Fanno flow is the
adiabatic flow through a constant area duct where the effect of
friction
Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. There are several types of friction:
*Dry friction is a force that opposes the relative lateral motion of t ...
is considered.
Compressibility
In thermodynamics and fluid mechanics, the compressibility (also known as the coefficient of compressibility or, if the temperature is held constant, the isothermal compressibility) is a measure of the instantaneous relative volume change of a fl ...
effects often come into consideration, although the Fanno flow model certainly also applies to
incompressible flow
In fluid mechanics or more generally continuum mechanics, incompressible flow ( isochoric flow) refers to a flow in which the material density is constant within a fluid parcel—an infinitesimal volume that moves with the flow velocity. An e ...
. For this model, the duct area remains constant, the flow is assumed to be steady and one-dimensional, and no mass is added within the duct. The Fanno flow model is considered an irreversible process due to viscous effects. The viscous friction causes the flow properties to change along the duct. The frictional effect is modeled as a shear stress at the wall acting on the fluid with uniform properties over any cross section of the duct.
For a flow with an upstream
Mach number
Mach number (M or Ma) (; ) is a dimensionless quantity in fluid dynamics representing the ratio of flow velocity past a boundary to the local speed of sound.
It is named after the Moravian physicist and philosopher Ernst Mach.
: \mathrm = \frac ...
greater than 1.0 in a sufficiently long enough duct, deceleration occurs and the flow can become
choked. On the other hand, for a flow with an upstream Mach number less than 1.0, acceleration occurs and the flow can become choked in a sufficiently long duct. It can be shown that for flow of calorically perfect gas the maximum
entropy
Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynam ...
occurs at
''M'' = 1.0. Fanno flow is named after
Gino Girolamo Fanno.
Theory
The Fanno flow model begins with a
differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
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
In thermal physics and thermodynamics, the heat capacity ratio, also known as the adiabatic index, the ratio of specific heats, or Laplace's coefficient, is the ratio of the heat capacity at constant pressure () to heat capacity at constant volu ...
, ''γ'', the
Fanning friction factor
The Fanning friction factor, named after John Thomas Fanning, is a dimensionless number used as a local parameter in continuum mechanics calculations. It is defined as the ratio between the local shear stress and the local flow kinetic energy d ...
, ''f'', and the
hydraulic diameter The hydraulic diameter, , is a commonly used term when handling flow in non-circular tubes and channels. Using this term, one can calculate many things in the same way as for a round tube. When the cross-section is uniform along the tube or channel ...
, ''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
Supersonic speed is the speed of an object that exceeds the speed of sound ( Mach 1). For objects traveling in dry air of a temperature of 20 °C (68 °F) at sea level, this speed is approximately . Speeds greater than five times ...
and especially
hypersonic
In aerodynamics, a hypersonic speed is one that exceeds 5 times the speed of sound, often stated as starting at speeds of Mach 5 and above.
The precise Mach number at which a craft can be said to be flying at hypersonic speed varies, since in ...
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.
: