TFF Calculator: Concentration, Diafiltration and Membrane Area

Batch concentration and membrane passage

The calculation follows a well-mixed retentate, with observed sieving $S=C_{permeate}/C_{retentate}$. Integrating the component balance $d(CV)=SC\,dV$ gives, for volume factor $F=V_0/V_c$:

During constant-volume washing, $N=V_{buffer}/V_c$ and $dC/dN=-SC$. Thus $Y_d=\exp(-S_{p,d}N)$ and overall product yield is $Y_uY_d$. Salt follows the same balances with its own coefficient: $C_{s,f}=C_{s,0}F^{1-S_s}\exp(-S_sN)$. Total salt mass remaining has the additional volume ratio $1/F$. A target wash fraction $r$ requires $N=-\ln(r)/S_s$ when $S_s>0$.

These balances are derived here from conservation; Millipore TB032, pp. 4 and 6, provides definitions and a published product-loss example. Pall's diafiltration guide, p. 3, provides independent rounded salt-clearance values.

Concentration-dependent flux and area

Use the classical limiting-flux approximation $J=k\ln(C_g/C)$ with a near-retained product (Pall, p. 5). Define $b_0=\ln(C_g/C_0)$, $b_c=\ln(C_g/C_c)$ and $u=\ln(V_0/V)$. Integrate area multiplied by time, rather than using an arithmetic mean flux:

Adaptive Gauss–Kronrod quadrature evaluates this integral. During washing, $C=C_c e^{-S_{p,d}N}$, so:

The ratio tends to one at zero product passage or zero wash, evaluated with a short series near zero. Add both area-time contributions: $A_{min}=[(At)_{UF}+(At)_{DF}]/t_{available}$; $A_{design}=f_{area}A_{min}$; installed-area time is $(At)_{total}/A_{installed}$. This stagewise sizing is consistent with Millipore TB032, p. 10.

Classical optimum

For complete product retention, fixed product mass and fixed wash target, DF area-time is proportional to $1/[C\ln(C_g/C)]$. Maximizing its denominator gives $C_{opt}=C_g/e$ (Ng, Lundblad and Mitra, 1976; Pall, p. 6). The normalized chart presents this ideal DF-only result independently of the selected batch recipe.

• Batch concentration first; then continuous buffer addition at the permeate withdrawal rate, at fixed mixed retentate volume. No final reconcentration.
• Uniform mixing and negligible concentration variation along the recirculation loop. Enter total mixed liquid inventory, including the loop.
• Product sieving is constant within each stage; salt sieving is constant throughout. Both are observed bulk-based passage ratios, not molecular-weight-cutoff ratings.
• Incoming buffer contains neither product nor the tracked salt. Salt removal means removal of that species, not total conductivity, pH or ionic strength.
• Constant fitted mass-transfer coefficient and gel concentration at the intended crossflow, buffer and temperature. The retained product controls concentration polarization; the tracked salt has no separate effect on flux.
• The gel-flux expression is a screening approximation for near-complete product retention. Product leakage is included exactly in the component balances and in the concentration history used by this approximate flux law.
• Defaults are illustrative: 100 L at 5 g/L, 10-fold volume reduction and 5 diavolumes. The default area factor of 1.2 follows the typical 20% allowance discussed in Millipore TB032, p. 10, but is user-adjustable.
• Flow markers show topology and direction only. They do not model circulation speed or represent individual product and salt molecules.

Require positive starting volume, product concentration, mass-transfer coefficient, filtration time and installed area; volume factor at least one; nonnegative diavolumes; sieving between zero and one. Initial and peak bulk product concentrations must be below the effective gel concentration. A salt-removal target requires nonzero salt passage.

Flux predictions are most appropriate in an experimentally demonstrated concentration-polarization-limited regime with small product sieving. At low concentrations the gel model can exceed the hydraulic flux limit; near the gel concentration, small fitting errors strongly affect time and area. Check the entire flux range against measurements. No pressure, pump, module-channel or fouling model is included.

Cg/e is the ideal unconstrained optimum for the DF stage, not a universal optimum for total processing. Concentration-dependent sieving, electrostatic interactions, viscosity, product stability and minimum recirculation volume can change the preferred concentration. Steele and Arias (2014), pp. 3–4, report a Donnan-effect example where the practical optimum moves substantially.

Product yield covers membrane passage only; adsorption, precipitation, degradation, sampling and unrecovered liquid can lower actual recovery. The area allowance does not validate scale-up or ensure product quality. Zero wash and unity volume factor are supported; selecting both gives zero theoretical filtration area and time.