# Correlations for Concentration Polarization and Pressure Drop in Spacer-Filled RO Membrane Modules Based on CFD Simulations

^{1}

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## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. CFD Simulations

**u**is the fluid velocity vector, ρ the density of the fluid, μ the viscosity of the fluid, p the pressure, c the solute concentration and D

_{sw}the diffusivity of solute in water. A dilute aqueous solution with a single solute is assumed and the fluid properties are approximated by those of pure water.

_{w}is the water flux, J

_{s}the solute flux, A the hydraulic conductivity, B the solute permeability, p

_{p}the permeate pressure, c

_{p}the permeate concentration, and κ the osmotic factor, which is assumed constant under an isothermal condition.

_{p}= 0) and (ii) permeate concentration is determined by the solute to water flux ratio (c

_{p}= J

_{s}/J

_{w}), water and solute fluxes can be expressed as a function of feed concentration and pressure [37].

**n**is the normal vector and

**u**and

_{l,up}**u**are the leaking velocities at the upper and lower membrane walls, respectively.

_{l,low}_{f}and filament spacing L

_{f}(details are provided later). The fluid domains are discretized into quadrilateral elements of varying sizes with denser mesh near the upper and lower membrane walls in order to ensure adequate resolution in the mass transfer boundary layer. Local mesh refinement is also performed in regions where circular filaments are in contact with membrane walls in the cavity and zigzag configurations. The number of mesh elements for the baseline cases (D

_{f}= 0.5 mm and L

_{f}= 4.5 mm) are 14,400, 16,200, and 9000 for the cavity, submerged and zigzag configurations, respectively. Mesh independence study is performed by increasing the mesh density until changes in averaged water and solute fluxes are less than 3 × 10

^{−5}%. Moreover, the local concentration in the vicinity of center filament and membrane wall, at x = L

_{f}+ 0.1 mm and y = 0.01 mm for the cavity and submerged configuration and y = H

_{c}– 0.01 mm for the zigzag configuration, is monitored and changes between the coarsest and finest meshes are below 0.02% for three configurations.

^{−6}for the relative error in the solution.

#### 2.2. Selection of Geometric and Operating Conditions

_{0}, solute diffusivity D

_{sw}, inlet concentration c

_{0}and outlet pressure p

_{0}, filament spacing L

_{f}, and filament diameter D

_{f}. Each operating parameter is varied one at a time while the other parameters are kept the same as in the corresponding baseline case. As shown in Figure 2, geometric variables L

_{f}and D

_{f}are simultaneously varied with the operating variables being fixed. For each configuration, a total of 76 cases are simulated as listed in Table 1. Spacers’ geometric parameters L

_{f}and D

_{f}are varied by selecting different feed channel height H

_{c}and L

_{f}to H

_{c}ratio, with D

_{f}and H

_{c}being related as: D

_{f}= 0.5H

_{c}. The baseline inlet concentration 603.45 mol/m

^{3}is equivalent to 35 kg/m

^{3}of an aqueous sodium chloride solution, which can approximate seawater. The hydraulic conductivity A and solute permeability B are selected within reported ranges [37,42,43,44]. The osmotic factor κ is obtained at temperature of 298.15 K.

#### 2.3. Dimensional Analysis and Derivation of Correlations

#### 2.3.1. Correlation for CP Modulus

_{CP}, fluid density ρ, dynamic viscosity μ, solute diffusivity in fluid D

_{sw}, cross-velocity u

_{c}, transmembrane velocity u

_{t}(or water flux at membrane walls J

_{w}), filament diameter D

_{f}, and filament spacing L

_{f}. In this study, a slightly different definition of CP modulus is used to facilitate its implementation in an RO process model.

_{m}is the feed concentration at membrane walls and c

_{0}the inlet concentration. According to the Buckingham Pi theorem, the above dimensional parameters can be grouped into five independent dimensionless groups (8 − 3 = 5). By choosing ρ, μ, D

_{f}as repeating variables, five dimensionless groups are formed, including M

_{CP}and 4 dimensionless groups commonly used in momentum and mass transfer, such as Re and Sc. The dimensionless groups Re

_{c}, Re

_{t}, Sc, and GR are defined as:

_{f}is used as a characteristic length in Re

_{c}and Re

_{t}. GR, the filament spacing to diameter ratio, allows the inclusion of an important geometric feature of a spacer-filled channel. Their functional relationship can be expressed as follows:

_{t}could be considered negligible compared to Re

_{c}due to transmembrane velocity being usually several order of magnitude smaller than cross-velocity, the term is kept in Equation (16) for the time being with an adjustment factor m. This is discussed later in Section 3.1.1 in relation with CFD results for the effects of transmembrane velocity on CP moduli.

#### 2.3.2. Correlation for Pressure Drop

_{c}, ρ, μ, D

_{f}, and L

_{f}. The transmembrane velocity u

_{t}is not included as its effect on axial pressure drop is negligible due to its small magnitude compared to the cross-velocity. Analysis using the Buckingham Pi theorem yields three dimensionless groups, which can be related as follows:

_{f}), which is the pressure drop per unit length in the fluid domain.

#### 2.3.3. Parameter Estimation

_{CP}, u

_{c}, u

_{t}, and Δp are averaged over the entire fluid domain, and spatially averaged values are used for data fitting. A wide range of values for the dimensionless parameters are considered by varying operating and geometric conditions (Sc = 111–4475, Re

_{c}= 1.12–274, Re

_{t}= 7.82 × 10

^{−4}–6.70 × 10

^{−3}, and GR = 4–12). Since transmembrane velocity, u

_{t}, is a key parameter that determines CP, a range of u

_{t}(3–12 × 10

^{−6}m/s) are obtained by varying feed concentration and pressure in the simulations. The coefficient and exponents in Equation (16) are estimated by solving the following:

_{CP}

^{CFD}is the CP modulus obtained from CFD simulations, M

_{CP}(C, α, β, γ, δ) the CP modulus calculated by Equation (16) with a set of parameters, n the index of data and N the total number of data. To find the values of C, α, β, γ, and δ that minimize the sum of the squared errors of CP moduli, the built-in simplex algorithm in MATLAB 2015b is utilized. Since there might exist multiple local minima, it is critical to provide a good initial guess, which is obtained by differentiating Equation (16) together with the corresponding CFD data.

#### 2.4. Implementation of Correlations in a Process Model

- Correlations for CP (obtained in this study) are coupled with the solution–diffusion model described in Equations (4) and (5).
- Existing correlations for mass transfer coefficient are implemented using the film theory.
- All these are incorporated in a spiral wound module model, where mass balances for the feed and permeate streams are performed separately.

_{c}have different characteristic lengths; d

_{h}and D

_{f}for Equations (20) and (21), respectively. The derived expression for d

_{h}of spacer-filled channels in [30] is expressed as:

_{SM}and f

_{K}are the friction factors in the respective correlations, and ν the kinematic viscosity. Two correlations, as in Equation (24), were derived using 3D CFD simulations with a constant D

_{f}(=1 mm). The main reason for selecting the aforementioned correlations is that they were derived under RO operating conditions, which usually yield a feed flow velocity below 0.4 m/s [23]. Two additional correlations for mass transfer coefficient [30,42,46,47] and pressure drop [42,48] are also included in the comparison of module performance, which are frequently used in membrane process models when spacer specifications are not available.

_{sp}represents wall roughness in a channel due to spacer filaments. The negative sign for pressure drop correlations in Equations (18), (23), (24) and (26) is omitted in order to make friction factors positive, but it is added when dp/dx is implemented in a process model. Equation (25) is for a flat channel without spacers, in which the hydraulic diameter, d

_{h}

_{,empty}, correspond to that of an empty channel.

_{sp}in Equation (26)) are also estimated by comparing predicted recovery, feed pressure drop and permeate concentration with those from ROSA (Reverse Osmosis System Analysis, The Dow Chemical Company), the latter serve as reference data for comparison purposes. The estimated permeabilities and friction coefficients values are listed in Table 2, as well as module and process specifications used to simulate RO processes.

## 3. Results and Discussions

#### 3.1. Analysis of CFD Results

#### 3.1.1. Water Flux and CP Modulus

_{t}in the CP modulus correlation in Equation (16); if Re

_{t}was neglected, the effects of transmembrane velocity caused by different feed concentrations and pressures on CP moduli would not be captured. On the other hand, increasing solute diffusivity and inlet flow velocity causes water flux to rise until it plateaus, as shown in Figure 3b,d. The increased water flux could be due to mitigated CP by accelerated diffusion of solutes back to the bulk flow at high solute diffusivity and by enhanced mixing near membrane walls at high flow velocity. This is supported by Figure 3f,h, where CP moduli are reduced as solute diffusivity or inlet flow velocity increases. The trend of water flux plateauing at high inlet velocity in Figure 3d also indicates that there is a limit to the impact of enhanced mixing.

_{f}/D

_{f}) ratios in Figure 4. The effect of channel height is obvious: water flux is reduced and CP modulus is elevated with an increase in channel height. Compared to the cavity configuration, the submerged configuration offers 2.1–13.5% higher water flux and 1.8–9.9% lower CP modulus by varying the geometric conditions. The results also indicate that denser spacer mesh (i.e., at small L

_{f}/D

_{f}ratios) achieves a slightly better performance.

_{f}/D

_{f}ratio in Figure 4d, where there is a logarithmic increase in CP modulus up to a channel height of around 0.8 mm and an almost linear increase thereafter for the cavity and zigzag configurations. Combined with the observation described previously in relation to Figure 3h, it appears that in the cavity and zigzag configurations CP moduli behave differently in two distinguished regimes, which can be separated using a critical Reynolds number, Re

_{crit}, based on the cross-flow velocity and characteristic length, D

_{f}(= 0.5H

_{f}). Therefore, a critical Reynolds number is introduced in this study to more accurately correlate CP moduli with operating and geometric conditions for the cavity and zigzag configurations.

#### 3.1.2. Pressure Drop and Friction Factor

_{f}/D

_{f}ratios (Figure 5b–d). As the channel height increases, pressure drop per unit length decreases rapidly. At the same channel height, higher L

_{f}/D

_{f}ratios result in less pronounced pressure drops per unit length. Since L

_{f}/D

_{f}ratio affects the void volume in a channel, it has a strong influence on the mean velocity and pressure drop. Friction factors show a similar trend to pressure drop per unit length except for varying inlet velocities. The definition of friction factor in Equation (18) dictates that it is inversely proportional to the square of cross-flow velocity, which results in the trend shown in Figure 5e.

#### 3.2. Correlations for CP Modulus and Friction Factor

^{4}in order to make the exponents of Re

_{t}and Re

_{c}be of a similar order of magnitude. Both CP modulus (M

_{CP}) and friction factor (f) correlations have a good fit to the CFD data with a mean APE less than 1.5% for all configurations. There are two sets of parameters for the M

_{CP}correlation for the cavity and zigzag configurations, depending on the critical Reynolds number, Re

_{crit}, which is found to be 24.4–31.0 for varying inlet velocities (Figure 3h) and geometric conditions (Figure 4d). As a result, Re

_{crit}is assumed to be 25.5 for both the cavity and zigzag configurations.

_{c}and Re

_{t}(i.e., α and β) reveal that CP modulus is more sensitive to the transmembrane velocity than cross-flow velocity. The value for β indicates that the relationship between Re

_{t}and M

_{CP}is almost linear, whereas the value for α suggests that Re

_{c}has a stronger influence on M

_{CP}at low Re

_{c}than at high Re

_{c}. It is also noted that the cavity and zigzag configurations have similar parameter values for the given regime, but there are differences between the two regimes. The value for γ, exponent of Sc, is assumed to be constant regardless of the flow regimes but to vary with the filament configuration. For the cavity and zigzag configurations, there is only a slight difference in α and β between the two regimes. However, values for δ are markedly different, leading to different effects: in the low Re

_{c}regime, M

_{CP}decreases slightly with an increase in L

_{f}/D

_{f}ratio, as displayed in Figure 4d–f. In the high Re

_{c}regime, δ tends to zero for both configurations, suggesting a negligible effect of L

_{f}/D

_{f}on M

_{CP}. For the submerged configuration, δ is very different from that for the other configurations; the positive sign of δ means that M

_{CP}increases with an increase in L

_{f}/D

_{f}, as shown in Figure 4d–f. The coefficient C varies with hydrodynamics and geometry, hence giving different values depending on configurations and flow regimes.

_{c}, implies an inverse relation between f and Re

_{c}as can be seen in Figure 5e. For the exponent of L

_{f}/D

_{f}ratio, δ′, its values capture the fact that f decreases more rapidly with an increase in L

_{f}/D

_{f}in the submerged configuration than in the cavity and zigzag configurations, as displayed in Figure 5f–h. The parameters C′ and ζ for the submerged configuration are found to be approximately 3.4–4.5 times larger than those for the cavity and zigzag configurations. This suggests the submerged configuration has a larger friction factor in most situations. Further comparisons between the CFD results and derived correlations can be found in Section C in the Supporting Information.

#### 3.3. Comparison with Existing Correlations

_{f}/D

_{f}ratio to commercial spacers (L

_{f}/D

_{f}= 8 and 12 and mesh angle = 90°) [26,27]. However, the filament diameter (1 mm) adopted in their work far exceeded that of commercial spacers (0.27–0.385 mm) based on the study by Schock and Miquel on four feed spacers in spiral wound RO modules [30].

#### 3.3.1. CP Modulus and Water Flux

#### 3.3.2. Pressure Gradient

_{f}/D

_{f}= 6 and L

_{f}/D

_{f}= 8 predict larger pressure drops than that of Schock and Miquel and our new correlations. For the particular spacer simulated here (Table 2), L

_{f}/D

_{f}= 7.3, which lies between the L

_{f}/D

_{f}ratios covered by the correlations of Koutsou et al.; the latter appear to over-predict dp/dx by 2.3 times compared to the Schock and Miquel correlation for the range of flow velocities examined. This is consistent with the findings reported in Koutsou et al. [26]. The mean and maximum percentage differences between the new and existing correlations are summarized in Table 5.

#### 3.4. Simulation Results from Process Model

_{f}/D

_{f}= 8 is used. Since this correlation was derived over a narrow flow velocity range (up to 0.15 m/s), it is not surprising that larger differences are found at higher feed flow velocities, although the differences are not significant.

#### 3.5. Limitations of the CFD Model and Derived Correlations

## 4. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Three-dimensional spacer geometries (

**a**–

**c**) and their cross-sections along the main flow path (

**d**–

**f**). (

**a**) Nonwoven spacer, (

**b**) fully woven spacer, (

**c**) partially woven spacer. 2D cross-sections: (

**d**) cavity, (

**e**) submerged, and (

**f**) zigzag configuration.

**Figure 2.**Schematic illustration of strategy for varying operating and geometric conditions. (

**a**) at a fixed geometry (

**b**) at a fixed operating condition.

**Figure 3.**Average water fluxes (top) and CP moduli (bottom) in the cavity, submerged and zigzag configurations for varying inlet concentration c

_{0}(

**a**,

**e**), diffusivity of solute in water D

_{sw}(

**b**,

**f**), outlet pressure p

_{0}(

**c**,

**g**), and inlet velocity u

_{0}(

**d**,

**h**).

**Figure 4.**Average water fluxes (top) and CP moduli (bottom) in the cavity, submerged and zigzag configurations for varying channel height at selected filament length to diameter (L

_{f}/D

_{f}) ratios. (

**a**,

**d**): L

_{f}/D

_{f}=4; (

**b**,

**e**): L

_{f}/D

_{f}=8; (

**c**,

**f**): L

_{f}/D

_{f}=12.

**Figure 5.**Pressure drop per unit length (top) and friction factor (bottom) in the cavity, submerged and zigzag configurations with respect to inlet velocity (

**a**,

**e**) and channel height with L

_{f}/D

_{f}=4 (

**b**,

**f**), L

_{f}/D

_{f}=/8 (

**c**,

**g**), L

_{f}/D

_{f}=12 (

**d**,

**h**).

**Table 1.**Model parameters and simulation conditions where inlet concentration c

_{0}, solute diffusivity in water D

_{sw}, outlet pressure p

_{0}, inlet velocity u

_{0}, channel height H

_{c}, and filament spacing to channel height ratio L

_{f}-to-H

_{c}are varied.

Parameters and Conditions | Case Number | |||||
---|---|---|---|---|---|---|

Base | 1–7 | 8–15 | 16–19 | 20–30 | 31–75 | |

Hydraulic conductivity, A (m/(s·Pa)) | 2.50 × 10^{−12} | |||||

Solute permeability, B (m/s) | 2.50 × 10^{−8} | |||||

Osmotic factor, κ (m^{3}·Pa/mol) | 4958 | |||||

Density, ρ (kg/m^{3}) | 998.20 | |||||

Viscosity, μ (Pa·s) | 8.93 × 10^{−4} | |||||

Inlet concentration, c_{0} (mol/m^{3}) | 603.45 | 362.07–905.17 | 603.45 | 603.45 | 603.45 | 603.45 |

Solute diffusivity, D_{sw} (10^{−9} m^{2}/s) | 1 | 1 | 0.2–8 | 1 | 1 | 1 |

Outlet pressure, p_{0} (10^{5} Pa) | 60 | 60 | 60 | 50–90 | 60 | 60 |

Mean inlet velocity, u_{0} (m/s) | 0.1 | 0.1 | 0.1 | 0.1 | 0.01–0.35 | 0.1 |

Channel height, H_{c} (10^{−3} m) | 1 | 1 | 1 | 1 | 1 | 0.2–1.4 |

Filament spacing to channel height ratio, L_{f}/H_{c} (-) | 4.5 | 4.5 | 4.5 | 4.5 | 4.5 | 2–6 |

Symbol (Unit) | Name | Value | Source |
---|---|---|---|

N_{ele} (-) | Number of modules arrayed in series | 6 | [49,50,51] |

N_{leaf} (-) | Number of leaves | 16 | [42,52] |

L_{i} (m) | Membrane length | 1.016 | [52] |

W_{i} (m) | Membrane width | 1.505 | [42,52] |

H_{c} (m) | Feed spacer thickness | 7.1 × 10^{−4} | [52] |

H_{c}_{,p} (m) | Permeate spacer thickness | 4.0 × 10^{−4} | [52] |

L_{g} (m) | Glue line width in length direction | 4 × 10^{−2} | [53] |

W_{g} (m) | Glue line width in width direction | 17 × 10^{−2} | [53] |

D_{f} (m) | Filament diameter | 3.85 × 10^{−4} | [30] |

L_{f} (m) | Filament spacing | 2.8 × 10^{−3} | [30] |

ε (-) | Voidage of spacer-filled channel | 0.89 | [30] |

d_{h} (m) | Hydraulic diameter of spacer-filled channel | 0.95 × 10^{−3} | [30] |

A (m^{3}/m^{2} s Pa) | Pure water permeability constant | 3.26 × 10^{−12} | Estimated |

B (m^{3}/m^{2} s) | Salt permeability constant | 1.15 × 10^{−8} | Estimated |

k_{sp}_{,f} (-) | Frictional coefficient in feed channel | 1.58 | Estimated |

k_{sp}_{,p} (-) | Frictional coefficient in permeate channel | 30 | Estimated |

η_{HPP} (-) | Efficiency of high pressure pump | 0.85 | [42,49,50,51] |

η_{PX} (-) | Efficiency of pressure exchanger | 0.98 | [42,49,50,51] |

**Table 3.**Estimated parameter values for CP modulus and pressure drop correlations in three configurations and APEs of the correlations compared to the corresponding CFD results.

Regime | Cavity | Submerged | Zigzag | ||||
---|---|---|---|---|---|---|---|

Re_{c} ≥ Re_{crit} | Re_{c} < Re_{crit} | Re_{c} ≥ Re_{crit} | Re_{c} < Re_{crit} | ||||

CP modulus, M_{CP}in Equation (16) | Estimated parameter values | C | 2.55 × 10^{−4} | 7.06 × 10^{−4} | 5.55 × 10^{−3} | 3.24 × 10^{−4} | 7.63 × 10^{−4} |

α | −0.350 | −0.382 | −0.422 | −0.394 | −0.365 | ||

β | 1.11 | 0.931 | 1.09 | 1.12 | 0.909 | ||

γ | 0.611 | 0.611 | 0.672 | 0.597 | 0.597 | ||

δ | 5.40 × 10^{−4} | −0.143 | 0.536 | 2.18 × 10^{−4} | −0.112 | ||

Mean APE [%] | 0.375 | 0.147 | 0.516 | ||||

Friction factor, f in Equation (17) | Estimated parameter values | C′ | 14.8 | 63.3 | 14.0 | ||

α′ | −0.910 | −0.994 | −0.898 | ||||

δ′ | −0.525 | −0.810 | −0.510 | ||||

ζ | 0.0256 | 0.0868 | 0.0200 | ||||

Mean APE [%] | 1.15 | 1.48 | 0.974 |

**Table 4.**Average and maximum absolute percentage difference between the new correlations and existing correlations for CP moduli and water fluxes.

Correlation | Absolute Percentage Difference, |Y – Y_{ex}| /Y_{ex} (%) | ||||||
---|---|---|---|---|---|---|---|

Variable | Existing | Schock and Miquel [30] | Koutsou et al. [27] | ||||

New | Cavity | Submerged | Zigzag | Cavity | Submerged | Zigzag | |

CP modulus | Mean | 1.9 | 4.5 | 1.8 | 5.5 | 1.8 | 5.4 |

Max. | 6.7 | 10.3 | 6.5 | 5.6 | 1.9 | 5.6 | |

Water flux | Mean | 2.7 | 6.4 | 2.6 | 5.6 | 1.6 | 5.5 |

Max. | 11.0 | 16.5 | 10.7 | 5.8 | 1.7 | 6.1 |

**Table 5.**Average and maximum absolute percentage differences between our new correlations and existing correlations for pressure gradient.

Correlation | Absolute Percentage Difference (%) | ||||||||
---|---|---|---|---|---|---|---|---|---|

Existing | Schock and Miquel [30] | Koutsou et al. [26] (L_{f}/D_{f} = 8) | Koutsou et al. [26] (L_{f}/D_{f} = 6) | ||||||

New | Cavity | Submerged | Zigzag | Cavity | Submerged | Zigzag | Cavity | Submerged | Zigzag |

Mean | 54 | 12 | 55 | 66 | 30 | 67 | 80 | 58 | 80 |

Max. | 63 | 28 | 64 | 74 | 44 | 75 | 84 | 64 | 84 |

**Table 6.**Simulation results for water recovery R

_{w}, specific energy consumption E

_{sp}and feed pressure drop P

_{d}, with different combinations of CP and pressure drop correlations under three simulation conditions (with feed concentration c

_{f}= 35kg/m

^{3}).

Case No. | Correlation | Condition 1 | Condition 2 | Condition 3 | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|

CP | dp/dx | Q_{f} = 2 × 10^{−3} m^{3}/s (u_{f} = 0.13 m/s), P_{f} = 60 bar | Q_{f} = 3 × 10^{−3} m^{3}/s (u_{f} = 0.20 m/s), P_{f} = 70 bar | Q_{f} = 4.5 × 10^{−3} m^{3}/s (u_{f} = 0.30 m/s), P_{f} = 80 bar | |||||||

R_{w}(%) | E_{sp}(kWh/m ^{3}) | P_{d}(×10 ^{5} Pa) | R_{w}(%) | E_{sp}(kWh/m ^{3}) | P_{d}(×10 ^{5} Pa) | R_{w}(%) | E_{sp}(kWh/m ^{3}) | P_{d}(×10 ^{5} Pa) | |||

1 | Equation (25) | Equation (26) | 50.03 | 2.06 | 0.72 | 53.43 | 2.39 | 1.08 | 52.92 | 2.74 | 1.67 |

Relative difference [%] | Condition 1 | Condition 2 | Condition 3 | ||||||||

Δ_{Rw} | Δ_{Esp} | Δ_{Pd} | Δ_{Rw} | Δ_{Esp} | Δ_{Pd} | Δ_{Rw} | Δ_{Esp} | Δ_{Pd} | |||

2 | Equation (25) | Zig * | 0.53 | −0.64 | −53 | 0.56 | −0.66 | −49 | 0.65 | −0.81 | −44 |

3 | Equation (25) | Sub | 0.066 | −0.096 | −8.0 | −0.056 | 0.034 | 2.3 | −0.36 | 0.37 | 19 |

4 | Equation (25) | S&M ^{+} | 0.31 | −0.40 | −34 | 0.075 | −0.15 | −12 | −0.43 | 0.39 | 20 |

5 | Equation (25) | KT8 ^{$} | 0.031 | −0.097 | −8.4 | −0.42 | 0.39 | 28 | −1.5 | 1.6 | 83 |

6 | Zig * | Equation (26) | 0.12 | −0.0076 | 0.34 | −1.0 | 0.10 | 1.1 | −2.6 | 0.28 | 1.6 |

7 | Sub | Equation (26) | 1.6 | −0.15 | −1.0 | 1.1 | −0.10 | -0.49 | 0.25 | −0.023 | 0.081 |

8 | S&M ^{++} | Equation (26) | -4.7 | 0.49 | 3.4 | −4.7 | 0.47 | 2.9 | −4.3 | 0.47 | 2.1 |

9 | KT ^{$$} | Equation (26) | 1.9 | −0.18 | −1.5 | 1.9 | −0.18 | −1.2 | 1.7 | −0.17 | −0.78 |

^{+}The correlation of Schock and Miquel for pressure gradient [30] in Equation (23).

^{++}The correlation of Schock and Miquel [30] for mass transfer coefficient in Equation (20).

^{$}The correlation of Koutsou et al. [26] for pressure gradient when L

_{f}/D

_{f}= 8 in Equation (24);

^{$$}The correlation of Koutsou et al. [27] for mass transfer coefficient in Equation (21).

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## Share and Cite

**MDPI and ACS Style**

Gu, B.; Adjiman, C.S.; Xu, X.Y.
Correlations for Concentration Polarization and Pressure Drop in Spacer-Filled RO Membrane Modules Based on CFD Simulations. *Membranes* **2021**, *11*, 338.
https://doi.org/10.3390/membranes11050338

**AMA Style**

Gu B, Adjiman CS, Xu XY.
Correlations for Concentration Polarization and Pressure Drop in Spacer-Filled RO Membrane Modules Based on CFD Simulations. *Membranes*. 2021; 11(5):338.
https://doi.org/10.3390/membranes11050338

**Chicago/Turabian Style**

Gu, Boram, Claire S. Adjiman, and Xiao Yun Xu.
2021. "Correlations for Concentration Polarization and Pressure Drop in Spacer-Filled RO Membrane Modules Based on CFD Simulations" *Membranes* 11, no. 5: 338.
https://doi.org/10.3390/membranes11050338