# Dean Number

Dean number, abbreviated as De, a dimensionless number, used in fluid dynamics to describe the momentum transfer for the flow in curved pipes and channels. This number characterizes the relative importance of inertial forces to centrifugal forces in curved pipe flows. At low numbers, the flow is characterized by a stable, axisymmetric vortex core, while at high numbers, the flow becomes unstable and develops a complex secondary flow structure. The Dean number is commonly used in the analysis of heat and mass transfer in curved pipes and in the design of microfluidic devices.

### Dean number categorizes fluids into different regimes

**Low Dean numbers**- Correspond to laminar flow, where secondary flow patterns are minimal.**Intermediate Dean numbers**- Is associated with the onset of transition from laminar to turbulent flow.**High Dean numbers**- Typically correspond to turbulent flow, with pronounced secondary flow patterns.

Dean numbers are used in fluid dynamics research to analyze and predict flow patterns in curved pipes, especially in applications such as heat exchangers, chemical reactors, and microfluidic devices. Understanding the Dean number helps engineers design and optimize systems to achieve desired flow characteristics and minimize undesirable effects associated with secondary flow phenomena.

## Dean Number formula |
||

\(\large{ De = \sqrt{ \frac{d}{2 \; r} } \; \frac{\rho \; v \; d}{ \mu } }\) | ||

Symbol |
English |
Metric |

\(\large{ De }\) = Dean number | \(\large{dimensionless}\) | |

\(\large{ d }\) = diameter | \(\large{in}\) | \(\large{mm}\) |

\(\large{ r }\) = radius of curviture of the path of channel | \(\large{in}\) | \(\large{mm}\) |

\(\large{ \rho }\) (Greek symbol rho) = density of fluid | \(\large{\frac{lbm}{ft^3}}\) | \(\large{\frac{kg}{m^3}}\) |

\(\large{ v }\) = axial velocity scale | \(\large{\frac{ft}{sec}}\) | \(\large{\frac{m}{s}}\) |

\(\large{ \mu }\) (Greek symbol mu) = dynamic viscosity | \(\large{\frac{lbf-sec}{ft^2}}\) | \(\large{ Pa-s }\) |

## Dean Number formula |
||

\(\large{ De = \sqrt{ \frac{ d }{ 2 \; r } } \; Re }\) | ||

Symbol |
English |
Metric |

\(\large{ De }\) = Dean number | \(\large{dimensionless}\) | |

\(\large{ d }\) = diameter | \(\large{in}\) | \(\large{mm}\) |

\(\large{ r }\) = radius of curviture of the path of channel | \(\large{in}\) | \(\large{mm}\) |

\(\large{ Re }\) = Reynolds number | \(\large{dimensionless}\) |