Multiscale Analysis of the Performance of Micro/nano Porous Filtration Membranes with Double Concentric Cylindrical Pores: Part II-Optimization Results

membranes with double concentric cylindrical pores: Part I-Analysis development, 28(3): The multiscale calculation results for the values of the optimized parameters are presented for the micro/nano porous filtration membranes with double concentric cylindrical pores where multiscale flows occur. For a weak liq-uid-pore wall interaction, the optimum ratio of the radius of the bigger pore to the radius of the smaller pore (i.e. the filtration pore) can be calculated from the classical equation, however the corresponding lowest flow resistance of the membrane should still be calculated from the multiscale analysis owing to the adsorbed layer effect when the ratio λ x of the thickness of the adsorbed layer to the radius of the smaller pore is greater than 0.1. For medium and strong liquid-pore wall interactions, the optimum ratios of the pore radii should be calculated from the multiscale analysis, and the corresponding lowest flow resistances of the membrane are much higher than the classical calculation showing the significant effect of the adsorbed layer when


Introduction
Nano porous filtration membranes have important applications in super purification, desalination, hemofiltration, DNA analysis, drug delivery, and bisosensors etc. [1][2][3][4][5][6][7][8][9]. Their advantage is to use the very small filtration pore with the nanometer scale radius. Consequently, one of their shortcomings is the much-reduced flux. When the radius of the filtration pore is so small that the flowing liquid across the pore radius all become non-continuum, the flow equation for nanoscale flow can solely solve the flow problem in the membranes [10]. However, when the membrane is used to separate the particles with relatively big size such as the impurities, bacteria, virus and organic macro molecules, the radius of the filtration pore may be relatively large such as on the scales of 10nm and even 100nm, the flowing liquid through the membrane may be mainly continuum inside the pores, but the adhering layer on the pore wall may have a considerable effect on the flux of the membrane owing to its thickness comparable to the pore radius. For this case, the multiscale approach is required for analyzing the flow and flux of the membrane by simultaneously incorporating the flows of the adsorbed layer and the continuum liquid.
The studies on nanoporous filtration membranes were mainly experimental with emphasis on their manufacturing [1,2,[5][6][7][11][12][13][14]. Theoretical researches on them are scarce mainly because of the difficulty in simulating the nanoscale flow in an engineering problem caused by molecular dynamics simulation (MDS) [15,16]. When facing the multiscale flow in the membrane, the challenging in the simulation problem exists the same owing to using MDS to simulate the adsorbed layer flow [17,18].
In the first part [19], we presented the multiscale analysis results for the flow resistance of the nanoporous filtration membrane where multiscale flow occurs, by using the flow equation for nanoscale flow to simulate the adsorbed layer flow and using the fluid Newtonian model to simulate the continuum liquid flow. The advantage of the present multiscale analysis is to give the fast results directly by inputting the parameter values. For improving the flux, in the studied membrane is also designed the bigger flow resistance-reducing pore, where may also occur the multiscale flow. We have noticed the best ratio of the radius of the bigger pore to the radius of the filtration pore which gives the maximum flux of the membrane. The present study is the successive work of the first part. In the present paper, we present the detailed optimization analysis and calculation for the studied membrane, and compare the obtained results from the multiscale analysis with the classical theory calculations. Important conclusions have been obtained regarding how to optimize the studied membrane from the viewpoint of the multiscale flow.

Studied membrane
The studied membrane has been shown in the first part [19]. Here, for easy understanding, it is shown again in Figs. 1a and b. There are two concentric cylindrical pores across the membrane thickness, one is smaller with the radius R0 and for filtration, and the other is bigger with the radius R1 (R1 > R0) and for reducing the flow resistance. The axial lengths of the two pores and the membrane thickness are respectively l0, l1 and l. As shown in Fig. 1b, the radius of the filtration pore is so large that near the pore wall occurs the flow of the adsorbed layer with the layer thickness hbf and in the central region of the pore occurs the continuum liquid flow. Both of these two flows may significantly contribute to the flux of the membrane. a b Fig. 1 The studied nanoporous filtration membrane with multiscale flow. a) The membrane profile; b) Flowing media inside the magnified filtration pore [19] 3. Analysis

Multiscale analysis
The multiscale analysis for the multiscale flow in the studied membrane has been shown in the first part [19], by neglecting the interfacial slippage on any interface. Here, for brevity, the detailed derivation and analytical results for the present membrane are not repeated.

Classical analysis
The classical analysis for the liquid flow in the membrane neglected both the adsorbed layer on the pore wall and any interfacial slippage and assumed the whole liquid as continuum whenever the pore radius is small or large. It calculates the dimensionless flow resistance of the membrane from the following equation [10]: where: F measures the dimensionless flow resistance of the membrane and was defined in the first part (by Eq. (8)), and λ0 = l0/l. The classical analysis gives the following optimum value of R1/R0 for the maximum flux of the membrane [10]: It gives the following equation for calculating the correspondingly resulting lowest flow resistance of the membrane when R1/R0 is optimum [10]:

Results
The calculations were made for the dimensionless flow resistances of the membrane, the optimum values of R1/R0 and the corresponding lowest dimensionless flow resistance of the membrane by using the present multiscale scheme when the liquid-pore wall interaction is respectively weak, medium-level and strong. These calculations were respectively compared with those calculated from the classical analysis (Eqs. In the present multiscale calculation, it is formulated that [10,19] where: eff bf  is the average density of the adsorbed layer; ρ is the bulk density of the flowing liquid; m0, m1, m2 and m3 are respectively constant.
The equations for calculating the important parameters ε, F1 and F2 in the present multiscale analysis have been shown in the first part [19] and here are not repeated.
Medium interaction: m = 1.0; n = 5; q0 = 1.   of R1/R0 for the lowest flow resistance of the membrane calculated from the multiscale analysis when the liquidpore wall interaction is respectively weak, medium and strong; The calculations are also compared with those by the classical equation (Eq. (2)). For the weak liquid-pore wall interaction, the values of (R1/R0)opt) calculated from the multiscale analysis are very close to those calculated by the classical equation for the practical values of λx and λ0.
For the medium liquid-pore wall interaction, only when 0.1 x   , the present multiscale calculation is close to the classical calculation; Otherwise, it is lower than the classical calculation especially for greater values of λx, showing the significant multiscale effect owing to the adsorbed layer. For the strong liquid-pore wall interaction, when λ0 = =0.2, the present multiscale calculation is close to the classical calculation.
When λ0 < 0.2, it is lower than the classical calculation; While for λ0 > 0.2, it is higher than the classical calculation. The figures show the significant multiscale influence on the value of (R1/R0)opt) which is stronger for greater values of λx and stronger liquid-pore wall interactions. The results strongly indicate that the value of (R1/R0)opt should be calculated from the multiscale scheme when 0.1 x   and the liquid-pore wall interaction is medium or strong; Nevertheless, for a weak liquid-pore wall interaction, the value of (R1/R0)opt) can still be calculated from the classical equation. x   , for the same operating condition, the lowest flow resistance of the membrane calculated from the multiscale analysis is considerably higher than the classical calculation even for the weak liquid-pore wall interaction; When the liquid-pore wall interaction is medium or strong, the present multiscale calculation is much greater than the classical calculation especially for large λ0 values. a) Weak interaction

Conclusions
The dimensionless flow resistances and the optimum values of R1/R0 of the nanoporous filtration membrane with double concentric cylindrical pores where the multiscale flow occurs are calculated from the developed multiscale scheme for widely varying operational parameter values when the liquid-pore wall interaction is respectively weak, medium and strong. The results show that when 1 x   , the flow resistance or flux of the membrane should be calculated from the multiscale approach owing to the adsorbed layer effect, as the classical calculation will be significantly erroneous. When the liquid-pore wall interaction is weak, the optimum value (i.e. (R1/R0)opt)) of R1/R0 for the lowest flow resistance of the membrane can be calculated from the classical equation even for the membrane in the multiscale flow; However for medium and strong liquid-pore wall interactions, when 1 x   , the values of (R1/R0)opt should be calculated from the multiscale scheme as the classical calculation ignores the significant multiscale effect in this case and thus brings considerable errors. The multiscale calculation results for the values of the optimized parameters are presented for the micro/nano porous filtration membranes with double concentric cylindrical pores where multiscale flows occur. For a weak liquid-pore wall interaction, the optimum ratio of the radius of the bigger pore to the radius of the smaller pore (i.e. the filtration pore) can be calculated from the classical equation, however the corresponding lowest flow resistance of the membrane should still be calculated from the multiscale analysis owing to the adsorbed layer effect when the ratio λx of the thickness of the adsorbed layer to the radius of the smaller pore is greater than 0.1. For medium and strong liquid-pore wall interactions, the optimum ratios of the pore radii should be calculated from the multiscale analysis, and the corresponding lowest flow resistances of the membrane are much higher than the classical calculation showing the significant effect of the adsorbed layer when 0.1 x