Reference ID: MET-536D | Process Engineering Reference Sheets Calculation Guide
Introduction & Context
Ultrafiltration (UF) is a pressure-driven membrane process widely used in the dairy industry to concentrate proteins while allowing lactose, minerals, and water to permeate. The Volume Reduction Ratio (VRR) quantifies how much the feed stream is concentrated; it directly sets the final retentate volume and—because proteins are almost totally rejected by the membrane—the final protein concentration. Accurate VRR calculations are essential for:
Designing membrane systems (area, pumps, energy)
Predicting product yield and composition
Avoiding gel-polarisation and fouling regimes that limit flux and damage membranes
Methodology & Formulas
The membrane is treated as perfectly rejecting protein while all other components pass freely. With this assumption, a simple mass balance on the feed tank gives:
These limits are checked after the algebraic calculation; warnings are issued if either threshold is exceeded.
The concentration factor (CF) in dairy ultrafiltration is usually between 2× and 5× for milk protein standardization, and up to 10× for specialized concentrates. CF is calculated as the ratio of initial feed volume (V₀) to retentate volume (Vᵣ): CF = V₀ / Vᵣ. Alternatively, it can be derived from component mass balance using the formula CF = (Cᵣ – Cₚ) / (C₀ – Cₚ), where C₀, Cᵣ, and Cₚ are the solute concentrations in the feed, retentate, and permeate, respectively.
The practical limit is governed by:
Retentate viscosity exceeding 80–100 cP at 50 °C, which drops shear-induced back-migration and raises ΔP > 8 bar
Membrane shear rate capability of the plant—spiral-wound modules max out around 4–5 % protein, while ceramic tubular systems can reach 8–10 % protein at 5–7 m s⁻¹ cross-flow
Fouling layer resistance (Rᶠ) growth rate; beyond CF 6×, protein aggregation and mineral precipitation increase Rᶠ exponentially, forcing flux < 10 L m⁻² h⁻¹
As CF rises, lactose and soluble minerals (Na, K, chloride) continue to permeate, but the percentage removal plateaus:
At CF 2×, ~55 % of feed lactose leaves in permeate; at CF 5×, cumulative removal only climbs to ~75 %
Calcium and phosphate partially precipitate as colloidal calcium phosphate, so permeation is < 15 % regardless of CF
Higher CF therefore increases protein-to-lactose ratio, but ash-on-protein also rises; target MPC70 powders require CF ~4.5× plus diafiltration to keep ash ≤ 7 % w/w
Raising temperature to 55–60 °C lowers viscosity and can add ~0.5× CF, but risks:
β-lactoglobulin denaturation > 10 %, causing gelation and higher fouling
Exceeding membrane polyethersulfone (PES) temperature rating, accelerating membrane hydrolysis
Mid-process diafiltration (adding 20–30 % water) can reduce viscosity and allow an extra 0.8–1× CF, but increases evaporative load downstream and dilates permeate, reducing overall plant capacity by 8–12 %
Worked Example: Calculating the Concentration Factor in Dairy Ultrafiltration
A small dairy plant wants to concentrate skim milk from 3.2 % protein to 16 % protein using a spiral-wound ultrafiltration module. The feed tank is charged with 1000 L of skim milk. The operator sets the control system to achieve a volume-reduction ratio (VRR) of 5, which is within the module’s maximum allowable VRR of 6. Determine the final retentate volume and verify that the target protein concentration is reached.
Knowns
Feed volume, \(V_{\text{feed}}\) = 1000 L
Feed protein concentration, \(C_{\text{prot,feed}}\) = 3.2 % (w/w)
Target VRR = 5
Maximum module VRR = 6
Maximum protein concentration (retentate) = 20 % (w/w)
