Membrane Separations & Bioprocess Engineering
Ultrafiltration Gel-Layer Limiting Flux Formula
Calculate Volumetric Concentration Factor Retentate Concentration C_bulk = C_0 * VCF, Blatt Stagnant-Film Limiting Permeate Flux J_lim = k_m * LN(C_gel / C_bulk), Total Permeate Flow, and Gel Headroom.
Raise crossflow shear k_m—not transmembrane pressure—in gel-polarized ultrafiltration!
During whey-protein, enzyme, or monoclonal-antibody ultrafiltration (UF), convective solvent permeation drags retained macromolecules against the membrane wall until back-diffusion balances convection! Under Blatt et al.’s (1970) stagnant-film Concentration Polarization model, once wall concentration hits protein gelling solubility C_gel (g/L), cranking transmembrane pressure higher thickens the gel layer without increasing flux! Pressure-independent Limiting Permeate Flux (in LMH = L / (m^2*h)) obeys J_lim = k_m * LN(C_gel / C_bulk), where k_m is the Sherwood crossflow mass-transfer coefficient and C_bulk = C_0 * VCF is bulk retentate concentration at Volumetric Concentration Factor VCF >= 1!
Verify on Mass_Transfer_Coeff k_m = 50.0 LMH (B2), Initial_Feed_Protein C_0 = 10.0 g/L (B3), Volumetric_Conc_Factor VCF = 3.0 (B4 -> Bulk Retentate C_bulk = 30.00 g/L in E2), Protein_Gel_Concentration C_gel = 240.0 g/L (B5 -> C_gel / C_bulk = 8.00x headroom in E5), and Cassette_Area = 10.0 m^2 (B6): Limiting Permeate Flux J_lim is 50 * LN(240 / 30) = 50 * LN(8) = 103.97 LMH (E3), producing 1,039.70 L/h of permeate flow (E4)!
| UF Gel-Polarization Metric (km=50 LMH, C0=10g/L, VCF=3x, Cgel=240g/L, Area=10m^2) | Formula | Verified Output |
|---|---|---|
| Retentate Bulk Conc C_bulk | C_0 * VCF (100% Retention) | 30.00 g/L (240.00 g/L @ VCF=24) |
| Limiting Permeate Flux J_lim | k_m * LN(C_gel / C_bulk) | 103.97 LMH (0.00 LMH @ VCF=24) |
| Total Permeate Flow Rate (L/h) | J_lim * Membrane_Area_m2 | 1,039.70 L/h |
| Gelation Margin C_gel / C_bulk | C_gel / (C_0 * VCF) | 8.00x Above Gel Shutdown |
Stagnant-Film Concentration Polarization Limiting Flux Formula
Guard C_0 * VCF > C_gel ("Membrane Gel Fouled (C_bulk > C_gel)") and note that at exact gelation C_bulk = C_gel, LN(1) = 0 cleanly shuts permeate flux down to 0.00 LMH!
=LET(k_mt, B2, c_fd, B3, vc_f, B4, c_gl, B5, ar_m, B6, IF(OR(k_mt <= 0, c_fd <= 0, vc_f < 1, c_gl <= 0, ar_m <= 0), "Invalid UF Input", LET(c_bk, c_fd * vc_f, IF(c_bk > c_gl, "Membrane Gel Fouled (C_bulk > C_gel)", ROUND(k_mt * LN(c_gl / c_bk), 2)))))Zero-Flux Gelation Limit Test (VCF = 24 -> C_bulk = C_gel)
Concentrate 24x in cell B4 (C_bulk = 240 g/L = C_gel): J_lim hits 0.00 LMH, proving why high-concentration biologics switch to formulation viscosity-reducing excipients before C_gel!
=IF(ISTEXT(E3), E3, ROUND(B5 / E2, 2))Try the example
Use the original values or change them to test your understanding.