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In this paper, a mathematical model is provided to describe the dynamics of membrane diafiltration processes for desalting acid whey. A rich representation of the separation process is given due to the employment of concentration‐dependent solute rejections in the design equations. We propose an experimental design and a suitable empirical method for parameter estimation. This technique supports the disciplined use of experimental data and reduces the number of necessary a‐priori experiments. With the help of experimental data we demonstrate the power of the presented modeling method. PRACTICAL APPLICATIONS The annual volume of dairy whey produced globally exceeds 160 million tons. Demineralized whey is of large importance for the food industry. A suitable degree of demineralization can be achieved by diafiltration. Whey represents a complex multi‐component system. Modeling of nanofiltration of such complex systems is a difficult task. We provide a rigorous mathematical model to predict the dynamics of whey diafiltration. The presented robust simulation technique is able to predict the experimentally observed behavior. We propose a minimal experimental design and a suitable mathematical tool to convert the raw experimental data into useful information. The provided methodology might be particularly applicable for decision makers to choose an appropriate diafiltration technique for a given separation design problem.

The main goal of our work was to examine the possibility of using membrane separation to concentrate sweet whey, which could then be used as the basis of more products (e.g., ice cream, soft drinks containing whey).Whey from cheese making was concentrated by nanofiltration. The operating parameters were determined in permeate recirculation mode, and the concentration experiment was carried out under optimal conditions. The analytical assay of the permeate and concentrate such as total soluble solid content (TSS), protein content, and lactose content was determined using infrared measurements. The retention of lactose was over 95% up to 3.0 volume concentration ratio (VCR) at 20-bar transmembrane pressure. The analysis of the process was based on the resistance in series model and the van't Hoff law. The model parameters were determined on the basis of lactose content, assuming that lactose is the key component of whey. The values of these parameters are the concentration polarization (β=1.68) and total resistance (R TOT =1.60×10 14 m −1 ). The procedure was also repeated based on TSS, and the value of the total resistance was almost the same (R TOT = 1.59×10 14 m −1 ). Due to this, the applied models describe appropriately the batch membrane concentration process of whey. The lactose concentration can be used for the calculation instead of TSS, which simplifies the job of estimating the model parameters. Nomenclature β concentration polarization index [-] η viscosity of permeate [Pas] Δπ osmotic pressure difference [bar, Pa] A membrane area [m 2 ] b proportional coefficient [-] c M concentration in the polarization layer in the retentate [%, mol/m 3 ] c P concentration in the permeate [%, mol/m 3 ] c R concentration in the bulk in the retentate [%, mol/m 3 ] c R,lactose lactose concentration in the retentate [%, mol/m 3 ] c R,TSS total soluble solid concentration in the retentate [%] J permeate flux [m 3 /(m 2 s), L/(m 2 h)] J W pure water flux [m 3 /(m 2 s), L/(m 2 h)] J W,F pure water flux of fouled membrane [m 3 /(m 2 s), L/(m 2 h)] Δp TM transmembrane pressure [bar, Pa] R gas constant [J/(mol K)] Re Reynolds number [-] Ret retention of a given solute [%] R F fouling resistance [m −1 ] R F,end fouling resistance at the end of concentration [m −1 ] R M membrane resistance [m −1 ] R P resistance of polarization layer [m −1 ] R TOT total resistance [m −1 ] t time [s, h] T temperature [°C, K] V 0 initial volume of whey [m 3 ] V P volume of permeate [m 3 ] VCR volume concentration ratio [-]

Process duration and fresh water consumption were determined (experimentally and statistically) for variable volume diafiltration (VVD) of cottage cheese through a flat sheet membrane (0.046 m 2 ). The VVD process was performed at two volume decreasing ratios (a ¼ 0.75 and a ¼ 0.5). The VVD-0.75 process lasts much longer than the VVD-0.5 process if the same concentration degree is required. The VVD-0.5 process lasts longer than the VVD-0.75 process if it is aimed to achieve the same purification degree. At low purification degrees, both processes require similar quantities of fresh water, but better demineralization is possible after higher dilution, which is typical of the VVD-0.75 process. The mathematical model applied in estimating the duration of the process proved very accurate, which cannot be said about the fresh water consumption model.