Engineering & Science
Whitaker Sphere Convection Correlation: External Flow Nusselt & Viscosity Correction
Calculate forced-convection heat transfer over spheres Nu_D = 2 + (0.4*Re^(1/2) + 0.06*Re^(2/3))*Pr^0.4*(μ_∞/μ_s)^(1/4) for spray drying, pellet cooling, and pebble beds.
Separating attached laminar boundary-layer heat transfer (Re^1/2) from turbulent wake shedding (Re^2/3)!
Stephen Whitaker (1972) improved upon the classic Ranz-Marshall droplet correlation by splitting convective transport around an isolated sphere into two distinct fluid-mechanical zones plus a Sieder-Tate-style wall-viscosity ratio: Nu_D = 2 + (0.4 * Re_D^(1/2) + 0.06 * Re_D^(2/3)) * Pr^0.4 * (μ_∞ / μ_s)^(1/4), valid across 3.5 <= Re_D <= 7.6×10^4 and 0.71 <= Pr <= 380!
Here 2 is the stagnant pure-conduction asymptote, 0.4*Re^(1/2) models the attached forward laminar boundary layer, 0.06*Re^(2/3) captures separated aft-wake vortex mixing, and (μ_∞ / μ_s)^(1/4) accounts for boundary-layer thinning when heating a liquid! At Re_D = 4,500, Pr = 6.2, μ_∞ = 0.00100 Pa·s, and heated-surface μ_s = 0.00055 Pa·s, viscosity correction is 1.1612 and Nu_D = 106.04 (jumping to 183.33 at Re_D = 12,000)!
| Sphere Reynolds Number Re_D (Pr = 6.2, μ_∞/μ_s = 1.818) | Wall Viscosity Factor (μ_∞/μ_s)^0.25 | Whitaker Sphere Nu_D | Dominant Regime |
|---|---|---|---|
| Re_D = 4,500 (Quench Bath Sphere) | 1.1612 | 106.04 | Balanced boundary layer + wake |
| Re_D = 12,000 (High-Velocity Cooling) | 1.1612 | 183.33 | Separated wake vortex boost |
| Invalid Re_D < 0 or Viscosity <= 0 | Error Guard | "Invalid Sphere Flow Inputs" | Physical Bounds Guard |
Whitaker External Forced-Convection Sphere Nusselt Number Nu_D
Evaluates 2 + (0.4*SQRT(Re) + 0.06*Re^(2/3)) * Pr^0.4 * (μ_∞/μ_s)^0.25 from B2 (Re_D), B3 (Pr), free-stream viscosity B4 (Pa·s), and surface-temperature viscosity B5 (Pa·s).
=IF(OR(B2 < 0, B3 <= 0, B4 <= 0, B5 <= 0), "Invalid Sphere Flow Inputs", ROUND(2 + (0.4 * SQRT(B2) + 0.06 * (B2 ^ (2 / 3))) * (B3 ^ 0.4) * ((B4 / B5) ^ 0.25), 2))Thermal Boundary-Layer Viscosity Correction Factor (μ_∞ / μ_s)^0.25
Isolates the wall-temperature viscosity multiplier (> 1 when heating liquids, < 1 when cooling liquids).
=IF(OR(B2 < 0, B3 <= 0, B4 <= 0, B5 <= 0), "Invalid Sphere Flow Inputs", ROUND((B4 / B5) ^ 0.25, 4))Try the example
Use the original values or change them to test your understanding.