Supersonic Branch

Apparently, this analysis/model is over simplified for the supersonic branch and does not produce reasonable results since it neglects to take into account the heat transfer effects. A dimensionless analysis^8.4 demonstrates that all the common materials that the author is familiar which creates a large error in the fundamental assumption of the model and the model breaks. Nevertheless, this model can provide a better understanding to the trends and deviations of the Fanno flow model.

In the supersonic flow, the hydraulic entry length is very large as will be shown below. However, the feeding diverging nozzle somewhat reduces the required entry length (as opposed to converging feeding). The thermal entry length is in the order of the hydrodynamic entry length (look at the Prandtl number, (0.7-1.0), value for the common gases.). Most of the heat transfer is hampered in the sublayer thus the core assumption of isothermal flow (not enough heat transfer so the temperature isn't constant) breaks down^8.5.

The flow speed at the entrance is very large, over hundred of meters per second. For example, a gas flows in a tube with $\frac{4fL}{D} =10$ the required entry Mach number is over 200. Almost all the perfect gas model substances dealt with in this book, the speed of sound is a function of temperature. For this illustration, for most gas cases the speed of sound is about $300 [m/sec]$ . For example, even with low temperature like $200K$ the speed of sound of air is $283 [m/sec]$ . So, even for relatively small tubes with $\frac{4fL}{D} =10$ the inlet speed is over 56 [km/sec]. This requires that the entrance length to be larger than the actual length of the tub for air. Remember from Fluid Dynamic book

$$L_{entrance} = 0.06 { U D \over \nu}$$ (8.53)

The typical values of the the kinetic viscosity, $\nu$ , are 0.0000185 kg/m-sec at 300K and 0.0000130034 kg/m-sec at 200K. Combine this information with our case of $\frac{4fL}{D} =10$

$${L_{entrance} \over D} = 250746268.7$$

On the other hand a typical value of friction coefficient $f = 0.005$ results in

$${L_{max} \over D} = { 10 \over 4 \times 0.005} = 500$$

The fact that the actual tube length is only less than 1% of the entry length means that the assumption is that the isothermal flow also breaks (as in a large response time).

Now, if Mach number is changing from 10 to 1 the kinetic energy change is about $\mathbf{T_0 \over {T_0}^{*}} = 18.37$ which means that the maximum amount of energy is insufficient.

Now with limitation, this topic will be covered in the next version because it provide some insight and boundary to the Fanno Flow model.