Multiscale Analysis of the Performance of Micro/nano Porous Filtration Membranes with Double Concentric Cylindrical Pores: Part I-Analysis Development

of the is comparable with the flow of the continuum in the central region of the pore, the multiscale analysis is required for estimating the performance of the membranes. The present paper presents such a multiscale analysis for the performance of nanoporous filtration membranes with double concentric cylindrical pores. The dimensionless flow resistance of the membrane was calculated for a weak fluid-pore wall interaction for varying operational parameter values. It was found that there is the optimum ratio of the radius of the filtration pore to the radius of the other bigger pore for the highest flux of the membrane.

In practice, there are such nanoporous filtration membranes in which the radii of the filtration pores are on the scales of 10 nm, 100 nm or even bigger. When the liquids flow through the filtration pores of these membranes, a molecular scale layer physically adsorbed to the pore wall will form, it will have an important influence on the flow inside the pore, while in the central region of the pore indeed occurs the continuum liquid flow. This is the multiscale flow occurring inside the nanoporous filtration membranes, which actually have been little addressed in the research. Although experiments may have been carried out a lot on these membranes with particular flow regimes, theoretical studies on them are more difficult, and the understandings on the performances of these membranes currently seem still scarce.
Recently, Zhang presented the theoretical derivation results for the multiscale flow in micro/nano cylindrical pores where the adsorbed layer effect plays, by ignoring the interfacial slippage on any interface [15].
Conventionally, a nanoporous filtration membrane consists of the very thin filtration layer containing dense nanoscale filtration pores and the supporting layer containing much bigger micro pores [12,13]. The micro pores in the supporting layer is for increasing the flux, but their radii and distributions on the membrane surface are normally not optimized.
The present paper presents a theoretical study on the performance of the nanoporous filtration membrane where the above mentioned multiscale flow occurs. Across the membrane thickness is distributed in sequence two concentric cylindrical pores with different radii; The small nano pore is for filtration, while the bigger micro/nano pore is for reducing the flow resistance and increasing the flux of the membrane. Although the filtration membrane with such a geometrical design has been addressed by Zhang, he studied this kind of membrane by assuming the liquid flow in the filtration pore as completely noncontinuum across the pore radius [14]. His analysis is valid when the radius of the filtration pore is sufficiently small so that all the liquid becomes non-continuum across the pore radius. However, the radius of the filtration pore in the present membrane may be much bigger so that the multiscale flow occurs inside the filtration pore; Inside the other bigger pore, which is for reducing the flow resistance, may also occur the multiscale flow. For the present membrane, the theoretical analysis is radically different from the previous ones, and the obtained results may also be fresh and of particular interest to the engineering design. Fig. 1, a shows the studied micro/nano porous filtration membrane where the radii of the pores are not so small that near the pore wall occurs the flow of the adjacent layers (i.e. the adsorbed layer flow) and in the central region of the pore occurs the continuum liquid flow as shown in Fig. 1, b. The flow regime in this membrane has been recognized and is actually multiscale [16]; it governs the flux of the membrane. The present membrane may be used to filter out impurities such as tiny particles, bacteria, virus or organic macro molecules, however the total flow rate through the membrane may consist of both the adsorbed layer flow and the continuum liquid flow because of the pore size. The surfaces of all the pore walls are identical. The pore with the radius R0 is for filtration, and its axial length is 0 l . The pore with the radius 1 R is bigger ( 10 RR  ) and for reducing the flow resistance and increasing the flux of the membrane, and its axial length is 1 l . The thickness of the membrane is l . The configuration of the present membrane is the same with that of the membrane studied before by Zhang, except that the radii of the pores in the present membrane are larger or even much larger than those in Zhang's earlier membrane [14]. Although Zhang has given the optimization results for the optimum value of 10 / RR for the highest flux of the membrane based on the non-continuum flow assumption inside the filtration pore in the condition of very small radii of the filtration pore [14], it is obvious that the new analysis should be developed for the present membrane, the design optimization of which should be based on the corresponding multiscale flow theory. a b Fig. 1 The studied nanoporous filtration membrane with multiscale flow: a) the membrane profile; b) flowing media inside the magnified filtration pore

The multiscale flow analysis for the studied membrane
When the pore wall is hydrophobic, the wall slippage may occur in the pore and it can greatly increase the flux of the membrane [17][18][19]. When the pore wall is hydrophilic with the nanoscale pore radius, the wall slippage can also occur in the pore [20]. The wall slippage in the membrane is determined by the power loss on the membrane. In the case of the wall slippage, both the flow resistance and flux of the membrane are intimately related to the power loss on the membrane; Greater the power loss on the membrane, smaller the flow resistance of the membrane, and higher the flux of the membrane. In the condition of the wall slippage, the optimization of the geometrical parameter values of the present membrane should be dependent on the power loss on the membrane. As a first work, the present study assumes no interfacial slippage on any interface and derives the corresponding optimization analysis results. The obtained results can be further examined to be applicable or not for the case of the wall slippage.

Mass flow rate through the pore
The mass flow rate through the filtration pore in the present membrane is [15]: where: hbf is the thickness of the adsorbed layer; D is the diameter of the liquid molecule; Δp0 is the pressure drop on the filtration pore; x  is the separation between the neighboring liquid molecules in the flow direction in the adsorbed layer, , ,  ( 1) − th j molecules across the adsorbed layer thickness, and j and ( 1) j − are respectively the order numbers of the molecules across the adsorbed layer thickness shown in Fig.1, b. The mass flow rate through the flow resistancereducing pore (with the radius 1 R ) in the present membrane is equated as [15]:

Flow resistance
The flow resistances of the single pores with the radii 1 R and 0 R in the present membrane are respectively [14]: According to Eqs. (1) and (2), it is formulated that: The flow resistance of the whole membrane is [14]: ( , ) The dimensionless flow resistance of the membrane is defined as: 2 / (4 ) where r R is a constant reference radius [14]. The dimensionless flow resistance of the present membrane is:    shows that for given values of r R and 0 R , the function F can measure the dimensionless flow resistance of the present membrane.

Calculation
Exemplary calculations were made for a weak liquid-pore wall interaction. In these calculations, it was taken that  [14,15].
According to the above input parameter values, the thickness of the adsorbed layer was calculated to be h,bf = 1.32 nm.

Conclusions
A multiscale analysis is presented for the flux and flow resistance of micro/nano porous filtration membranes where multiscale flows occur. Principally, when a liquid flows through these membranes, there is a layer formed on and physically adsorbed to the pore wall surface. When the radius of the filtration pore is in such a range that the thickness of the adsorbed layer is on the same scale with the radius of the filtration pore, the adsorbed layer flow is comparable with the continuum liquid flow in the central region of the pore, and the multiscale analysis is required for calculating the flux of the membranes by incorporating both the adsorbed layer flow and the continuum liquid flow.
The present study uses the flow factor approach model for nanoscale flow to simulate the adsorbed layer flow and uses the Newtonian fluid model to simulate the continuum liquid flow. The equations are respectively given for the mass flow rates of these two flows. The flow resistances of the pores and the membrane are defined.
The analysis was particularly carried out for micro/nano porous filtration membranes with double concentric cylindrical pores, where across the membrane thickness are distributed in sequence two cylindrical pores with different radii, the smaller pore with the radius R0 is for filtration and the larger pore with the radius R1 is for reducing the flow resistance. The dimensionless flow resistance of this membrane was formulated by the closed-form explicit equation based on the derived multiscale flow equations. Exemplary calculations were made for a weak liquid-pore wall interaction.
The multiscale calculation results show that there is the optimum value of the ratio R1/R0 for the lowest flow resistance and thus the highest flux of the membrane. The increase of 1 R is not beneficial for the flux of the membrane when R1/R0 is over the optimum value. An over large 1 R will result in the sensitive variation of the flow resistance of the membrane with the variations of both 1 R and the axial length 0 l of the filtration pore. From the engineering viewpoint, the ratio R1/R0 should be designed as the optimum value, allowing the tolerable variations of both 0 l and 1 R . In nanoporous filtration membranes, when the radius of the filtration pore is so big that the flow of the adsorbed layer on the pore wall is comparable with the flow of the continuum fluid in the central region of the pore, the multiscale analysis is required for estimating the performance of the membranes. The present paper presents such a multiscale analysis for the performance of nanoporous filtration membranes with double concentric cylindrical pores. The dimensionless flow resistance of the membrane was calculated for a weak fluid-pore wall interaction for varying operational parameter values. It was found that there is the optimum ratio of the radius of the filtration pore to the radius of the other bigger pore for the highest flux of the membrane.