### 1. Introduction

### 2. Simulation Set-up

### 2.1. Governing Equations

*k-ɛ*) model can analyze flow properties for turbulence conditions in CFD. In this model, there are two variables: turbulence kinetic energy (

*k*) and the rate of dissipation of turbulence energy (ɛ). The turbulent viscosity is assumed to be isotropic, as the ratio between the Reynolds stress and deformation rate is the same in all directions [16]. Note that

*k-ɛ*equations may have many unknown terms; however, the standard

*k-ɛ*model minimizes the unknown factors, and thus can be used in turbulent flow applications. The equations of turbulent kinetic energy (

*k*) and dissipation (

*ɛ*) are follows [17].

##### (1)

$$k:\frac{\partial (\rho k)}{\partial t}+\frac{\partial (\rho k{u}_{i})}{\partial {x}_{i}}=\frac{\partial}{\partial {x}_{j}}\left[\frac{{\mu}_{t}}{{\sigma}_{k}}\frac{\partial k}{\partial {x}_{j}}\right]+2{\mu}_{t}{E}_{ij}{E}_{ij}-\rho \varepsilon $$##### (2)

$$\varepsilon :\frac{\partial (\rho \varepsilon )}{\partial t}+\frac{\partial (\rho \varepsilon {u}_{i})}{\partial {x}_{i}}=\frac{\partial}{\partial {x}_{j}}\left[\frac{{\mu}_{t}}{{\sigma}_{\varepsilon}}\frac{\partial \varepsilon}{\partial {x}_{j}}\right]+{C}_{1\varepsilon}2{\mu}_{t}{E}_{ij}{E}_{ij}-{C}_{2\varepsilon}\rho \frac{{\varepsilon}^{2}}{k}$$*u*

*is the velocity component in the corresponding direction,*

_{i}*E*

*is the component of the deformation rate, and*

_{ij}*μ*

*is the eddy viscosity.*

_{t}*σ*

*,*

_{k}*σ*

*,*

_{ɛ}*C*

*, and*

_{1ɛ}*C*

*are adjustable and have been determined by many iterations for turbulent flow conditions. From previous results, the values of these constants are follows:*

_{2ɛ}*C*

*= 0.09,*

_{μ}*σ*

*= 1.00,*

_{k}*σ*

*= 1.30,*

_{ɛ}*C*

*= 1.44, and*

_{1ɛ}*C*

*= 1.92 [17].*

_{2ɛ}### 2.2. Module Structure

### 2.3. Boundary Conditions

### 2.4. Non-uniformity Coefficient and Energy Utilization

*N*, that considers the correlation can be estimated from the flux distribution;

*N*is the standard deviation of the local flux, and indicates there is less uniform distribution when it is larger. The non-uniformity coefficient can be calculated from Eq. (3) and (4) [13].

*J*is the flux at the specific location, and

*ĵ*is the length-averaged local flux.

*η*, is the pressure drop in transmembrane pressure for filtration as fluid is transported to the place;

*η*can be calculated from Eq. (5) and (6).

*p*

*is the pressure at the inlet,*

_{0}*p*

*is the pressure at the specific location, and*

_{L}*J*

*is the ratio between the pressure drop and dynamic viscosity.*

_{lim}*η*indicates the energy utilization of a module and can be applied to most pressurized modules regardless of their shapes. Hence,

*N*and

*η*are parameters commonly used to verify the effect of a variety of geometrical configurations [13].

*N*and

*η*were derived from five planes, e.g., the inlet, and at 2.8 cm, 9.0 cm, 11.0 cm, and 12 cm from inlet, in order to determine the even flow distribution at the inflow part. The locations of the five points measured are shown in Fig. S1.

### 3. Results and Discussion

### 3.1. Fluid Dynamics at Inlet and Outlet Plane

^{3}and 2136 cm

^{3}, whereas the Non-distributor was 2,140 cm

^{3}. Hence, it was considered that the inlet distributor greatly divided the inflow part and acted as a large barrier of the fluid flow.

### 3.2. Velocity and Pressure at Cross-sectional Plane

^{2}, smaller than the other circles; the area of the 2

^{nd}circle was 628 cm

^{2}and the edge circle was 1,256 cm

^{2}. Overall, the fluid velocity at the cross-sectional plane in the inflow part was high and displayed the most even distribution in Case 2, indicating the ideal flow.

### 3.3. Fluid Dynamics at Sections on the Outlet

#### 3.3.1. Velocity and pressure variation at sections on the outlet

#### 3.3.2. Flux variation at sections on the outlet

^{2}·h) (max. 505,948 L/(m

^{2}·h) in Section 1, with a minimum of 0 L/(m

^{2}·h) in Sections 4 to 9). In Case 1, all fluid went through Sections 1, 3, 8, and 9, though there was no flux in Sections 4 to 7 because the inlet distributor had a wide funnel shape at the end part, and the fluid did not reach the outlet plane. Here, the standard deviation was 86,945 L/(m

^{2}·h) (max. 263,634 L/(m

^{2}·h) at Section 1, with a minimum of 0 L/(m

^{2}·h) in Sections 4 to 7); the standard deviation for Case 1 is lower than for the Non-distributor. In Case 2, the fluid passed through all sections of the outlet plane except for Section 2, and the standard deviation was the lowest at 27,599 L/(m

^{2}·h) (max. 83,687 L/(m

^{2}·h) at Section 7, with a minimum of 0 L/(m

^{2}·h) in Section 2). These results confirm that the fluid is relatively uniformly distributed in the inflow part in Case 2, as the inlet distributor has two rounded shapes. Overall, the flux distribution in sections of the outlet plane was the best in Case 2, and it had the potential to reduce the local flux and fouling by maintaining an even flux due to the shape of the inlet distributor.

### 3.4. Non-uniformity Coefficient and Energy Utilization at Each Plane

*N*is the standard deviation of the local fluxes at the specific plane, which indicated that the evenness of flow is higher when the value is smaller. Fig. 7 presents the values of the non-uniformity coefficient at five planes, i.e., the inlet, and at 2.8 cm, 9 cm, 11 cm and 12 cm (outlet) from inlet, to identify the evenness of flow within the inflow part.

*N*of the Non-distributor is the largest in all planes, with Cases 1 and 2 being relatively small, proving that the flow was uniform on each plane due to the inlet distributors. In addition, the

*N*at the inlet was the largest compared to other planes, with the

*N*values at 2.8 cm, 9 cm, and 11 cm remaining almost constant; there was only a slight increase at the outlet plane (12 cm). Here, the fluid flow from the inlet was stabilized as it passed the 2.8 cm, 9 cm, and 11 cm planes, even though the diameter of inflow part increased, and the

*N*at the outlet plane was influenced by the presence or absence of inlet distributors.

*N*values at 2.8 cm for the Non-distributor, Case 1, and Case 2 were 0.023, 0.013, and 0.009, respectively; Cases 1 and 2 were lower compared to the Non-distributor, indicating an even flow distribution at this plane due to the effect of the inlet distributors.

*N*values for Case 2 at 9 cm and 11 cm were the lowest at 0.0091 and 0.0088, respectively, whereas the

*N*values at these planes were 0.015 and 0.014 for the Non-distributor, and 0.016 and 0.012 for Case 1. The total length of the inlet distributor in Case 1 was shorter, at 8 cm, than for Case 2, which reduced its influence. However, the

*N*values at the outlet plane were 0.039 for the Non-distributor, 0.030 for Case 1, and 0.017 for Case 2. Overall, Case 2 displayed the lowest non-uniformity coefficient, and Case 1 was slightly lower than the Non-distributor. These results confirm that the fluid from the inlet was evenly distributed to the outlet plane within the inflow part. Here, the non-uniformity coefficient illustrated the fact that the inlet distributor in Cases 1 and 2 induced an even flow distribution during fluid flow from the inlet to the outlet.

*η*, refers to the pressure drop as a fluid is transported to a place; a high

*η*denotes a low pressure drop and energy utilization of the module newly designed. Fig. 7(b) shows the values of energy utilization at four planes (2.8 cm, 9 cm, 11 cm, and 12 cm (outlet)) based on the initial pressure at the inlet.

*η*at 2.8 cm was 0.0019 for the Non-distributor, 0.0046 for Case 1, and 0.0033 for Case 2; Case 1 was the highest and the Non-distributor was the lowest. It is posited here that the pressure drop decreased because the flow was evenly distributed by the inlet distributor. The

*η*at 9 cm, 11 cm, and 12 cm steeply decreased, as there was a rapid pressure drop compared to 2.8 cm in all conditions. The

*η*values at 12 cm (outlet plane) were 1.64e−05 for the Non-distributor, 1.35e−0.5 for Case 1, and 1.46e−05 for Case 2; the Non-distributor showed a slightly higher value, with the lowest pressure drop, because the fluid was not disturbed by inlet distributor.

*N*and high

*η*when the shell void fraction (ɛ) in the membrane module is large. Under this condition, there is generally a uniform flow and high energy efficiency due to the low pressure drop when the water channel is increased in the membrane module. In our study, Case 2 showed a more uniform flow than Case 1 and the Non-distributor has higher energy efficiency than Case 1 with its complicated configuration. Hence, Case 2 appears to have a more ideal flow pattern.

### 4. Conclusions

^{2}·h) in the Non-distributor. On the other hand, Case 2 showed that the fluid passed through all sections of the outlet plane except for Section 2, and its standard deviation was the lowest at 27,599 L/(m

^{2}·h), indicating an even flow distribution. Based on these results, Case 2 has the potential to reduce the local flux and fouling due to its round-shaped inlet distributor, which is divided into 3 parts.

*N*, of the Non-distributor was the largest in all planes. As Cases 1 and 2 were smaller, the flow was uniform on each plane due to the inlet distributors. The energy utilization parameter,

*η*, of Case 1 was the highest and that of the Non-distributor was the lowest. It seems that the pressure drop of Case 1 decreases because the flow is evenly distributed by the inlet distributor.