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Fanno flow refers to adiabatic flow through a constant area duct where the effect of friction is considered. This model is used extensively in thermodynamics and fluid dynamics to understand the behavior of gas flows in pipelines, especially with respect to changes in temperature, pressure, density, stagnation pressure, and velocity. The concept is linked with two crucial parameters: heat capacity ratio and Mach number.

Fanno Flow Temperature Ratio (T/T_{0}) = |

Fanno Flow Pressure Ratio (P/P_{0}) = |

Fanno Flow Density Ratio (ρ/ρ_{0}) = |

Fanno Flow Stagnation Pressure Ratio (P_{0}/P_{0* }) = |

Fanno Flow Velocity Ratio (u/a_{0}) = |

In the context of Fanno flow, different formulas are used to calculate the various ratios, all of which are functions of the Mach number (M) and the heat capacity ratio (γ).

1. Temperature Ratio (T/T_{0})

T/T_{0} = 1 + (γ - 1)/2 × M^{2}

2. Pressure Ratio (P/P_{0})

P/P_{0} = [1 + (γ - 1)/2 × M^{2}]^{-γ/(γ-1)}

3. Density Ratio (ρ/ρ_{0})

ρ/ρ_{0} = [1 + (γ - 1)/2 × M^{2}]^{-1/(γ-1)}

4. Stagnation Pressure Ratio (P_{0}/P_{0* })

P_{0}/P_{0* } = [1 + (γ - 1)/2 × M^{2}]^{γ/(γ-1)}

5. Velocity Ratio (u/a_{0})

u/a_{0} = M × [1 + (γ - 1)/2 × M^{2}]^{-1/2}

Where:

- T: Temperature of the flow.
- T
_{0}: Stagnation temperature. - P: Pressure of the flow.
- P
_{0}: Stagnation pressure. - ρ: Density of the flow.
- ρ
_{0}: Stagnation density. - P
_{0* }: Stagnation pressure at sonic state (M=1). - u: Flow velocity.
- a
_{0}: Speed of sound at the stagnation state. - M: Mach number.
- γ: Heat capacity ratio (ratio of specific heat at constant pressure to specific heat at constant volume).

The Fanno flow model was developed by the Italian engineer Gino Girolamo Fanno. The formulas were further refined with the development of computational fluid dynamics and the advent of high-speed computing, allowing for more accurate modeling of complex flow situations.

Fanno flow has significant applications in various engineering domains, especially in the design and analysis of propulsion systems. For example, it is used in the analysis of flow in nozzles, diffusers, and the internal flow of jet engines. Moreover, it is also applied in HVAC systems to predict how pressure and temperature change in ducts due to friction.

Gino Girolamo Fanno, after whom Fanno flow is named, made significant contributions to the study of fluid dynamics. His work on the effects of friction on flow through a constant area duct has had lasting impacts on the fields of thermodynamics and fluid mechanics. Fanno's work has enabled significant advances in propulsion system design and analysis.

- Fanno flow model is extensively used in the aerospace industry to understand and design the internal flow of jet engines, supersonic nozzles, and diffusers.
- The principles of Fanno flow are not only limited to gas dynamics but can also be observed in everyday phenomena like the flow of water through pipes and the circulation of air in HVAC systems.
- The application of Fanno flow has revolutionized the field of fluid dynamics by considering the real-world effect of friction, leading to more efficient and effective designs in various engineering disciplines.

Understanding Fanno flow and its associated calculations is essential for those studying or working in the fields of thermodynamics and fluid dynamics. The impact of these principles is far-reaching, with applications ranging from aerospace engineering to HVAC systems. As technology continues to evolve, the principles of Fanno flow will continue to play a crucial role in the design and operation of various systems.

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