Civil, Structural & Geotechnical Engineering
Fair-Hatch Rapid Sand Filter Clean-Bed Head Loss Formula: Porosity & Sphericity
Calculate clean-bed hydraulic head loss h_L (m) through drinking-water rapid sand and dual-media anthracite filters using Gordon M. Fair and L. P. Hatch’s (1933) laminar porous-media drag equation.
Why filter head loss scales inversely with porosity^3 / (1 - porosity)^2 and grain diameter^2!
Before turbidity floc clogs a municipal drinking-water rapid gravity filter, operators verify Clean-Bed Hydraulic Head Loss h_L (typically 0.15 to 0.50 m of water) right after backwashing! Formulated at Harvard by sanitary engineering pioneer Gordon Maskew Fair and L. P. Hatch (1933, Journal of the American Water Works Association) for creeping laminar pore flow (Re_p < 1–5), Clean-Bed Head Loss across bed depth L (m) at superficial filtration approach velocity v_a (m/s, e.g., 5 to 15 m/h = 0.0014 to 0.0042 m/s) is:
h_L = (k_FH * nu * ((1 - eps)^2) * v_a * L) / (g * (eps^3) * ((psi * d_g)^2)), where Kozeny-Fair laminar constant k_FH = 5.0 * (6^2) = 180, nu is water kinematic viscosity (~= 1.004e-6 m^2/s at 20°C, rising +55% to 1.519e-6 m^2/s in 5°C winter water!), eps is bed porosity (~= 0.40–0.45 for silica sand, 0.50 for crushed anthracite), psi is grain sphericity (~= 0.82 for rounded river sand, 0.72 for angular anthracite), and d_g is geometric mean grain diameter (m)!
| Filter Media Design Scenario (v_a = 0.0020 m/s = 7.2 m/h, L = 0.75 m) | Porosity eps & Sphericity psi | 20°C Summer Head Loss | 5°C Winter Head Loss (nu = 1.52e-6) |
|---|---|---|---|
| Fine Rounded Silica Sand (d_g = 0.00055 m) | eps = 0.42, psi = 0.82 | 0.617 m | 0.934 m (+51% Cold Viscosity!) |
| Coarse Silica Filter Sand (d_g = 0.00085 m) | eps = 0.42, psi = 0.82 | 0.258 m | 0.391 m |
| Crushed Anthracite Cap (d_g = 0.00110 m) | eps = 0.50, psi = 0.72 | 0.101 m | 0.153 m |
Fair-Hatch (1933) Clean-Bed Filter Head Loss h_L (m of Water)
Evaluates (180 * nu * (1 - eps)^2 * v_a * L) / (9.80665 * eps^3 * (psi * d_g)^2) across approach velocity v_a (B2 m/s), bed depth L (B3 m), porosity eps in (0.2, 0.8) (B4), sphericity psi in (0.3, 1.0] (B5), grain diameter d_g (B6 m), and kinematic viscosity nu (B7 m^2/s).
=IF(OR(B2 <= 0, B3 <= 0, B4 <= 0.2, B4 >= 0.8, B5 <= 0.3, B5 > 1, B6 <= 0, B7 <= 0), "Invalid Filter Velocity, Depth, Porosity (0.2–0.8), Sphericity (0.3–1.0), d_g, or Viscosity", ROUND((180 * B7 * ((1 - B4) ^ 2) * B2 * B3) / (9.80665 * (B4 ^ 3) * ((B5 * B6) ^ 2)), 4))Pore-Scale Grain Reynolds Number Re_p = v_a * d_g / nu (Laminar Darcy Check)
Verifies Re_p < 5 so laminar Fair-Hatch scaling holds without turbulent Burke-Plummer inertial terms.
=IF(OR(B2 <= 0, B6 <= 0, B7 <= 0), "Invalid Reynolds Inputs", ROUND((B2 * B6) / B7, 3))Try the example
Use the original values or change them to test your understanding.