Water Treatment Hydraulics, Granular Media Filtration & Fluidization
Filter Backwash Bed Expansion % & Fluidized Head-Loss Formula
Calculate Fluidized Backwash Expanded Bed Depth L_e (m), Percentage Bed Expansion (%), Constant Fluidized Buoyant Head Loss h_f (m water), and Underdrain Buoyant Pressure (kPa).
Once sand fluidizes during backwash, head loss LOCKS at buoyant grain weight—even as bed expansion rises!
During upward water backwashing of a clogged rapid sand or anthracite filter bed (static depth L_0, static porosity epsilon_0, grain specific gravity SG_s), solid grain volume L_0 * (1 - epsilon_0) is conserved as upward drag lifts the grains to expanded porosity epsilon_e: Expanded_Depth L_e = L_0 * (1 - epsilon_0) / (1 - epsilon_e) and Bed_Expansion_Pct = ((L_e - L_0) / L_0) * 100 (targeting 20%–35% expansion to scour trapped floc without washing media over the washwater troughs)! A classic hydraulics exam trap asks how much head loss increases when backwash flow increases expansion from 25% to 45%: the answer is ZERO—above minimum fluidization velocity, Fluidized Head Loss equals submerged buoyant media weight per unit area h_f = L_0 * (1 - epsilon_0) * (SG_s - 1), completely independent of epsilon_e!
Verify on an 0.80 m Silica Sand Filter Bed—Static_Bed_Depth L_0 = 0.80 m (B2), Static_Porosity epsilon_0 = 0.42 (B3), Expanded_Backwash_Porosity epsilon_e = 0.536 (B4), and Sand_Specific_Gravity SG_s = 2.65 (B5): Expanded Bed Depth reaches 0.80 * 0.58 / 0.464 = 1.00 m (E2), Bed Expansion hits 25.00% (E3), Fluidized Buoyant Head Loss equals 0.80 * 0.58 * 1.65 = 0.7656 m of water (E4), and Underdrain Buoyant Pressure equals 0.7656 * 9.81 = 7.51 kPa (E5)!
| Filter Backwash Fluidization KPI (L0=0.80 m, eps0=0.42, epse=0.536, SG=2.65) | Formula | Verified Output |
|---|---|---|
| Expanded Bed Depth L_e (m) | L0 * (1 - eps_0) / (1 - eps_e) | 1.00 m (1.16 m @ eps_e = 0.60) |
| Filter Bed Expansion (%) | (((1 - eps_0) / (1 - eps_e)) - 1) * 100 | 25.00% (45.00% @ eps_e = 0.60) |
| Fluidized Buoyant Head Loss (m) | L0 * (1 - eps_0) * (SG_sand - 1) | 0.7656 m (UNCHANGED @ eps_e=0.60!) |
| Buoyant Bed Pressure Drop (kPa) | Fluidized_Head_Loss_m * 9.81 | 7.51 kPa |
Solid-Volume Conservation Backwash Expansion & Buoyant Head-Loss Formula
Validate Static_Depth L_0 > 0, 0.25 <= Static_Porosity < Expanded_Porosity < 0.85, and Media_SG > 1.05!
=LET(l0_m, B2, ep_0, B3, ep_e, B4, sg_s, B5, IF(OR(l0_m <= 0, ep_0 < 0.25, ep_e <= ep_0, ep_e >= 0.85, sg_s <= 1.05), "Invalid Backwash Input", ROUND((((1 - ep_0) / (1 - ep_e)) - 1) * 100, 2)))Summer Warm-Water vs High-Rate Wash Trough Overflow Check (Set Expanded Porosity = 0.60)
Increase Expanded_Porosity B4 from 0.536 to 0.600 (higher backwash pump rate): Expanded Depth E2 climbs to 1.16 m (45.00% Expansion in E3, risking media carryover into wash troughs!), while Fluidized Head Loss E4 remains invariant at 0.7656 m!
=IF(ISTEXT(E3), E3, ROUND(B2 * (1 - B3) * (B5 - 1), 4))Try the example
Use the original values or change them to test your understanding.