Determining fouling-independent component of critical flux in protein ultrafiltration using the free-solvent-based (FSB) model

In protein ultrafiltration (UF), critical flux is generally considered the point on the flux-pressure curve where the permeate flux begins to significantly deviate from the pure solvent flux line. This concept has generally been associated with membrane fouling. Recently, however, we showed that a free-solvent-based model (FSB model) demonstrated that osmotic pressure can be the primary and dominant factor for the nonlinear flux behavior observed in protein UF. Consequently, the critical flux may be the results of two phenomena, the osmotic pressure effect and potential irreversible fouling. Here we show that the fouling-independent component of the critical flux can be redefined as the critical point of the second derivative of the flux-pressure profile determined from the FSB model. This definition of the fouling-independent component of critical flux was compared to values traditionally determined qualitatively on the flux-pressure curve and was found to be in excellent agreement with these experimental observations. During protein ultrafiltration (UF), critical flux has been generally considered the mark of the deviation of the permeate flux from the linear pressure dependence of the pure solvent flux. It has been defined as the flux below which no fouling occurs. Critical flux has seen growing interest from the researchers over the past 10 years due to the importance in understanding the nonlinear flux-pressure dependence during UF processes. Bacchin et al. recently published a comprehensive review that summarizes the research on the critical flux. Despite the variation in theories, experimental methods, and mathematical models developed by different researchers, the general agreement is that the nonlinear flux behavior is the result of the hydraulic resistance associated with various membrane fouling phenomena. Hence, a family of critical fluxes (strong form, weak form, and critical flux for irreversibility) is defined based on different types of fouling behaviors. Recently, we have developed a new flux model, the freesolvent-based (FSB) model (based on the free-solvent model for osmotic pressure).The free-solvent model for concentrated solutions has been around of some time but has, surprisingly been overlooked for its application to concentrated protein solutions. As early as 1916, Frazer and Myrick analyzed the nonidealities in aqueous sucrose solutions using a free-solvent model. More recently, other researchers, in a similar approach, based their models on the van’t Hoff equation, but only had limited success for protein solutions up to moderate concentrations. However, we have subsequently demonstrated that a free-solvent model corrected for ion binding demonstrates excellent predictability for osmotic pressure of concentrated solutions of lysozyme, ovalbumin, albumin and immunogammaglobulins in a range of moderate salt solutions with varying pH. In fact, the free-solvent model demonstrated excellent prediction for protein concentrations up to saturation concentrations. We have recently shown that this model provides excellent prediction and characterization of the flux-pressure behavior in protein UF. We further showed that when membrane fouling contributions are experimentally minimized, the permeate flux continues to demonstrate nonlinear, reversible pressure dependence, and reaches limiting flux at sufficiently high pressure. These observations are in excellent agreement with the FSB model prediction. The unique feature of the free-solvent model for this work is that one Correspondence concerning this article should be addressed to V. G. J. Rodgers at victor.rodgers@ucr.edu.