Haider & Levenspiel (1989) Drag Coefficient C_D Formula: Spherical & Non-Spherical Particles

Predict drag on cubes, flakes, crushed minerals, and catalyst pellets via Wadell Sphericity phi!

Real industrial fluidized-bed catalyst pellets, coal particles, polymer flakes, and mineral crystals are NOT smooth spheres — their Wadell Sphericity phi = (Surface Area of Equal-Volume Sphere) / (Actual Particle Surface Area) lies well below 1.0 (e.g., Cube phi = 0.806, Cylinder L=D phi = 0.874, Crushed Sand/Coal phi ~= 0.65–0.75, Mica Flakes phi ~= 0.28)! Synthesizing 419 spherical and non-spherical drag measurements up to sub-critical Reynolds Number Re = 2.6e5, A. Haider and Octave Levenspiel (1989, Powder Technology) published the universal 4-parameter Drag Correlation:

C_D = (24 / Re) * (1 + A * (Re^B)) + C / (1 + D / Re), where shape functions of Sphericity phi in (0, 1] are: A = EXP(2.3288 - 6.4581 * phi + 2.4486 * (phi^2)), B = 0.0964 + 0.5565 * phi, C = EXP(4.905 - 13.8944 * phi + 18.4222 * (phi^2) - 10.2599 * (phi^3)), and D = EXP(1.4681 + 12.2584 * phi - 20.7322 * (phi^2) + 15.8855 * (phi^3))! Look at the dramatic physics at Re = 1,000: a smooth sphere (phi = 1.00) has C_D = 0.472, whereas an isometric cube (phi = 0.806) jumps more than 2.5x in wake drag to C_D = 1.217!

Smooth-Sphere Limit (phi = 1.0) Haider-Levenspiel Drag Coefficient C_D

Evaluates exact phi = 1 constants (A = 0.1806, B = 0.6529, C = 0.4251, D = 6880.95): (24 / Re) * (1 + 0.1806 * Re^0.6529) + 0.4251 / (1 + 6880.95 / Re) for Re in (0, 260000).

=IF(OR(B2 <= 0, B2 >= 260000), "Requires Sub-Critical Particle Reynolds Number 0 < Re < 260,000", ROUND((24 / B2) * (1 + 0.1806 * (B2 ^ 0.6529)) + 0.4251 / (1 + 6880.95 / B2), 4))

General Non-Spherical Haider-Levenspiel (1989) Drag Coefficient C_D(Re, phi)

Evaluates full A(phi), B(phi), C(phi), D(phi) exponential polynomials across Particle Reynolds Number Re (B2) and Wadell Sphericity phi in [0.20, 1.00] (B3).

=IF(OR(B2 <= 0, B2 >= 260000, B3 < 0.2, B3 > 1), "Requires 0 < Re < 260,000 and Sphericity 0.20 <= phi <= 1.00", ROUND((24 / B2) * (1 + EXP(2.3288 - 6.4581 * B3 + 2.4486 * (B3 ^ 2)) * (B2 ^ (0.0964 + 0.5565 * B3))) + EXP(4.905 - 13.8944 * B3 + 18.4222 * (B3 ^ 2) - 10.2599 * (B3 ^ 3)) / (1 + EXP(1.4681 + 12.2584 * B3 - 20.7322 * (B3 ^ 2) + 15.8855 * (B3 ^ 3)) / B2), 4))

Further reading

• NETL / MFiX Multiphase Flow Documentation — Haider-Levenspiel Non-Spherical Gas-Solid Drag Law