**Theory**

The Fanno flow model begins with a differential equation that relates the change in Mach number with respect to the length of the duct, *dM/dx*. Other terms in the differential equation are the heat capacity ratio, *γ*, the Fanning friction factor, *f*, and the hydraulic diameter, *D*_{h}:

Assuming the Fanning friction factor is a constant along the duct wall, the differential equation can be solved easily. One must keep in mind, however, that the value of the Fanning friction factor can be difficult to determine for supersonic and especially hypersonic flow velocities. The resulting relation is shown below where *L** is the required duct length to choke the flow assuming the upstream Mach number is supersonic. The left-hand side is often called the Fanno parameter.

Equally important to the Fanno flow model is the dimensionless ratio of the change in entropy over the heat capacity at constant pressure, *c*_{p}.

The above equation can be rewritten in terms of a static to stagnation temperature ratio, which, for a calorically perfect gas, is equal to the dimensionless enthalpy ratio, *H*:

The equation above can be used to plot the Fanno line, which represents a locus of states for given Fanno flow conditions on an *H*-*ΔS* diagram. In the diagram, the Fanno line reaches maximum entropy at *H* = 0.833 and the flow is choked. According to the Second law of thermodynamics, entropy must always increase for Fanno flow. This means that a subsonic flow entering a duct with friction will have an increase in its Mach number until the flow is choked. Conversely, the Mach number of a supersonic flow will decrease until the flow is choked. Each point on the Fanno line corresponds with a different Mach number, and the movement to choked flow is shown in the diagram.

The Fanno line defines the possible states for a gas when the mass flow rate and total enthalpy are held constant, but the momentum varies. Each point on the Fanno line will have a different momentum value, and the change in momentum is attributable to the effects of friction.

Read more about this topic: Fanno Flow

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### Famous quotes containing the word theory:

“We commonly say that the rich man can speak the truth, can afford honesty, can afford independence of opinion and action;—and that is the *theory* of nobility. But it is the rich man in a true sense, that is to say, not the man of large income and large expenditure, but solely the man whose outlay is less than his income and is steadily kept so.”

—Ralph Waldo Emerson (1803–1882)

“The *theory* of truth is a series of truisms.”

—J.L. (John Langshaw)

“A *theory* of the middle class: that it is not to be determined by its financial situation but rather by its relation to government. That is, one could shade down from an actual ruling or governing class to a class hopelessly out of relation to government, thinking of gov’t as beyond its control, of itself as wholly controlled by gov’t. Somewhere in between and in gradations is the group that has the sense that gov’t exists for it, and shapes its consciousness accordingly.”

—Lionel Trilling (1905–1975)