The analytic solution of interfacial concentration with observed rejection ratio during dead-end membrane filtration
The analytic solution of interfacial concentration with observed rejection ratio during dead-end membrane filtration
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DOI：
10.1016/j.desal.2023.117006
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发表时间：
2023-09
期刊：
影响因子：
9.9
通讯作者：
Albert S. Kim
中科院分区：
文献类型：
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作者：
Albert S. Kim
在本研究中，我们回顾了恒定通量死端过滤的基本原理，开发了界面浓度φ m τ R o作为无因次时间τ和观察到的抑制R o的函数的解析解，并将该解与先前为恒定内在抑制R i开发的解进行了比较。过度浓度φ m τ R o−1由三个非线性τ项组成，并在τ> 1/2的渐近极限下达到4 R o τ。我们分别在膜表面和远进料口处应用了Robin（混合）和Dirichlet边界条件。利用误差函数和互补误差函数的拉普拉斯变换的线性组合，并应用卷积定理，解决了拉普拉斯逆变换的数学难题。利用整体质量平衡分析得到了压力释放后界面浓度的非定常变化，并数值计算了将界面浓度降低到某一特定极限所需的时间。更重要的是，我们发现了观测到的拒斥率与本征拒斥率之间的关系，即R o≃R i，并利用文献中的实验数据进行了验证。
In this study, we revisit the fundamentals of constant–flux dead–end filtration, develop an analytical solution of the interfacial concentration ϕ m τ R o as a function of dimensionless time τ and observed rejection R o, and compare the solution with previous work developed for constant intrinsic rejection, R i. The excessive concentration, ϕ m τ R o− 1, consists of three nonlinear terms of τ and reaches 4 R o τ in an asymptotic limit of τ> 1/2. We apply the Robin (mixed) and Dirichlet boundary conditions on the membrane surface and at a far feed–entrance, respectively. The mathematical difficulties for the inverse Laplace transform are resolved using a linear combination of the Laplace transform of error and complementary error functions and applying the convolution theorem. We analytically obtain the unsteady variation of the interfacial concentration after the pressure release using the global mass balance and numerically calculate the required time to reduce the interfacial concentration to a specific limit. More importantly, a relationship between observed and intrinsic rejection ratios is found, such as, R o≃ R i, and verified using experimental data from the literature.