Fanno Flow - Additional Fanno Flow Relations

Additional Fanno Flow Relations

As was stated earlier, the area and mass flow rate in the duct are held constant for Fanno flow. Additionally, the stagnation temperature remains constant. These relations are shown below with the * symbol representing the throat location where choking can occur. A stagnation property contains a 0 subscript.

\begin{align} A &= A^* = \mbox{constant} \\ T_0 &= T_0^* = \mbox{constant} \\ \dot{m} &= \dot{m}^* = \mbox{constant} \end{align}

Differential equations can also be developed and solved to describe Fanno flow property ratios with respect to the values at the choking location. The ratios for the pressure, density, temperature, velocity and stagnation pressure are shown below, respectively. They are represented graphically along with the Fanno parameter.

\begin{align} \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{V}{V^*} &= 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^\frac{\gamma + 1}{2\left(\gamma - 1\right)} \end{align}

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