A systematical quantitative understanding of different mechanisms, though of fundamental importance for better fouling control, is still unavailable for the microfiltration (MF) of humic acid (HA) and protein mixtures. Based on extended Derjaguin–Landau–Verwey–Overbeek (xDLVO) theory, the major fouling mechanisms, i.e., Lifshitz–van der Waals (LW), electrostatic (EL), and acid–base (AB) interactions, were for the first time quantitatively analyzed for model HA–bovine serum albumin (BSA) mixtures at different solution conditions. Results indicated that the pH, ionic strength, and calcium ion concentration of the solution significantly affected the physicochemical properties and the interaction energy between the polyethersulfone (PES) membrane and HA–BSA mixtures. The free energy of cohesion of the HA–BSA mixtures was minimum at pH = 3.0, ionic strength = 100 mM, and c(Ca2+) = 1.0 mM. The AB interaction energy was a key contributor to the total interaction energy when the separation distance between the membrane surface and HA–BSA mixtures was less than 3 nm, while the influence of EL interaction energy was of less importance to the total interaction energy. The attractive interaction energies of membrane–foulant and foulant–foulant increased at low pH, high ionic strength, and calcium ion concentration, thus aggravating membrane fouling, which was supported by the fouling experimental results. The obtained findings would provide valuable insights for the quantitative understanding of membrane fouling mechanisms of mixed organics during MF.

Summary In this paper, we compare two block triangular preconditioners for different linearizations of the Rayleigh–Bénard convection problem discretized with finite element methods. The two preconditioners differ in the nested or nonnested use of a certain approximation of the Schur complement associated to the Navier–Stokes block. First, bounds on the generalized eigenvalues are obtained for the preconditioned systems linearized with both Picard and Newton methods. Then, the performance of the proposed preconditioners is studied in terms of computational time. This investigation reveals some inconsistencies in the literature that are hereby discussed. We observe that the nonnested preconditioner works best both for the Picard and for the Newton cases. Therefore, we further investigate its performance by extending its application to a mixed Picard–Newton scheme. Numerical results of two‐ and three‐dimensional cases show that the convergence is robust with respect to the mesh size. We also give a characterization of the performance of the various preconditioned linearization schemes in terms of the Rayleigh number.

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