In this article, we study a mathematical model that represents the concentration polarization and osmosis effects in a reverse osmosis cross-flow channel with dense membranes at some of its boundaries. The fluid is modeled using the Navier–Stokes equations and the solution-diffusion is used to impose the momentum balance on the membrane. The scheme consist of a conforming finite element method with the velocity–pressure formulation for the Navier–Stokes equations, together with a primal scheme for the convection–diffusion equations. The Nitsche's method is used to impose the permeability condition across the membrane. Several numerical experiments are performed to show the robustness of the method. The resulting model accurately replicates the analytical models and predicts similar results to previous works. It is found that the submerged configuration has the highest permeate production, but also has the greatest pressure loss of all three configurations studied.

中文翻译：

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膜通道中浓差极化和渗透效应的有限元模型

在本文中，我们研究了一个数学模型，该模型表示反渗透错流通道中的浓差极化和渗透效应，该通道的某些边界具有致密膜。使用纳维-斯托克斯方程对流体进行建模，并使用溶液扩散对膜施加动量平衡。该方案由符合纳维-斯托克斯方程的速度-压力公式的有限元方法以及对流-扩散方程的原始方案组成。 Nitsche 方法用于施加跨膜的渗透性条件。进行了多次数值实验以证明该方法的鲁棒性。生成的模型准确地复制了分析模型，并预测了与之前的工作类似的结果。结果发现，浸没式配置具有最高的渗透产量，但在所研究的所有三种配置中也具有最大的压力损失。