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Fanno Curve and Adiabatic Gas Flow

In the following blog post we will have a look at the Fanno curve. Therefore, I will distinguish static and stagnation state and explain their relation. This will enable us to understand the basic concept of compressible flow and apply it.

Assumptions

  • Stationary, 1-dimensional flow
  • Single phase, pure component gas flow. The two-phase equations for the stagnation states differ strongly.
  • Adiabatic flow with friction. This means, there is no heat or mass transfer with the environment.
  • Horizontal pipe with a constant diameter

Basics

First, we start with some necessary basics. We can calculate the mass flux G of a fluid through a certain geometry using the mass flow \dot m and the area of the geometry A:

(1)   \begin{equation*} G = \frac{\dot m}{A}. \end{equation*}

The velocity u can be calculated from the mass flux G and the density \rho as follows:

(2)   \begin{equation*} u = \frac{G}{\rho}. \end{equation*}

Static and Stagnation State

A stagnation state is a thermodynamic state of a fluid, where the fluid velocity is zero. This means, the fluid is decelerated isentropically to stagnation. This is what happens for instance when a fluid flow enters a large vessel from a pipe. Hence, the entropy between the static and the stagnation state stays constant. The stagnation enthalpy for gaseous flow h_0 can be calculated from the static enthalpy h and the velocity u as follows:

(3)   \begin{equation*} h_0 = h + 0.5 \cdot u^2. \end{equation*}

Depending on the reference point we can state that the stagnation enthalpy stays constant whenever there is no change in the geodetic height z

(4)   \begin{equation*} h_0 = const. \Rightarrow \Delta z = 0. \end{equation*}

Static and Stagnation Pressure

The difference between the static and the stagnation pressure can be measured using a pitot tube. We can calculate the stagnation pressure p_0 using the following equations.

Stagnation pressure – incompressible:
In case the fluid flow can be presumed as incompressible we can use Bernoulli’s equation to calculate p_0:

(5)   \begin{equation*} p_0 = p + 0.5 \cdot u^2 \cdot \rho. \end{equation*}

Stagnation pressure – compressible (perfect gas):
In case the fluid flow is compressible and can be treated as a perfect gas, p_0 can be calculated as (see [1, p. 132]):

(6)   \begin{equation*} p_0 = p \cdot \left( 1 + \frac{\kappa-1}{2} Ma^2 \right)^\frac{\kappa}{\kappa-1}. \end{equation*}

Stagnation pressure – compressible (real gas) :
To calculate the stagnation pressure of a real gas, we need fluid properties from an equation of state. Hence, we can use the entropy of the static state s and the stagnation enthalpy h_0 to calculate the stagnation pressure p_0

(7)   \begin{equation*} p_0 = p_0(h_0, s). \end{equation*}

Fanno Curve

A Fanno flow is a adiabatic flow in a pipe or duct with a constant diameter, where friction is considered. We can use the Fanno curve concept for compressible and incompressible flow.

Some observations:

  • The mass flux G is constant.
  • The stagnation enthalpy h_0 in the horizontal pipe is constant
  • The static entropy 1_s is equal to the stagnation entropy 1_0.
  • The stagnation pressure is always greater or equal to the static pressure.
  • The difference between the stagnation and static enthalpy can be calculated from 0.5u²
  • In subsonic flow the friction causes an increase in velocity.
  • This means the entropy in subsonic flow can only increase
  • The enthalpy in subsonic flow can only decrease (without a change of the area).
  • The point of maximum entropy is where the Mach number approaches 1 (choking point).
  • This means basically, that a Mach number of 1 can only occur at the end of the pipe. Elsewise the fluid flow would choke.

Fanno curve

Conclusion

The concept of stagnation and static states is an important concept in fluid dynamics. It enables us to model thermo- and fluid dynamical relations. For pipes or ducts with a constant diameter, we can apply the Fanno curve.

Since there are various possibilities to calculate the stagnation states, it is very important to use appropriate equations.

Furthermore, the 2nd law of thermodynamics states, that the entropy in an irreversible process always increases. A choking point occurs when the Mach number reaches 1. It it the point of maximum entropy. Therefore, it can only occur at the end of a pipe. For further details on choked flow have a look at this.

Sources

[1] Thévenin, D., Janiga G.: Fluid Dynamics for Engineers. 2014.

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