Reference ID: MET-2B8C | Process Engineering Reference Sheets Calculation Guide
Introduction & Context
Diafiltration is a membrane-based operation in which fresh solvent (usually water) is continuously added to the retentate side while filtrate is removed, thereby washing out low-molecular-weight solutes without changing the retentate volume. Accurate prediction of the required diafiltration volume is essential for designing buffer preparation skids, sizing hold tanks, scheduling CIP cycles, and demonstrating clearance of host-cell proteins, antibiotics, or other small-molecule contaminants in biopharmaceutical downstream processing.
Methodology & Formulas
Ideal well-mixed assumption
The retentate compartment is assumed perfectly mixed and the rejected species exhibits a constant true rejection coefficient \(R=1\). Under these conditions, the mass balance on the impurity leads to a first-order wash-out curve.
Governing differential equation
\[
\frac{\mathrm{d}C}{\mathrm{d}V_{\text{dia}}}} = -\frac{C}{V_{\text{ret}}}
\]
where
\(C\) = instantaneous impurity concentration in retentate (mass/volume)
\(V_{\text{dia}}\) = cumulative diafiltration volume added (same volume units as \(V_{\text{ret}}\))
\(V_{\text{ret}}\) = constant retentate volume (volume)
Analytical integration
Separate variables and integrate from the initial concentration \(C_{\text{i}}\) to the target concentration \(C_{\text{f}}\):
\[
\int_{C_{\text{i}}}^{C_{\text{f}}}\frac{\mathrm{d}C}{C}= -\frac{1}{V_{\text{ret}}}\int_{0}^{V_{\text{dia}}}\mathrm{d}V_{\text{dia}}
\quad\Longrightarrow\quad
\ln\left(\frac{C_{\text{i}}}{C_{\text{f}}}\right)=\frac{V_{\text{dia}}}{V_{\text{ret}}}
\]
Design formula
\[
V_{\text{dia}}=V_{\text{ret}}\ln\left(\frac{C_{\text{i}}}{C_{\text{f}}}\right)
\]
Validity regime
Condition
Range
Consequence if exceeded
Ratio \(C_{\text{i}}/C_{\text{f}}\)
≤ 1000
Simple logarithmic formula remains accurate; higher ratios may require correction for non-ideal rejection or volume losses.
The required diavolumes (N) are calculated from the natural logarithm of the ratio of initial to final concentration.
For concentration: N = ln(C₀/Cf) where C₀ is the initial concentration and Cf is the final concentration.
For impurity reduction: N = ln(I₀/If) where I₀ is the initial impurity level and If is the desired level.
Remember that 1 diavolume equals the current retentate volume; multiply N by the retentate volume to obtain the total diafiltration buffer volume.
Constant-volume diafiltration keeps the retentate volume fixed; the diavolume formula above applies directly.
Variable-volume diafiltration simultaneously concentrates and diafilters; the effective diavolume at any instant is the cumulative buffer added divided by the instantaneous retentate volume.
When both concentration and buffer addition occur, integrate the instantaneous diavolume over the process or use the combined equation: N = ln(C₀/Cf) + (V₀–Vf)/Vf where V₀ and Vf are the initial and final retentate volumes.
Real membranes do not achieve 100 % rejection; use the observed rejection coefficient (R) to adjust the calculation.
Effective diavolumes N_eff = N / R where N is the ideal value.
If R varies with concentration, integrate 1/R over the concentration range or use small step-wise calculations.
Validate with inline concentration or conductivity measurements to confirm the target is met.
Prepare 10–20 % excess buffer beyond the theoretical diavolume to cover tubing, pump heads, and membrane hold-up.
Measure the actual retentate volume at the start of diafiltration; do not use tank graduation alone.
Subtract the system void volume from the buffer tank volume to avoid underestimating the diavolumes delivered to the product.
Program the feed pump to automatically switch to water-for-injection for the final 0.2–0.3 diavolumes to minimize product dilution while flushing lines.
Worked Example – Diafiltration Volume Calculation
A biopharmaceutical plant is purifying a monoclonal-antibody solution. The retentate tank currently holds 100 L of a 5 % (w/v) buffer containing low-molecular-weight impurities. To meet final purity specifications, the impurity concentration must be reduced to 0.5 %. The process engineer decides to use constant-volume diafiltration, adding buffer while permeate is removed at the same rate. Determine how much diafiltration buffer must be prepared.
Knowns
Retentate volume, Vret = 100 L
Initial impurity concentration, Ci = 5 %
Target impurity concentration, Cf = 0.5 %
Step-by-Step Calculation
Impurities are washed out by buffer exchange; the impurity concentration ratio equals the fraction of the original volume remaining after each diafiltration turn. For constant-volume diafiltration, the relationship is
\[
\frac{C_{\text{f}}}{C_{\text{i}}} = e^{-N}
\]
where N is the number of diavolumes (buffer volume processed per retentate volume).
Total diafiltration buffer volume is the number of diavolumes multiplied by the retentate volume:
\[
V_{\text{diafilter}} = N \times V_{\text{ret}} = 2.303 \times 100\ \text{L} = 230.259\ \text{L}
\]
Final Answer
Approximately 230 L of diafiltration buffer must be processed to reduce the impurity concentration from 5 % to 0.5 % while keeping the retentate volume constant at 100 L.
"Un projet n'est jamais trop grand s'il est bien conçu."— André Citroën
"La difficulté attire l'homme de caractère, car c'est en l'étreignant qu'il se réalise."— Charles de Gaulle
