Preprint / Version 1

A new algebraic water flux equation for the streamlined evaluation and design of membranes and membrane systems


  • Alberto Tiraferri
  • Marco Malaguti
  • Madina Mohamed
  • Mattia Giagnorio
  • Fynn Jerome Aschmoneit Department of Mathematical Sciences, Aalborg University



algebraic water flux equation, membrane characterization, non-dimensional water flux, process design, process optimization, reverse osmosis


The development of new membranes, membrane materials, and membrane-based separation processes should be accompanied by a standardization of the protocols applied for membrane characterization and for system design by the community of academic and industrial stakeholders. For example, one of the obstacles to the simple and effective use of the permeate flux equation across dense membranes is the fact that the magnitude of concentration polarization depends on the flux itself, thus complicating the estimation of the latter, given a certain membrane permeance and driving force (and, on the other hand, the estimation of the membrane permeance from flux data in the presence of solutes in the feed solution). Here, a new, streamlined equation for the calculation of the water flux in pressure-driven dense membrane processes is presented. In contrast to the classic mathematical expression of the water flux, the proposed equation is algebraic. This characteristic poses the advantage of simple calculation, whereas the classic equation needs to be solved iteratively. Non-dimensional variables for water flux, driving pressure, and mass transfer are introduced as parameters of the new equation. It is shown that membrane characterization and process design are significantly simplified by deployment of the new algebraic equation and by manipulation of the non-dimensional variables. In particular, the algebraic water flux equation and the non-dimensional variables address the effect of concentration polarization and relate this phenomenon directly to a decline in water flux, allowing for the definition of a filtration efficiency. In addition, a robust protocol for the experimental characterization of the intrinsic properties of dense membranes is discussed and the results are compared to those expected from the pure solution-diffusion model of species transport. The use of the non-dimensional variables introduced in the new algebraic equation allows simpler calculation of the solute permeability coefficient of the membranes without the need to estimate the solute concentration and the membrane-feed interface or knowledge of the feed channel mass transfer coefficient.


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