## Abstract

Hollow fiber (HF) membranes with circular geometry, are used in many separation processes such as water and wastewater treatment. Since optimization of energy efficiency is important for wastewater treatment, the aim of this study was to investigate the effect of non-circular geometry of the inner surface of the HF on the separation performance. To this purpose, the HF bundle has been assumed as a double porous media having two porosities and permeabilities. Since these two parameters are defined by the geometry of the porous medium, any change in the geometry affects their values and the media performance. Therefore, in this study a mathematical modeling has been divided into five categories, including circular, oval, square, rectangular and triangular geometries, and their geometric properties have been calculated based on three different strategies. The results have been compared with the data obtained from literature and showed that the membrane inner surface to cross-section area ratio (*a*), axial permeability, and porosity in the inner region for the non-circular HF are larger than that of the circular HF and *a* increased 16%, 27%, 35% and 65% in ellipse, square, rectangle and triangle geometry, respectively, in comparison with the circle. Axial permeability increased 98%, 68%, 63%, and 26% for a triangle, rectangle, ellipse, and square respectively in the third strategy when compared to the circle. Due to 50% feed flow rate reduction, maximum transmembrane pressure (TMP) reduction was 85% related to the rectangular geometry in the first strategy and minimum was 55% corresponding to the triangle in the third strategy. As *a* increased up to 65%, TMP reduced by up to 200% and consequently energy consumption and operating costs of the system are decreased.

## INTRODUCTION

When water shortages arise, the three main sources considered are; water conservation, water recycling, and desalination. Because water conservation has its limitations, the most effective alternatives to obtain water resources are the water recycling and desalination in terms of price. In this regard, the use of membranes is the best solution (EPA 2005). Among the different polymeric membranes, hollow fiber (HF) membranes are used in several processes, such as water and wastewater treatment. Numerous studies have been done to better understand the fiber production with a higher performance (Widjojo & Chung 2006; Jiang *et al*. 2008; Arthanareeswaran *et al*. 2010; Deng *et al*. 2011; Wan *et al*. 2017). The advantage of these membranes is their larger inner surface to cross-section ratio. In all applications that perform with these membranes, the fibers are circular cylindrical shape, but the reason for choosing this geometry is not the maximum inner surface area per unit cross-section ratio of the membrane but rather the existing methods of manufacture. The external or internal circular shape of the fibers can be changed to improve the surface area. It also reduces the module production cost significantly, which is highly desired. A few researchers (Buer & Cumin 2010; Culfaz *et al.* 2011a) investigated the fouling resistance of non-circular HFs in constant flux experimentally. Their results showed that the fouling resistance for these fibers increases more rapidly than the circular fibers and fouling has better reversibility. The actual permeate flow was 60% higher than the round fibers of equal length. In another study, Culfaz *et al.* (2011b) produced non-circular fibers that improved surface area to cross-section ratio. They concluded that the flux of membrane increased up to 16%. Wang *et al*. (2014) produced a new multi-bore HF with triangle outer geometry; the inner region to cross-sectional area ratio of this HF was more than the circular fiber and dramatically increased pure water permeability compared with the circular fibers. News (2016) proposed a new geometry to match the outer diameter of the circular fiber with the diameter of the outer circumference of the non-circular fiber. These fibers showed a 19% higher surface area and flow compared to the circular fibers. In fact, they showed that in their approach, the surface area of a fiber increases at the fixed external circumference. However, in many industries, separation and purification of the product have the highest cost. That is why finding a simpler and lower-cost separation method should be considered. There is a lot of information about the effects of operating conditions and membrane material on the membrane performance, but few studies on the influence of geometrical membrane characteristics on the filtration performance (Hashino *et al.* 2011; Shi *et al*. 2016; Sungil *et al.* 2016; Chen *et al.* 2017; Guo *et al*. 2017; Wu *et al*. 2018). Since in the past few decades, microfiltration and ultrafiltration systems have been used in various industries, such as water and wastewater treatment, it is important to gain a better understanding of their performance and the effect of various parameters on it by considering the systems' key parameter, i.e. transmembrane pressure (TMP), because of progressive fouling on the membrane (Dario *et al*. 2010). HF is typically made circular. So the following questions arise. If the HF membrane has a non-circular cross-section, what will be its performance in the process? Does it have advantages over a conventional circular HF and improve the process? What will happen if the HF has a non-circular cross-section? In the literature, there are a number of studies investigated the changing membrane characteristics but a very limited number of them have been conducted on the inner surface of the HF. Therefore, by constructing non-circular geometries, there are more choices for the shape of the fiber cross-section. Regarding the above mentioned, in most published articles, the outer surface of the fibers has changed and the internal surface remains unnoticed. Triangles, squares, rectangles and ovals have not been considered. The geometries examined in other papers have a complex form, and most of them have a smaller surface area to cross-section ratio (*a*) than the geometries of this paper.

The effect of the change in the surface geometry of the fiber on parameters such as the TMP has also not been investigated. Therefore, by modeling the previous works, and also for a more comprehensive examination of the subject and having a complete view, and considering the mathematical relations between geometries, three different strategies are considered in this study. The aim of this study is to investigate the performance of an HF membrane system with four internal non-circular shapes. To calculate parameters related to these geometries, three strategies have been used. The HF performance with non-circular geometries has been compared to the circular one and the effect of various parameters and operating conditions on them investigated. In this regard, two-dimensional simulations were performed using the finite element method.

## METHODS

Two regions were defined for the modeling: lumen and shell regions. The lumen region is the inner part of the HF membranes and the shell region is the space between the HFs, i.e. extra capillary space (ECS) (Labecki *et al*. 1995). Since the parameters of the media are defined by their geometry, any change in geometry affects these values and the media performance. In separation processes by the HF membrane, mass conservation and mass transport equations are used. A 2D model based on solving these equations using computational fluid dynamics and finite element method using MatLab for laminar flow condition to simulate the process was used.

### HF characterization and modeling

In this study, an HF sulfone polymeric membrane and UPVC inner layer module within a circular cartridge containing pressure house and dead-end HF have been studied. A Newtonian fluid with constant density and viscosity flows in the module from outside to inside, i.e. feed flows into the shell. To evaluate the performance of a full-scale system containing thousands of HFs that take part in the separation process simultaneously, the module has been assumed as a double porosity and double permeability porous medium (Mohebbi-Kalhori *et al*. 2012). The primary parameters of the model from industrial practice (Borsi & Lorain 2012) are inner radius of hollow fibers, *r _{i}* 0.15 (mm), length,

*L*907.5 (mm), inflow,

*Q*83 (l/h), outer radius of hollow fibers,

_{in}*r*0.35 (mm), number of fibers,

_{o}*N*3,000. Figure 1 shows the computational mesh in the simulation. As can be seen, mesh cells on the output and input of the module are smaller than the other points. The largest mesh size was 4 mm and its smallest size was 0.15 mm and the type of mesh was triangular, and also a time-dependent modeling was used.

### Strategies and determination of the HF characterization

The HF fabrication parameters affect the fiber's mechanical and transport properties, including pore morphology and permeability (Diban & Stamatialis 2014). For this reason, in this study, it has been assumed that the only permeability and porosity changes are due to the geometric parameters, and other parameters and properties are constant. Three strategies that are considered in this study are as follows: Strategy 1 (S1): the radius of the surrounding circle of non-circular geometry is equivalent to the inner radius of the circular HF; Strategy 2 (S2): the perimeter of the new geometry is equivalent to the inner region perimeter of the circular HF; Strategy 3 (S3): the radius of the inscribed circle of non-circular geometry is equivalent to the inner radius of a circular HF. In fact, inner radius in S1 and S3 are equal and only the other geometric specifications for the non-circular geometries are different. Triangle side is calculated for each strategy as follows, for example:

S1:

S2:

S3:

The module with triangle inner shape and cross-sections in all considered strategies are shown in Figure 2, for example.

In all three strategies, it is assumed that the non-circular HF membrane physical characteristics (porosity, pore size distribution) are similar to those of the circular one. To investigate the effect of non-circular HF on process performance, as shown in Figure 2, it is assumed that the inner shape of the HF converts to the mentioned geometry (for example, triangle in this figure) and the outer shape remains in circular HF. The geometries of the inner and outer surfaces are concentric and the outer cylinder and inner element are sufficiently long in the axial direction that the end effects can be ignored. This is due to the end effects of the finite length cylinder which produce a boundary layer that is slightly less resistant than the other regions of the cylinder; i.e. the end effects of the cylinder are ignored, so that the problem takes the form of an analysis of a cross-section for a short cylinder segment.. Because of the axial symmetry of the module and to reduce computational cost and time, a 2D model was applied.

### The governing equations and boundary conditions

*q* is specific discharge (m/s), *k* is permeability of the media: *k _{s,x}* and

*k*permeability for axial and radial in shell region, respectively,

_{s,r}*k*permeability for lumen region (m

_{l}^{2}),

*p*pressure (kPa), φ fraction of the volume occupied by the fibers,

*ɛ*porosity,

*c*pollutant concentration (chloride in this study) in the volume of water (kg/m

^{3}),

*D*dispersion coefficient (m

^{2}/s),

*α*attachment coefficient,

*c*cake concentration on membrane (kg/m

_{m}^{3}),

*R*cake resistance (m

_{c}^{−1}), γ

*cake permeability (m*

_{c}^{2}),

*R*module radius (m),

*ρ*cake density (kg/m

_{c}^{3}),

*R*membrane resistance (m

_{m}^{−1}),

*c*mass fraction of adsorbed pollutant in membrane,

_{p}*A*filtering area (m

_{filt}^{2}),

*N*number of pores,

_{p}*η*adsorption constant (m

^{−2}),

*P*inlet pressure (kPa),

_{in}*P*outlet pressure (kPa), dm membrane thickness (m),

_{out}*Г*rate of water loss per unit volume,

*μ*dynamic viscosity of the water. The conditions of the solid walls were set as non-slip. The fluid flow boundary conditions include a uniform velocity at the module inlet and pressure outlet condition at the outlets of the module. The direction of velocity was parallel to the boundary. The fluid used in the modeling, was water with chloride pollutant in it (kg/m

^{3}) and it was assumed to be Newtonian with constant density and viscosity. The density and viscosity of the medium were set as 998.2 g/m

^{3}and 1.003 × 10

^{−3}Pa.s, respectively. The fluid flows into the module from the inlet and currents into the ECS from the hollow membrane porous wall. A pressure gradient was applied between the outlet and the inlet of the hollow fibers. The filtered fluid flows through the module outlet. The module houses the hollow fibers totally. In the present numerical study the boundary conditions for the mass transport used in the model are given in Table 1.

Inlet (ECS) | Outlet (lumen) |
---|---|

Inlet (ECS) | Outlet (lumen) |
---|---|

*J*is the flux (m/s),

*A*area of module section (m

^{2}),

*Q*feed flow rate (l/s),

*R*module radius (m),

*n*normal vector outward the section,

*c*inlet concentration of pollution. The ratio of the inner perimeter and total cross-section area (

_{in}*a*) of the HF is equivalent to the membrane area per volume (m

^{2}/m

^{3}) (Culfaz Wessling

*et al.*2011b). For non-circular HF compared to circular HF,

*a*is calculated and mentioned for all strategies in Table 2. As can be seen in S3, all geometries have a higher

*a*than circle and triangle has the maximum value. This ratio

*a*is an indicator of inner surface area enhancement. In fact, an inner surface of this geometry is 65% more than the circle.

Geometry | a | ||
---|---|---|---|

S3 | S2 | S1 | |

Triangle | 1.65 | 1 | 0.83 |

Rectangle | 1.35 | 1 | 0.9 |

Square | 1.27 | 1 | 0.9 |

Ellipse | 1.16 | 1 | 0.79 |

Circle | 1 | 1 | 1 |

Geometry | a | ||
---|---|---|---|

S3 | S2 | S1 | |

Triangle | 1.65 | 1 | 0.83 |

Rectangle | 1.35 | 1 | 0.9 |

Square | 1.27 | 1 | 0.9 |

Ellipse | 1.16 | 1 | 0.79 |

Circle | 1 | 1 | 1 |

*et al*. 2005; Costa 2006) has been used. According to the three mentioned strategies, the equations were calculated and are reported in Tables 3 and 4. Porosity equation for triangle shape, for example in S3, is calculated as follows:

Geometry | S1 | S2 | S3 |
---|---|---|---|

Triangle | |||

Rectangle | |||

Square | |||

Ellipse |

Geometry | S1 | S2 | S3 |
---|---|---|---|

Triangle | |||

Rectangle | |||

Square | |||

Ellipse |

Geometry | S1 | S2 | S3 |
---|---|---|---|

Triangle | |||

Rectangle | |||

Square | |||

Ellipse |

Geometry | S1 | S2 | S3 |
---|---|---|---|

Triangle | |||

Rectangle | |||

Square | |||

Ellipse |

*et al.*2005; Costa 2006), the permeability expression is given in the following equation:

*α*is a dimensionless geometric factor,

*A*is the generic cross-sectional area of the HF,

*ɛ*is the porosity and

*τ*is tortuosity, which is 1 in this study. For example, permeability for triangle shape was calculated in S3:

According to this equation, *k _{l}* depends on the triangle's side and it is calculated for the three strategies. For example in S3:

For other geometries, porosity and permeability equations are calculated similarly.

Figure 3 indicates the axial permeability of the inner region for HF for all geometries in all three strategies. Maximum and minimum values for permeability are for S3 and S1, which are related to the triangle and rectangular geometries, respectively. Also, the axial permeability for all geometries reduced from S1 to S3. It is expected that according to Darcy's law, TMP variations reduce similarly. The ratio *a* increased 16%, 27%, 35% and 65% in ellipse, square, rectangle and triangle geometry, respectively, in comparison with the circle. Axial permeability increased 98%, 68%, 63%, and 26% for the triangle, rectangle, ellipse, and square, respectively, in S3, in comparison to circle.

## RESULTS AND DISCUSSION

### Model validation

In this study, an ultrafiltration (UF) module with dead-end HF at a constant flux was modeled. The validation of the model during the simulation with that of Borsi & Lorain (2012) is reported in Figure 4 and indicates a good agreement with the proposed trend mentioned by Borsi & Lorain (2012) after a few hours of runtime.

### Velocity and TMP distribution

Figure 5 shows the Darcy's velocity value of the HF membrane module. As seen in this figure, there is a sharp drop and a sudden increase in the Darcy's velocity at the beginning and end parts of the module, respectively, indicating the input and output manifolds of the module, which were not porous domains, while the middle part was considered as porous medium resulting in a very low Darcy's velocity.

After validation, geometry characteristics for non-circular cross-section HF are computed according to the mentioned three strategies and their impact on process performance are compared to circular HF during 4 days of filtration. For this purpose, TMP has been investigated for all geometries in the same conditions and also by changing the feed flow rate, feed concentration and HF number (packing density). The reduction of TMP leads to reduced energy consumption and operating costs in the industry (Ghidossi *et al*. 2006; Pearce 2008; Zhu *et al*. 2009; Howell *et al*. 2012; News 2016). As can be seen in Figure 6(a) and 6(b) in S1 and S2 there are similar and additive trends for TMP during the filtration process for all geometries. Maximum and minimum values of TMP belong to the rectangle and circle geometries (conventional HF), respectively. So in these strategies, the TMP for all considered geometries is more than that of a circle. Therefore, according to Darcy's law in porous media and as expected, pressure difference increases. So in these strategies, the system's operating costs would increase, whereas the use of HF with a circular cross-section would result in lower energy costs. Figure 6(c) shows the TMP changes versus time for all geometries in S3. As seen, all geometries have lower TMP values compared to the circle. Maximum and minimum values of TMP belong to the circle and triangle, respectively. So in this strategy, all geometries have a better performance than the circle. Consequently, using S3 to determine the HF characterization has less operating cost. Figure 6(d) compares maximum TMP after 9 hours, for all geometries in all three strategies. As shown, the lowest values of TMP are for the S3 and maximum values are for the S1. TMP decreases about 200% for the triangle HF in comparison with the circle HF; therefore operation cost is decreased in S3.

Figure 7 shows the TMP distribution in the ECS region (lumen region) using the S3 for all considered geometries compared to the circle. In this figure, the TMP was increased from inlet section to the outlet section for all considered geometries. The value of this parameter in the triangular geometry (133 kPa) is less than that of the other considered geometries.

Figure 8 shows the change in the TMP with the membrane inner surface to cross-section ratio (*a*) for all considered geometries in S3. Most variations are observed for triangular HF. This figure gives an example of how the TMP decrease for all considered geometries is due to the improvement of the *a* ratio comparing to the circle, which is the potential benefit of using the proposed geometries in this study instead of circular HF.

### The effect of various operational parameters on TMP for HF with non-circular geometries

Reduction of energy consumption was achieved by changing the circular HF geometry, as proposed in the present study. To continue in this section, the feed concentration, flow rate, and packing density are investigated as the most important parameters affecting the performance of the HF membrane modules.

#### The effect of feed concentration

For this purpose, the effect of three different feed concentrations (0.35, 0.45, 0.6 kg/m^{3}) on the TMP variations were investigated. Because of observing a similar trend in the TMP variations in all three strategies, and also the advantage of S3 compared to the two other strategies according to the previous sections, the results of this strategy have been discussed only. As expected and shown in Figure 9(a), increasing (or decreasing) of the concentration would increase (or decrease) the TMP value in all considered strategies and geometries. The reason is that an increase in feed concentration changes the solution viscosity, which affects the energy consumption (Garcia & Chiu 2008) and creates a stiffer layer on the membrane surface. An additional layer causes an increase in the resistance for the water transport (Luo *et al.* 2015) and, as shown, the TMP increases as well.

#### The effect of HF number (packing density)

The packing density can be decreased by decreasing the number of HFs per bundle, which affects hydrodynamic conditions (Bütehorn 2011). To evaluate this effect, HF number was reduced to 2,000 and TMP variation investigated. As expected and shown in Figure 9(b), similar to a circle, reducing of the HF numbers leads to increase in TMP due to the inner surface to cross-section ratio reduction. Results show in the same packing density, for different geometries, the TMP decreases in S1 and S2 and increases in S3 compared to the circle.

#### The effect of feed flow rate

In order to evaluate the effect of feed flow rate on the process performance, this parameter was decreased to 50%. Due to the feed velocity reduction, the TMP decreases (Ghidossi *et al.* 2006; Peng *et al*. 2011) in all strategies and all geometries and leads to reduced energy consumption (Pearce 2008). As shown, the TMP variation is not equal for all geometries. According to Figure 10, a maximum reduction is 85% corresponding to rectangular HF in S1, and minimum reduction in the TMP is 55% corresponding to the triangle HF in S3. Therefore, for different geometries, the TMP variation is affected by the reduction in the feed flow rate at the same conditions.

## CONCLUSIONS

HF membranes are extensively used in the separation processes due to their dense size and capability to do separations that may not be technically or economically practicable via other technologies. In the present study, a set of modeling and simulations were done to evaluate the effect of the changes in the geometry of the inner surface of HF membrane with the same intrinsic properties used in the water and wastewater treatment. In this regard, three strategies and five geometries (circle, triangle, ellipse, square, and rectangle) were introduced to calculate the geometric parameters of the HF membrane. Using S1 and S2 showed that the inner surface area to cross-section ratio (*a*) in all cases was less than that of the circle, while in the S3, this ratio was significantly high. In S3, a maximum ratio *a* was 65% corresponding to the triangle. Also, the axial permeability for all considered geometries was greater than the circular HF. The ratio *a* increased 16%, 27%, 35% and 65% in ellipse, square, rectangle, and triangle geometry, respectively, in comparison to circle. Axial permeability increased 98%, 68%, 63%, and 26% for the triangle, rectangle, ellipse, and square respectively in S3 in comparison to the circle.

Changes in feed flow rate in the different geometries made unequal TMP variations. The maximum TMP reduction was 85% corresponding to HF with rectangle inner cross-section geometry in the S1 due to 50% feed flow rate reduction, and the minimum was 55% corresponding to the triangle in S3. In addition, the TMP variations were because of changes in the HF inner surface to the cross-section ratio and axial permeability. The obtained results showed that increasing the permeability would reduce the pressure drop, which was in accordance with Darcy's law in the porous media. Moreover, one of the most important advantages of the TMP reduction is to reduce the system's energy consumption and operating cost as well. The results showed that using the S3 for the HF characterization can improve the separation process performance, by increasing the inner surface to the cross-section ratio, and also decreasing TMP. The TMP decreases nearly 200% when the HF membrane geometry changes from a round inner shape to a triangle one. These findings quantitatively demonstrate the advantages of the non-circular inner shape HF membranes over the traditional round inner shape ones. Therefore, using these kinds of HF membrane is recommended for the separation process such as in water and wastewater treatment, especially for the wastewater manager or controller who focuses on operating-cost savings.

## REFERENCES

*Experimental and Numerical Investigation of the Hydrodynamics of Microfiltration Processes Using a Multi-Scale Approach*