Prediction of Permeate Flux in Ultrafiltration Processes: A Review of Modeling Approaches

Abstract

In any membrane filtration, the prediction of permeate flux is critical to calculate the membrane surface required, which is an essential parameter for scaling-up, equipment sizing, and cost determination. For this reason, several models based on phenomenological or theoretical derivation (such as gel-polarization, osmotic pressure, resistance-in-series, and fouling models) and non-phenomenological models have been developed and widely used to describe the limiting phenomena as well as to predict the permeate flux. In general, the development of models or their modifications is done for a particular synthetic model solution and membrane system that shows a good capacity of prediction. However, in more complex matrices, such as fruit juices, those models might not have the same performance. In this context, the present work shows a review of different phenomenological and non-phenomenological models for permeate flux prediction in UF, and a comparison, between selected models, of the permeate flux predictive capacity. Selected models were tested with data from our previous work reported for three fruit juices (bergamot, kiwi, and pomegranate) processed in a cross-flow system for 10 h. The validation of each selected model's capacity of prediction was performed through a robust statistical examination, including a residual analysis. The results obtained, within the statistically validated models, showed that phenomenological models present a high variability of prediction (values of R-square in the range of 75.91–99.78%), Mean Absolute Percentage Error (MAPE) in the range of 3.14–51.69, and Root Mean Square Error (RMSE) in the range of 0.22–2.01 among the investigated juices. The non-phenomenological models showed a great capacity to predict permeate flux with R-squares higher than 97% and lower MAPE (0.25–2.03) and RMSE (3.74–28.91). Even though the estimated parameters have no physical meaning and do not shed light into the fundamental mechanistic principles that govern these processes, these results suggest that non-phenomenological models are a useful tool from a practical point of view to predict the permeate flux, under defined operating conditions, in membrane separation processes. However, the phenomenological models are still a proper tool for scaling-up and for an understanding the UF process.

Permeate Flux in Ultrafiltration Processes—Understandings and Misunderstandings

“…We concur with [9] that CP is a reversible phenomenon. Indeed it is suggested that the statement "CP is considered as a reversible phenomenon" be strengthened so as to state that CP is a reversible phenomenon.…”

“…Also as pointed out by others the supposedly constant value of C gel . seems to vary during filtration [9,16,17].…”

“…This is absolutely consistent with the statement made by researchers from another laboratory [8] who distinguished between "(i) concentration polarization (i.e., accumulation of retained solutes, reversibly and immediately occurring) and (ii) fouling phenomena such as adsorption, pore-blocking and deposition of solidified solutes, a long-term, and more or less irreversible process." Whilst maintaining a distinction between concentration and fouling, some authors such as those who write, "Concentration polarization (CP), produced by the accumulation of solutes on the membrane surface, causes an increased resistance to solvent transport and possibly a change in the separation characteristics of the membrane" [9] clearly create room for a number of subtle misunderstandings. Firstly there should be no change to the intrinsic separation characteristics of the membrane as the pores have not been blocked or even partially blocked by CP and secondly there is no accumulation of solutes on the membrane surface; CP refers to the mass transfer layer adjacent to the upstream membrane surface and the associated equation includes the thickness of this mass transfer layer: As shown elsewhere (e.g., [10,11]) the relationship linking the concentrations of the solute in the bulk (C b ), at the interface with the membrane (C m ) and in the permeate (C p ) is:…”

“…The quotation from a recent 2021 paper given towards the end of the previous section continues, "CP is considered as a reversible phenomenon and is the primary reason of flux decline during the initial period of operation" [9]. Elsewhere it is not uncommon for authors to mistakenly attribute the flux decline over the first 10-30 min to CP.…”

Geometrical Influence on Particle Transport in Cross-Flow Ultrafiltration: Cylindrical and Flat Sheet Membranes

“…By using the generalized mBLA method, spatially resolved UF concentration and flux profiles are determined in addition to their spatial averages, based on theoretically precise and experimentally well-tested expressions for the dispersion properties. In case one is interested in spatial averages only, e.g., in order to compare with experimental data for the length-averaged permeate flux and TMP, a number of more simple methods are given in the literature as summarized in [2,46]. The mBLA method in its present form deals with generic CP layer effects only.…”