Fanno and Rayleigh Flow, calculating exit conditions

[roldy]

I've seen equations for Fanno flow and Rayleigh flow but I am confused on how to use them properly.

Fanno Flow

$$\frac{P}{P^{*}}=\frac{1}{M}\frac{1}{\sqrt{\left(\frac{2}{\gamma+1}}\right)\left(1+\frac{\gamma-1}{2}M^2\right)}$$

$$\frac{\rho}{\rho^{*}}=\frac{1}{M}{\sqrt{\left(\frac{2}{\gamma+1}}\right)\left(1+\frac{\gamma-1}{2}M^2\right)}$$

$$\frac{T}{T^{*}}=\frac{1}{\left(\frac{2}{\gamma+1}\right)\left(1+\frac{\gamma-1}{2}M^2\right)}$$

$$\frac{U}{U^{*}}=M\frac{1}{\sqrt{\left(\frac{2}{\gamma+1}}\right)\left(1+\frac{\gamma-1}{2}M^2\right)}$$

$$\frac{P_0}{P_0^{*}}=\frac{1}{M}\left[\left(\frac{2}{\gamma+1}}\right)\left(1+\frac{\gamma-1}{2}M^2\right)\right]^{\frac{\gamma+1}{2(\gamma-1)}}$$

$$T_0=T_0^{*}$$

Inlet conditions:

$$P_0=101325$$ Pa

$$T_0=288$$ K

$$M_0=0.1$$

$$Area=0.1 m^2$$

$$T_{wall}=3000$$K

Friction Coefficient($$C_f$$)=0.2

Duct Length=10 m

Adiabatic, no work interaction, constant area

I guess my confusion comes from the * parameters. If I know my inlet conditions only, how can I calculate the exit conditions from these equations? What do the numerator parameters represent? Are they the values at the exit? The Mach number in these equations, are they any the given Mach number (in this case 0.1)?

 

[boneh3ad]

The star quantities just represent the pressure at sonic condition (M = 1). It is just a reference condition. If you had, for example, $p_{1}$ and wanted $p_{2}$, you would simply do:
$$p_{2} = \frac{p_2}{p^*}\frac{p^*}{p_1}p_1$$

You can generally find those values in tables or just calculate them directly if you wish.