Conventional filter cake filtration flux model

Microporous filtration flux attenuation model

Filter flux: $J=\frac{1}{A}\times \frac{\text{d}V}{\text{d}t}=\frac{\Delta p}{\mu \left(R_{\text{m}}+R_{\text{c}}\right)}$ (1)

Membrane resistance: $R_{\text{m}}=\frac{k_{\text{m}}{\left(1-{\epsilon }_{\text{m}}\right)}^2{S}_{\text{m}}{l}_{\text{m}}}{{\epsilon }_{\text{m}}^3}$ (2)

Cake resistance: $R_{\text{c}}=\frac{k_{\text{c}}{\left(1-{\epsilon }_{\text{c}}\right)}^2{S}_{\text{c}}{l}_{\text{c}}}{{\epsilon }_{\text{c}}^3}$ (3)

Filter flux: $J=\frac{1}{A}\times \frac{\text{d}V}{\text{d}t}=\frac{\beta }{{\left(\alpha t+\beta \right)}^2}\times \frac{1}{A}$ (4)

Model parameters: $\alpha =\frac{4}{\pi h_{\text{m}}{n}_{\text{p}}{d}_{\text{p}}^2}\times \frac{N}{A{\rho }_{\text{s}}\left(1-\epsilon \right)}$ (5)

Model parameters: $\beta =\frac{h_{\text{m}}}{N}\times \frac{128\mu }{\pi d_{\text{p}}^{2}A\times \Delta p_{\text{TMP}}}$ (6)

J—Filtration flux (also called membrane filtration rate), m³·m⁻²·h⁻¹;

A—Filter area, m²;

V—Permeate filtrate volume, m³;

t—Filter time, h;

Δp—The pressure difference applied to the membrane and the filter cake, kPa;

μ—Viscosity of the suspension fluid, Pa·s;

R_m, R_c—Respectively the resistance of the membrane layer and the filter cake layer, N;

k_m, k_c—Respectively filter membrane constant (k_m = 2) and filter cake constant (k_c = 5);

ε_m, ε_c—Respectively, membrane porosity and filter cake porosity, dimensionless,

S_m, S_c—Respectively, the specific surface area of the filter membrane and the specific surface area of the filter cake, m³·m⁻²;

l_m, l_c—Respectively the thickness of the filter membrane and the thickness of the filter cake, m.

J—Filtration flux (also called membrane filtration rate), m³·m⁻²·h⁻¹;

A—Filter area, m²;

V—Permeate filtrate volume, m³;

t—Filter time, h;

α, β—Model parameters;

μ—Viscosity of the suspension fluid, Pa·s;

h_m—Circulation depth, m;

n_p—Number of membrane holes;

d_p—Porous outer diameter before filtration, m;

N—Concentration of suspended solids entering the porous, kg·m⁻³;

ε—The porosity of the pore surface layer structure is dimensionless,

Δp_TMP—The pressure difference applied to the membrane and the filter cake, kPa;

ρ_s—Solid particle density, kg·m⁻³.