Dean Number formula |
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\( De \;=\; Re \cdot \sqrt{ \dfrac{ d }{ 2 \cdot r } } \) (Dean Number) \( Re \;=\; De \cdot \sqrt{ 2 } \) \( d \;=\; \dfrac{ 2 \cdot r \cdot De^2 }{ Re^2 }\) \( r \;=\; \dfrac{ d }{ 2 \cdot \left( \dfrac{ Re }{ De} \right)^2 } \) |
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| Symbol | English | Metric |
| \( De \) = Dean Number | \(dimensionless\) | \( dimensionless \) |
| \( Re \) = Reynolds Number | \(dimensionless\) | \( dimensionless \) |
| \( d \) = Diameter | \(in\) | \(mm\) |
| \( r \) = Radius of Curviture of the Path of Channel | \(in\) | \(mm\) |
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 Interpretation
- Low Dean Number (De < 50-100): - Secondary flows (Dean vortices) are weak or negligible. The flow remains largely similar to that in a straight pipe, dominated by viscous forces smoothing out disturbances. Laminar flow is typically maintained unless the Reynolds number itself pushes the flow toward turbulence.
- Moderate Dean Number (De ≈ 100-1000) - Dean vortices become noticeable, forming a pair of counter-rotating flows in the cross-section of the pipe. These secondary flows enhance mixing and alter the velocity profile, with faster fluid pushed toward the outer wall of the curve. The flow is still laminar but shows significant curvature-induced effects.
- High Dean Number (De > 1000) - Strong secondary flows dominate, significantly distorting the primary flow. Depending on the Reynolds number, the flow may transition to turbulence earlier than in a straight pipe due to the destabilizing effect of curvature. This regime is common in tightly coiled pipes or high-velocity flows.
The exact thresholds depend on the system (pipe roughness, fluid properties, and curvature ratio), but a critical Dean number (often cited around 50-150) marks the onset of observable secondary flows. For transition to turbulence, the Dean number alone isn’t sufficient, it interacts with the Reynolds number. A common criterion is that turbulence may occur when Re > 2000 in straight pipes, but in curved pipes, this threshold can drop as the Dean number increases.
The Dean number combines the effects of the Reynolds number, which measures the relative importance of inertial to viscous forces, with a geometric factor \(\sqrt{\frac{ D }{ 2 \cdot R} }\), where "D" is the pipe diameter and "R" is the radius of curvature. It essentially indicates how curvature amplifies inertial effects over viscous damping in a flow. A higher Dean number suggests stronger centrifugal forces relative to viscous forces, leading to more pronounced secondary flows perpendicular to the main flow direction.
