CP Calculator — CentrivaLab

Cartridge & Rotor

Cartridge

Rotor

Initial feed volume V₀ (mL)

Permeate collected V_p (mL)

—

r_m = — mm ρ_feed = — kg/m³ r_s start = — mm r_s end = — mm

Operating point

set mean ΔP set rotor speed

Mean ΔP (bar)

Rotor speed (rpm)

Permeate flux J_v (LMH)

Temperature (°C)

Observed rejection R_obs,A (%)

J_v and R_obs: what you measure (or expect) at this ΔP. From them the tool backs out the intrinsic rejection and the solute permeability B, and predicts R_obs at other fluxes.

a_m = — × g ΔP = ½ρω²(r_m²−r_s²) ΔP start = — bar ΔP end = — bar (—)

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Mean ΔP = ½(ΔP_start + ΔP_end). Γ, the fluxes and the rejections below refer to this mean pressure, i.e. to the permeate collected over the whole run. ρ is the feed density at the set concentration and temperature.

Main solute A

The retained solute that sets the density contrast at the membrane and drives the buoyancy convection.

Type

Polarisation model

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Cation

Anion

Solute

Feed concentration

Unit

Solute: — · —

Tracer B (optional)

A second solute too dilute to affect the density field: it only adds its own diffusivity.

Type

Ion

Diffuses with its own ionic coefficient — valid when the main solute is an electrolyte in large excess.

Solute

Tracer: —

⚠

Γ_A — main solute

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Γ_B — tracer

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Rotor speed

—

mean ΔP — bar (—) · a_m = — g

Intrinsic rejection R_int,A

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Main solute A across the flux range — same mean ΔP, solution and cartridge, same B

J_v (LMH)	𝒢_A	k_A (µm/s)	J_v/k_A	Γ_A	c_m,A (g/L)	R_int,A	R_obs,A

Γ = c_m/c₀ − 1 from film theory with the predicted k_c. R_obs at each flux is what a membrane with the same solute permeability B (solution–diffusion, R_int = J_v/(J_v+B)) would show at that flux; the ◀ row reproduces your input. Orange rows: outside the range over which the correlation was validated against DNS (𝒢 7×10⁻⁹–5×10⁻⁶; Sc 700–2200), or J_v/k_c > 1.5 where film theory itself loses accuracy — values are extrapolated. With the UF model the flags are J_v/k_c > 7 or Γ > 100. Olive rows: Sc between 500 and 700, just below the validated range — k_c ∝ Sc^−¼ is a weak dependence (≈ 6 % at Sc = 555, NaCl at 25 °C), so these values are marginal rather than extrapolated; Sc < 500 is orange.

Tracer B across the flux range — same conditions, same B_B

J_v (LMH)	k_B (µm/s)	J_v/k_B	Γ_B	c_m,B (mg/L)	R_int,B	R_obs,B

k_B is set by the convective field of solute A (𝒢_A) and the tracer's own diffusivity (Eq. 47). Orange rows: Sc_B outside 500–3600 or J_v/k_B > 1.5, or the row is already extrapolated for A.

Solution properties used

Solute A — M_A — g/mol β_A — ×10⁻⁴ (g/L)⁻¹ D_A — ×10⁻⁹ m²/s ν (kin.) — ×10⁻⁶ m²/s Sc_A — U_A* — mm/s a_m — m/s² 𝒢_A — k_A — µm/s · J_v/k_A = — B_A — LMH π_feed / π_wall — bar (—) Tracer B — D_B — ×10⁻⁹ m²/s Sc_B — k_B — µm/s B_B — LMH

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Want to see how this compares with a stirred cell (Sterlitech HP4750) or a spiral-wound element at the same ΔP? Open the device comparison.

Model: Geraldes V., Fan H., Yang M., Rodrigues M., Elimelech M. — "Predicting concentration polarization in Ortho-Centrifugal Membrane Filters for NF/RO membrane screening" (2026): k_c/U* = 0.28 𝒢^1/4 Sc^−1/4, U* = (a_mD)^1/3, 𝒢_A = R_A β_A c_A0 J_v/U_A* (Eq. 35–36); a dilute tracer B follows k_B/U_B* = 0.28 𝒢_A^1/4 Sc_B^−1/4 (Eq. 47) — the convective field is set by A alone, B only contributes its own diffusivity. Electrolyte properties are estimated from ionic data: D from limiting ionic conductivities (Nernst–Hartley, with a Davies thermodynamic factor and a viscosity correction), viscosity from Jones–Dole ionic B-coefficients, β = (1/ρ) dρ/dc from conventional partial molar ionic volumes (Millero 1972); a trace ion in a supporting electrolyte uses its own ionic diffusivity. Neutral solutes: literature D° and partial molar volumes. All data at 25 °C (CRC Handbook; Jenkins & Marcus 1995), scaled to the set temperature. The framework was derived for neutral solutes; for electrolytes it is applied with the salt (Nernst–Hartley) diffusivity. For NaCl, MgSO₄ and lactose the device comparison uses measured property fits instead. UF model (macromolecules): the same law with its prefactor re-extracted from the DNS with the finite-permeation closure of Geraldes & Afonso (2006), Γ = R_obs φ (1 + 0.26 φ^1.4)^1.7, replacing film theory (valid only for φ ≲ 1.5), and with ν and D evaluated at the film concentration ½(c_m + c₀); validated on twelve reconstructed PEG 35 kDa ultrafiltration experiments (Pereira, MSc thesis, IST 2026; Sc ≈ 3×10⁴, Γ 2–66, φ 1.4–7, within ±20 % in Γ). Properties of PEG 35 kDa from measured fits at 20 °C (Pires et al.). Osmotic pressure π = φ ν m R T ρ_w with the osmotic coefficient φ from the Pitzer ion-interaction model (β⁽⁰⁾, β⁽¹⁾, β⁽²⁾, C^φ at 25 °C from Kim & Frederick, J. Chem. Eng. Data 33 (1988) 177, for ~140 salts; class-average parameters, flagged, otherwise) and φ = 1 + b·m for neutral solutes; it is used only to bound the input (π_feed ≤ 50 bar) and to check that the set flux is compatible with ΔP. Pressure: ΔP = ½ρω²(r_m² − r_s²) with the feed density ρ and the free-surface radius r_s(V) from the cartridge calibration; since r_s increases as the feed volume drops, the rotor speed is chosen for the arithmetic mean of ΔP at the initial volume V₀ and at the final volume V₀ − V_p (within ~0.3 bar of the time-averaged pressure for V_p up to ~half of V₀).