Chemical & Pharmaceutical Solid-Liquid Separation, Ruth Cake Filtration & Filter Press
Ruth Constant-Pressure Cake Filtration Time Formula
Calculate Total Dry Filter Cake Deposited (kg), Parabolic Cake-Resistance Filtration Time (min), Linear Filter-Cloth Resistance Time (min), and Total Cycle Filtration Time (min).
Doubling filtrate volume in a recessed-plate filter press takes FOUR TIMES longer—because every liter of filtrate builds thicker cake resistance!
Recessed-plate and membrane **Filter Presses** pump slurry of Dry-Cake-Mass-per-Filtrate-Volume `c` (`kg/m³`) and Filtrate Viscosity `mu` (`mPa·s` = `10^{-3}` Pa·s) across Filter Cloth Area `A` (`m²`) at Constant Pressure Drop `Delta_P` (`kPa` = `10^3` Pa). Unlike clean-liquid membrane microfiltration (`t ~ V`), accumulating filter cake adds hydraulic resistance proportional to Cumulative Filtrate Volume `V` (`m³`)! Integrating Darcy’s Law across **Specific Cake Resistance `alpha`** (`10^{10}` m/kg) in series with **Filter-Cloth Resistance `R_m`** (`10^{10}` m⁻¹) yields **B. F. Ruth’s Constant-Pressure Filtration Equation**: **`t = ((mu * alpha * c) / (2 * A^2 * Delta_P)) * V^2 + ((mu * R_m) / (A * Delta_P)) * V`** (`s`, `/ 60` = `min`)—separating **Quadratic Cake Time `t_{cake} ~ V^2`** from **Linear Cloth Time `t_{cloth} ~ V`**!
Verify on a `16.0` m² Pharmaceutical Precipitate Filter Press—Filter_Area `A` = 16.0 m² (B2), Feed_Pressure `Delta_P` = 450 kPa (B3), Mother-Liquor_Viscosity `mu` = 1.15 mPa·s (B4), Specific_Cake_Resistance `alpha` = 8.5 (`x 10^10` m/kg, B5), Dry_Solids_Concentration `c` = 65.0 kg/m³ (B6), Polypropylene_Cloth_Resistance `R_m` = 4.2 (`x 10^10` m⁻¹, B7), and Batch_Filtrate_Volume `V` = 4.80 m³ (B8): Deposited Dry Cake equals `312.0` kg (E2), Parabolic Cake Time takes `10.59` min (E3), Cloth Resistance adds `0.54` min (E4), and Total Filtration Time equals `11.13` min (E5)!
| Ruth Filter-Press KPI (A=16 m², dP=450 kPa, V=4.8 m³, c=65 kg/m³, alpha=8.5 vs 24.0e10) | Formula | Verified Output |
|---|---|---|
| Deposited Dry Filter Cake (kg) | Dry_Solids_Per_Filtrate_c * Filtrate_V_m3 | 312.0 kg |
| Quadratic Cake Time t_cake (min) | (((mu*alpha*c)/(2*A^2*dP)) * V^2) / 60 | 10.59 min (29.90 min @ un-flocculated fines!) |
| Linear Cloth Time t_cloth (min) | (((mu * Rm) / (A * dP)) * V) / 60 | 0.54 min |
| Total Filtration Cycle Time (min) | Cake_Time_min + Cloth_Time_min | 11.13 min (30.44 min bottleneck @ 24e10!) |
Ruth Parabolic (t/V vs V) Constant-Pressure Cake + Medium Filtration Formula
Validate `Area A > 0.1` m², `Pressure Delta_P > 10` kPa, `Viscosity mu > 0.1` mPa·s, `Specific_Cake_Resistance alpha > 0`, `Solids c > 0.5` kg/m³, `Cloth R_m >= 0`, and `Filtrate_Volume V > 0` m³!
=LET(af_m, B2, dp_p, B3 * 1000, mu_p, B4 * 0.001, al_m, B5 * 10000000000, cs_k, B6, rm_i, B7 * 10000000000, vf_m, B8, tc_s, ((mu_p * al_m * cs_k) / (2 * (af_m ^ 2) * dp_p)) * (vf_m ^ 2), tm_s, ((mu_p * rm_i) / (af_m * dp_p)) * vf_m, IF(OR(af_m <= 0.1, B3 <= 10, B4 <= 0.1, B5 <= 0, cs_k <= 0.5, B7 < 0, vf_m <= 0), "Invalid Filtration Input", ROUND((tc_s + tm_s) / 60, 2)))Over-Agitated Sheared Crystal Fines (alpha = 24.0 x 10^10 m/kg) Cycle Bottleneck Check (Set alpha B5 = 24.0)
Shear crystal agglomerates with a high-RPM centrifugal transfer pump (`alpha` jumps nearly `3x` to `B5 = 24.0`): Cake Time E3 balloons from `10.59` to `29.90` min (`Total = 30.44` min in E5)—proving why gentle low-shear diaphragm feed pumps and diatomaceous-earth body-feed filter aids pay back immediately!
=IF(ISTEXT(E5), E5, LET(dp_p, B3 * 1000, mu_p, B4 * 0.001, al_m, B5 * 10000000000, ROUND((((mu_p * al_m * B6) / (2 * (B2 ^ 2) * dp_p)) * (B8 ^ 2)) / 60, 2)))Try the example
Use the original values or change them to test your understanding.