Engineering & Science
Dean Number (De) Curved-Pipe Vortices & Ito Critical Reynolds Formula
Calculate W. R. Dean’s secondary-vortex number De = Re * SQRT(D / (2 * R_c)) in helical coils and pipe bends alongside Hidesato Ito’s curved-pipe laminar-to-turbulent stabilization threshold Re_crit = 2100 * (1 + 12 * SQRT(D / (2 * R_c))).
Why helical heat-exchanger coils stay laminar way past Re = 2,100 while boosting radial mixing!
Send fluid around a curved elbow or helical steam-generator coil of inner tube diameter D (m) and coil centerline radius R_c (m): centrifugal inertia flings fast core fluid outward toward the outer wall, driving a counter-rotating pair of secondary Dean Vortices quantified by W. R. Dean’s (1928) dimensionless Dean Number De = Re * SQRT(D / (2 * R_c))!
Those secondary vortices do two things simultaneously: (1) above De ~ 40–60, radial vortex mixing dramatically enhances heat and mass transfer without turbulent eddies, and (2) curvature stabilizes the boundary layer against turbulence, raising the laminar-to-turbulent transition far above straight-pipe Re = 2,100 according to Hidesato Ito’s (1959) correlation Re_crit = 2100 * (1 + 12 * SQRT(D / (2 * R_c)))! In a D = 10 mm (0.010 m) tube wound on an R_c = 125 mm (0.125 m) helical coil at Re = 1,800, De = 360.0 (strong secondary Dean vortices!) while laminar flow remains stable all the way up to Re_crit = 7,140!
| Coil Curvature Scenario (D = 10 mm, Re = 1,800) | Curvature Ratio D / (2 * Rc) | Dean Number De | Ito Critical Reynolds Re_crit |
|---|---|---|---|
| Compact Helical Coil (Rc = 125 mm) | 0.0400 (SQRT = 0.20) | 360.0 (Strong Vortices) | 7,140 (3.4x Straight Pipe!) |
| Gentle Pipeline Bend (Rc = 1,250 mm) | 0.0040 (SQRT = 0.0632) | 113.8 | 3,694 |
| Invalid Geometry (Rc <= D / 2 or D <= 0) | Physical Kink Guard | "Invalid Coil Geometry" | Requires 2 * Rc > D > 0 |
Dimensionless Curved-Pipe Dean Number De = Re * SQRT(D / (2 * R_c))
Validates physically realizable bend geometry 2 * R_c > D > 0 and positive Reynolds number B2.
=IF(OR(B2 <= 0, B3 <= 0, 2 * B4 <= B3), "Invalid Coil Geometry", ROUND(B2 * SQRT(B3 / (2 * B4)), 1))Ito (1959) Curved-Tube Critical Laminar-to-Turbulent Reynolds Number Re_crit
Computes 2100 * (1 + 12 * SQRT(D / (2 * R_c))) across curvature ratios 0.001 <= D/(2*Rc) <= 0.07.
=IF(OR(B3 <= 0, 2 * B4 <= B3), "Invalid Coil Geometry", ROUND(2100 * (1 + 12 * SQRT(B3 / (2 * B4))), 0))Try the example
Use the original values or change them to test your understanding.