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.

 

[NatSusan]

Hi there,

I understand your confusion with the Fanno flow and Rayleigh flow equations. Let me try to explain it to you.

The * parameters represent the values at the sonic conditions, which is the point where the flow reaches Mach 1. So, for example, P* represents the pressure at the sonic conditions, which is when the flow reaches Mach 1. Similarly, for the other parameters, * represents the sonic conditions.

To calculate the exit conditions, you need to use the equations in conjunction with each other. You can start by calculating the Mach number at the exit using the given Mach number at the inlet and the friction coefficient. Then, plug in the calculated Mach number in the equations to find the exit conditions.

The Mach number in the equations refers to the local Mach number at any given point in the duct. So, in this case, it will not be the same as the given Mach number of 0.1, as the flow will accelerate or decelerate depending on the duct geometry and conditions.

I hope this helps clear up your confusion. Let me know if you have any other questions.