# Membrane Deformation and Its Effects on Flow and Mass Transfer in the Electromembrane Processes

^{*}

## Abstract

**:**

## 1. Introduction

^{2}) installed within the REAPower project [28], pressure drops from ~0.2 to ~0.9 bar were measured at flow velocities up to 1 cm/s [29]. Despite some of the pressure drop can be supposed to occur in the manifolds, a significant part of it is expected to occur in the channels, thus causing the onset of non-negligible TMP values. Moreover, the compartments were asymmetric, because the viscosity of the concentrated solution (brine) was almost twice that of the dilute feed, thus causing an unbalanced pressure distribution in the two solutions. Larger TMP values (up to ~1.5 bar) were measured by Hong et al. [27] in a cross-flow RED stack (35.5 × 35.5 cm

^{2}) fed with inlet velocities up to ~5 cm/s, which provided a significantly lower electrical power (less than half) compared to an equivalent parallel-flow stack. Although the authors attributed this decline in performance to issues of internal leakage, an important effect of deformation can be supposed.

^{3}fed with parallel flow in a stack 50 cm long, the optimum thickness and fluid velocity are ~400 μm and ~1.4 cm/s for the concentrate and ~217 μm and ~2.6 cm/s for the diluate. The pressure drop predicted by Computational Fluid Dynamics (CFD) correlations is 0.07 bar for the concentrate and 0.46 bar for the diluate, thus giving a maximum TMP located at the inlet equal to ~0.39 bar.

## 2. Results and Discussion

#### 2.1. Mechanical Results

#### 2.1.1. Influence of Pitch to Height Ratio (P/H) and Limiting Values

#### 2.1.2. Membrane and Channel Deformation for the Selected Geometry (P/H = 8)

#### 2.2. CFD Results for P/H = 8

#### 2.2.1. Undeformed Configuration

_{b}/c

_{w}(bulk to wall concentration ratio) in the undeformed configuration characterized by P/H = 8 for a friction velocity Reynolds number Re

_{τ}= 5.2 (bulk Reynolds number Re ≈ 17.6, approach velocity ~4 cm/s) and all three values of the flow attack angle (angle formed by the flow direction with the membrane ridges belonging to the upper wall) investigated (γ = 0°, 45° and 90°). The flow direction is indicated by arrows. Definitions of approach velocity and friction velocity Reynolds number are provided in Section 3.3.2, Equations (11) and (13).

_{b}/c

_{w}in the bottom row show that the case γ = 45° provides a more uniform distribution of the wall salt concentration, while the other two cases exhibit a very strong spanwise non-uniformity; the concentration is lower in the central region of the wall, where it becomes less than the bulk value despite the net overall salt flux being into the channel, and larger in the lateral regions of the channel walls, where low fluid velocities (stagnation zones) occur. Please note that the distribution of θ on the upper wall for γ = 0°, once rotated by 90°, would become the corresponding lower wall distribution for γ = 90° and vice versa. Also, remember that the values of the polarization coefficient depend on the flux imposed at the boundary and on the bulk concentration considered. Therefore, for example, much lower values would be obtained for dilute solutions.

#### 2.2.2. Deformed Configurations

_{τ}= 5.2 (corresponding to bulk Reynolds numbers between ~7 and ~35, approach velocity ~1.6 and ~7.8 cm/s, depending on the load conditions) and a flow attack angle γ = 90° (flow orthogonal to the profile ridges adjacent to the upper wall of the fluid channel), as evidenced in the inset.

_{b}/c

_{w}on both the upper and the lower wall of the fluid-filled channel, as clarified by the sketches in the rightmost part of the figure. The corresponding values of the F ratio (friction coefficient normalized by that for laminar flow in an undeformed void plane channel of indefinite width, 96/Re) and of the Sherwood number are also reported.

_{τ}is ~5.2, corresponding to an inlet-outlet pressure drop in a unit cell (1.6 mm in side) of ~34.36 Pa.

_{b}/c

_{w}are deeply affected by deformation. In the compressed configuration, both on the upper and on the lower wall the region of high θ (i.e., low concentration) observed in the undeformed case splits into two smaller regions, symmetrically located about the midline parallel to the flow direction, whereas the central region of the wall close to this midline exhibits low values of θ (i.e., high values of concentration). In the expanded configuration, the concentration distribution on the lower wall remains similar to that observed in the undeformed case, with a single large central strip where c

_{w}< c

_{b}, which is consistent with the fact that the longitudinal velocity exhibits a single central maximum as in the undeformed case (see top row). The θ distribution on the upper wall becomes flat, with two shallow θ maxima (i.e., c

_{w}minima) symmetrically located about the longitudinal midline.

#### 2.2.3. Global Parameters

_{τ}appear as an inclined row of symbols since they correspond to different values of Re.

## 3. Materials and Methods

#### 3.1. Simulation Strategy

- First, the influence of the pitch-to-height ratio (P/H) was addressed by mechanical simulations. A TMP of 0.8 bar was applied, and the geometry with the largest value of P/H still able to withstand this load without collapsing (i.e., without exhibiting a contact between opposite membranes) was identified. The figure of 0.8 bar was conservatively chosen as a value comfortably larger than the highest TMP actually expected in real RED/ED applications. The search for the largest admissible P/H was motivated by the fact that small values of P/H are associated with large pressure drops: many studies [13,16,30,53,54,55] have highlighted the importance of reducing pressure drop and thus mechanical power losses in the channels, especially in RED applications. It is true that the increase of P/H may also cause a reduction in mass transfer coefficients, but its effect on stack performance is usually less important.
- The geometry thus identified was then investigated under expansion and compression conditions corresponding to TMP varying from −0.4 to +0.4 bar. As discussed in the Introduction, this range encompasses most of the conditions that are likely to occur in actual ED/RED applications. For each load condition, the deformed configuration was computed by mechanical simulations.
- Finally, for each deformed configuration, fluid flow and mass transfer in the expanded or compressed channel were numerically simulated by CFD; in particular, friction coefficients and Sherwood numbers were computed as functions of the Reynolds number.

#### 3.2. The Mechanical Problem

#### 3.2.1. Governing Equations

_{i}are body forces, ${\epsilon}_{i}$ are normal strains, ${\gamma}_{ij}$ are shear strains, and G = E/[2 × (1 + ν)] (shear modulus). The small deformation approximation was not used.

#### 3.2.2. Computational Domain and Boundary Conditions

- Each of the four segments representing the external vertical edges of the domain (1) was clamped, i.e., zero displacement and rotation were imposed to all points belonging to it.
- Each of the four side faces of the domain (2) was imposed zero displacement in the direction normal to itself, so that a single computational domain is representative of a periodic array of repetitive units.
- The trans-membrane pressure TMP (relative to that of the internal fluid channel) was applied to the whole outer surface of the domain (3). Please note that TMP > 0 for compression conditions, while TMP < 0 for expansion conditions.

#### 3.2.3. FE mesh for Mechanical Simulations

^{3}elements (OCF-II) were used in all following simulations as a compromise between accuracy and computational effort.

#### 3.2.4. Mechanical Model Validation

^{2}square samples of flat ion exchange membranes. Details of the experiments are reported in Section 3 of the Supplementary Material (Figure S2). Figure 11a compares the predicted and experimental maximum displacements (placed at the central point of the membrane) as functions of the trans-membrane pressure. Error bars are reported for the experimental data. A good agreement can be observed, with a maximum relative discrepancy of a few percent.

#### 3.3. The Fluid Dynamics and Mass Transfer Problem

#### 3.3.1. Computational Approach

^{®}. From the numerical solution of these equations, velocity, pressure and electrolyte concentration fields are obtained. Raw results are then elaborated in order to calculate friction factor and Sherwood number. This simulation method is particularly suitable for the implementation of integrated multi-scale process simulators [1], where basic data produced by CFD are merged with higher-scale simulation tools [8,16,62].

#### 3.3.2. Governing Equations and Definitions

_{b}/c

_{w}ranging from 0.97 to 1.03, see Figure 5 and Figure 6). Under these assumptions, we used the most complete possible model, i.e., that made up the full, three-dimensional continuity, Navier-Stokes and scalar transport equations.

_{i}is the i-th velocity component of the fluid.

**s**, due to frictional losses and solute inflow or outflow through the channel walls, respectively. These apparently contradictory requirements are reconciled as follows.

- Consider pressure p first. In the fully developed region of a channel, p can be decomposed into a periodic component $\tilde{p}$, whose spatial distribution repeats itself identically in each unit cell, and a large-scale component −K
_{p}(**x**·**s**) which decreases linearly along the main flow direction whose unit vector is**s**(**x**is the position vector of components x_{i}). By substituting $\tilde{p}-{K}_{p}\left(\mathit{x}\xb7\mathit{s}\right)=\tilde{p}-{K}_{p}{x}_{i}{s}_{i}$ for p in the i-th steady-state Navier-Stokes equation:$$\frac{\partial \rho {u}_{j}{u}_{i}}{\partial {x}_{j}}=-\frac{\partial p}{\partial {x}_{i}}+\frac{\partial}{\partial {x}_{j}}\mu \frac{\partial {u}_{i}}{\partial {x}_{j}}$$$$\frac{\partial \rho {u}_{j}{u}_{i}}{\partial {x}_{j}}=-\frac{\partial \tilde{p}}{\partial {x}_{i}}+\frac{\partial}{\partial {x}_{j}}\mu \frac{\partial {u}_{i}}{\partial {x}_{j}}+{K}_{p}{s}_{i}$$Equation (6) is similar to Equation (5), but (a) the “true” pressure p is replaced by its periodic component $\tilde{p}$, and (b) a body force per unit volume (mean pressure gradient) acting along the main flow direction**s**appears at the right hand side. If required, the “true” pressure p can always be reconstructed from the simulation results as p = $\tilde{p}-{K}_{p}\left(\mathit{x}\xb7\mathit{s}\right)$. - In regard to the concentration c, by definition of fully developed conditions it can be decomposed into a periodic component $\tilde{c}$ and a large-scale component K
_{c}(**x**·**s**), where K_{c}can now be either positive (net inflow of electrolyte into the channel) or negative (net outflow of electrolyte from the channel). By substituting $\tilde{c}+{K}_{c}\left(\mathit{x}\cdot \mathit{s}\right)=\tilde{c}+{K}_{c}{x}_{i}{s}_{i}$ for c in the transport equation$$\frac{\partial {u}_{j}c}{\partial {x}_{j}}=\frac{\partial}{\partial {x}_{j}}D\frac{\partial c}{\partial {x}_{j}}$$$$\frac{\partial {u}_{j}\tilde{c}}{\partial {x}_{j}}=\frac{\partial}{\partial {x}_{j}}D\frac{\partial \tilde{c}}{\partial {x}_{j}}-{K}_{c}{u}_{s}$$_{s}=**u**·**s**is the local velocity component along the main flow direction**s**. The large-scale gradient K_{c}can be obtained by an elementary balance as:$${K}_{c}=\frac{A}{V}\frac{j}{{u}_{s}}$$_{s}〉 is the volume average of u_{s}.

_{Darcy}was defined with reference to the above approach velocity U and hydraulic diameter 2H, i.e., as:

_{p}= |dp/ds| in Equation (6) was imposed, while the flow rate was obtained as part of the solution. Please note that |dp/ds| can be expressed in terms of the friction velocity Reynolds number

_{τ}is the friction velocity,

_{τ}(friction velocity Reynolds number) rather than for a given Re (bulk Reynolds number). Please note that according to the present definitions, between Re, Re

_{τ}and f

_{Darcy}the following relation holds:

_{b}is the molar bulk concentration and c

_{w}is the local molar concentration at the membrane surface. Please note that defining the average polarization coefficient in such a way that it is lower than 1 [10,16,51], the local polarization coefficient in Equation (17) refers to the case of either a dilute channel of RED or a concentrate channel of ED, where the flux enters from membrane’s walls.

_{proj}is the projected membrane surface area and (c

_{w}) is the area average of c

_{w}on the same membrane. Please note that the Sherwood numbers on the two membranes facing a channel may differ depending on the flow direction.

#### 3.3.3. Flow Attack Angle, Boundary Conditions and Simulation Settings

**u**, p and c between opposite inlet-outlet boundaries. At the membrane surfaces, no slip conditions were imposed for velocity and a uniform value of 2.6 × 10

^{−4}mol/(m

^{2}s) for the molar salt flux entering the fluid, corresponding to a current density of 50 A/m

^{2}. An NaCl aqueous solution at a bulk concentration of 500 mol/m

^{3}was considered (i.e., seawater, see physical properties reported in Table 4). Please note that these choices on flux and bulk concentration affect directly the polarization coefficient (Equation (17)), while, the Sherwood number depends only on geometry, Re and Sc, due to the linearity of the transport Equation (8).

#### 3.3.4. FV Mesh for CFD Simulations

_{τ}≈ 5, corresponding to a bulk Reynolds number of ~20. Therefore, the test case selected for the grid-independence assessment lies well above the creeping flow range and close to the highest Reynolds numbers investigated. Results are shown in Table 5, where the computed values of the Darcy friction coefficient f

_{Darcy}and of the Sherwood number Sh are reported as functions of the number of finite volumes.

^{−10}.

## 4. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

CFD | Computational Fluid Dynamics |

ED | ElectroDialysis |

FE | Finite Element |

FV | Finite Volume |

OCF | Overlapped Crossed Filaments |

RED | Reverse ElectroDialysis |

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**Figure 1.**Profiled membranes of the Overlapped Crossed Filaments (OCF) type. The repetitive periodic unit of a cell pair is shown, enlarged, in the inset. The geometric parameters H (channel thickness), P (pitch) and γ (flow attack angle) are indicated.

**Figure 2.**Deformation of membranes with different P/H ratios under TMP = +0.8 bar. The quantity shown is the displacement in the direction orthogonal to the undeformed membranes (y). Top: external view; bottom: view after sectioning by a mid-plane A-A.

**Figure 3.**Deformation of membranes with P/H = 8 for the compressed and the expanded cases at TMP = ±0.4 bar. The quantity shown is the displacement in the direction orthogonal to the undeformed membranes (y). The corresponding deformed fluid volume is shown in the insets.

**Figure 4.**Fluid volume (normalized by the undeformed volume) as a function of trans-membrane pressure for P/H = 8.

**Figure 5.**CFD results for the undeformed configuration with P/H = 8 at Re

_{τ}= 5.2 (approach velocity ~4 cm/s). Top row: sketches illustrating the flow direction; middle row: 3-D streamlines; bottom row: maps of the concentration polarization coefficient θ = c

_{b}/c

_{w}on the upper wall. c

_{b}= 500 mol/m

^{3}, flux corresponding to a current density of 50 A/m

^{2}entering the fluid domain (dilute channel of RED or concentrate channel of ED).

**Figure 6.**Influence of deformation on flow and mass transfer for P/H = 8, γ = 90°. Left column: compressed (TMP = +0.4 bar); middle column: undeformed; right column: expanded (TMP = −0.4 bar). Top row: distribution of the streamwise velocity component in the central cross section of the channel (for symmetry reasons, only half map is shown); middle and bottom rows: distribution of the polarization coefficient on the upper and lower walls (see sketches on the right). c

_{b}= 500 mol/m

^{3}, flux corresponding to a current density of 50 A/m

^{2}entering the fluid domain (dilute channel of RED or a concentrate channel of ED). F ratio and Sherwood number are also reported.

**Figure 7.**Normalized Darcy friction coefficient (F ratio) as a function of Re for P/H = 8, different values of the trans-membrane pressure TMP and two values of the flow attack angle γ. (

**a**) γ = 0° or 90°; (

**b**) γ = 45°.

**Figure 8.**Sherwood number on the upper wall as a function of the Reynolds number for P/H = 8 and different values of the trans-membrane pressure and of the flow attack angles. (

**a**) γ = 0°; (

**b**) γ = 45°; (

**c**) γ = 90°.

**Figure 10.**Computational domain. Numbers 1–3 indicate the mechanical boundary conditions (see text). A detail of the finite element mesh is shown on the right.

**Figure 11.**Maximum displacement for a square, edge-clamped membrane as a function of the trans-membrane pressure. Comparison of FE predictions (solid line) with (

**a**) experimental results (symbols) of bulge tests on a 10 × 10 cm

^{2}sample and (

**b**) the first-order analytical solution by Iyengar and Naqvi [58] for a 2 × 2 mm

^{2}membrane (dashed line).

**Table 1.**Approach velocity and mass transfer coefficients for the load conditions in Figure 6.

Quantity | Compressed +0.4 Bar | Undeformed | Expanded −0.4 Bar |
---|---|---|---|

[cm/s] | ~1.6 | ~4 | ~7.8 |

〈k〉, upper wall [m/s] | ~3.72 × 10^{−5} | ~2.84 × 10^{−5} | ~2.89 × 10^{−5} |

〈k〉, lower wall [m/s] | ~2.47 × 10^{−5} | ~1.78 × 10^{−5} | ~2.00 × 10^{−5} |

Quantity | Value | Units |
---|---|---|

Membrane Young modulus, E | 150 | MPa |

Membrane Poisson ratio, ν | 0.4 | - |

Membrane thickness | 120 | μm |

Channel thickness, H | 200 | μm |

Pitch-to-height ratio, P/H | 7–9 | - |

Angle between filaments | 90 | deg |

FE Mechanical Mesh | No. Elements (P/H = 8) | Maximum Displacement at Membrane Surface [μm] |
---|---|---|

OCF-I | 200 × 10^{3} | 67.04 |

OCF-II | 500 × 10^{3} | 67.38 |

OCF-III | 1 million | 67.53 |

Property | Value | Units |
---|---|---|

Density, ρ | 1017 | kg m^{−3} |

Viscosity, µ | 0.931 × 10^{−3} | N s m^{−2} |

Salt diffusivity, D | 1.47 × 10^{−9} | m^{2} s^{−1} |

Schmidt number, (µ/ρ)/D | 622 | - |

FV CFD Mesh | No. Finite Volumes (P/H = 8, Re_{τ} = 5.2, γ = 0°) | Darcy Friction Coefficient | Sherwood Number (Upper Wall) | Sherwood Number (Lower Wall) |
---|---|---|---|---|

OCF-A | 2.252 × 10^{6} | 10.985 | 5.685 | 9.122 |

OCF-B | 3.833 × 10^{6} | 11.062 | 5.519 | 8.771 |

OCF-C | 7.502 × 10^{6} | 11.117 | 5.491 | 8.596 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Battaglia, G.; Gurreri, L.; Airò Farulla, G.; Cipollina, A.; Pirrotta, A.; Micale, G.; Ciofalo, M.
Membrane Deformation and Its Effects on Flow and Mass Transfer in the Electromembrane Processes. *Int. J. Mol. Sci.* **2019**, *20*, 1840.
https://doi.org/10.3390/ijms20081840

**AMA Style**

Battaglia G, Gurreri L, Airò Farulla G, Cipollina A, Pirrotta A, Micale G, Ciofalo M.
Membrane Deformation and Its Effects on Flow and Mass Transfer in the Electromembrane Processes. *International Journal of Molecular Sciences*. 2019; 20(8):1840.
https://doi.org/10.3390/ijms20081840

**Chicago/Turabian Style**

Battaglia, Giuseppe, Luigi Gurreri, Girolama Airò Farulla, Andrea Cipollina, Antonina Pirrotta, Giorgio Micale, and Michele Ciofalo.
2019. "Membrane Deformation and Its Effects on Flow and Mass Transfer in the Electromembrane Processes" *International Journal of Molecular Sciences* 20, no. 8: 1840.
https://doi.org/10.3390/ijms20081840