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.
Highlights
IntroductionMembrane processes have become major techniques in the food industry over the last few decades, thanks to their ability to provide gentle treatment of products at low-tomoderate temperatures
Membrane processes have become major techniques in the food industry over the last few decades, thanks to their ability to provide gentle treatment of products at low-tomoderate temperatures.Membrane applications in the food industry have focused on separation, fractionation, purification, clarification, and concentration of several food products and by-products such as whey, milk, wine, beer, vinegar fruit, and vegetable juices [1]
Phenomenological models present a capacity of prediction ranging from 75.91 to 99.78% (R-squares), whereas the Mean Absolute Percentage Error (MAPE) ranged from 3.14 to 51.69, and Root Mean Square Error (RMSE) from 0.22 to 2.01
Summary
Membrane processes have become major techniques in the food industry over the last few decades, thanks to their ability to provide gentle treatment of products at low-tomoderate temperatures. The natural question for people working in the field of membrane technology is related to the efficiency of these models in terms of permeate flux prediction, with more complex matrices such as fruit juices, dairy products, and by-products, oil derived effluents, and wastewaters in long-term operations In this context, this work aims to provide an extensive review illustrating the models considered as the foundation of the analysis of phenomenology in membrane separation (e.g., Carman–Kozeny equation, film theory, Darcy law) and relate them to how modeling continuously became more accurate in order to improve the capacity of explaining the complexity of membrane separation. An analysis of the capacity to predict permeate flux was developed for selected models (based on the criteria of a number of citations and validation within others) and tested for data related to the clarification of bergamot, kiwi, and pomegranate juices with UF membranes in long-term operations, as reported in previous studies [69]
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