the pipe length results in mass flow rate decreases. It is worth noting that entrance Mach number is reduced (as some might explain it to reduce the flow rate). The entrance temperature increases as can be seen from Figure 11.24. The velocity therefore must decrease because the loss of the enthalpy (stagnation temperature) is "used.'' The density decrease because $$\rho = \dfrac{P }{ R\, T}$$ and when pressure is remains almost constant the density decreases. Thus, the mass flow rate must decrease. These results are applicable to the converging nozzle. In the case of the converging–diverging feeding nozzle, increase of the dimensionless friction, $$ld$$, results in a similar flow pattern as in the converging nozzle. Once the flow becomes choked a different flow pattern emerges.

### 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.