# Pressure-Induced Deformation of Pillar-Type Profiled Membranes and Its Effects on Flow and Mass Transfer

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

^{2}µm) and have a low stiffness (Young modulus 10

^{1}–10

^{3}MPa), they may undergo significant deformations if even small transmembrane pressures (TMP) of the order of 10

^{−1}bar are applied. Such values of TMP commonly arise in real stacks due to different frictional pressure drops in the two channels or to other causes, depending also on the flow arrangement selected (e.g., parallel flow yields the lowest TMP and counter flow the highest). In fact, a review of the literature on different membrane-based processes shows that the effects of membrane/channel deformation and their relevance have been detected in the context of different processes: forward osmosis and pressure assisted osmosis [6], pressure retarded osmosis [7], reverse osmosis [8], liquid-to-air membrane energy exchanger [9], proton exchange membrane fuel cells [10,11], and membrane microcontactors [12]. Time-dependent membrane deformation has recently been considered as a possible means to improve process performances in the “breathing cell” concept for reverse electrodialysis systems [13]. Nevertheless, experiments, models, and simulations concerning fluid–structure interactions and its effects are still few, especially for electro-membrane processes such as ED and RED.

## 2. Materials and Methods

#### 2.1. Computational Strategy

_{max}causes the first contact between opposite membranes. The value of 0.8 bar was chosen for (TMP)

_{max}as conservatively larger than the highest TMP realistically expected in real RED-ED applications. Note that actual electro-membrane systems will operate at lower trans-membrane pressures (0.05–0.2 bar), since, at a higher TMP, leakage and partial mixing of the solutions may occur.

#### 2.2. Mechanical Model

- The four external vertical edges of the domain were clamped, i.e., zero displacement and zero rotation were imposed.
- The four side faces of the domain were imposed zero normal displacement so that the computational domain was representative of a periodic array of repetitive units.
- The transmembrane pressure relative to that of the internal fluid channel (and thus >0 for compression and <0 for expansion) was applied to the whole outer surface of the domain.

^{®}. A detail of the FE mesh, which was completely hexahedral and block-structured, is shown in Figure 2.

^{4}elements (RP-II) was chosen as a compromise between accuracy and computational effort. The same element size was adopted in all following simulations.

^{2}samples of real ion exchange membranes, and with analytical solutions presented by Iyengar and Naqvi [32] for the deformation of a square elastic body loaded with a uniform pressure and having all the edges clamped. A good agreement was demonstrated, with discrepancies of only a few percent in the maximum displacement.

#### 2.3. CFD Model

_{i}is the ith velocity component, $\tilde{p}$ is the periodic component of pressure, K

_{p}= |dp/ds| is the pressure loss per unit length, $\tilde{c}$ is the periodic component of concentration, K

_{c}is the bulk concentration variation per unit length, s

_{i}is the ith component of the unit vector

**s**directed along the main flow direction, and u

_{s}=

**u**·

**s**is the local velocity component along the same direction.

^{3}(i.e., seawater), are reported in Table 3.

_{p}in Equation (2) was imposed in the simulations, while U and Re were computed as parts of the solution. Note that K

_{p}is related to the friction velocity Reynolds number:

_{τ}is the friction velocity:

_{b}is the molar bulk concentration and c

_{w}is the local molar concentration at the membrane surface. In the present simulations, the electrolyte flux is supposed to be into the fluid (i.e., the channel represents either a concentrate channel in ED or a dilute channel in RED); therefore, the average polarization coefficient is lower than 1 [5,33]. Finally, the Sherwood number was defined as

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

_{w}〉 is the area average of c

_{w}on the same membrane. In the present geometry, the distributions of wall concentration and, therefore, the Sherwood numbers, are practically identical on the upper and lower walls of a channel.

**u**, p, and c between opposite inlet–outlet boundaries. No slip conditions were assumed for velocity at the membrane surfaces. A uniform value of 2.6 × 10

^{−4}mol/(m

^{2}s) was imposed for the molar salt flux j entering the fluid, corresponding to an electrical current density of 50 A/m

^{2}. The molar flux was imposed to be null at the cylindrical side surfaces of the round pillars; mass transfer results do not significantly change if these surfaces are imposed the same molar flux as in the rest of the membranes. Note that the values adopted for bulk concentration, solution properties, and salt flux affect the values computed for the polarization coefficient (Equation (9)) while the Sherwood numbers depend only on geometry, Re, and Sc.

**s**of the applied pressure gradient K

_{p}.

_{τ}≈ 5, corresponding to a bulk Reynolds number of ~30, well above the creeping flow range and close to the highest Reynolds numbers investigated. The results are shown in Table 4, where the computed values of f and Sh are reported as functions of the number of finite volumes.

^{®}.

## 3. Results and Discussion

#### 3.1. Mechanical Simulations

#### 3.1.1. Influence of Pitch to Height ratio (P/H) and Limiting Value

#### 3.1.2. Membrane and Channel Deformation for the Selected Geometry (P/H = 10)

#### 3.2. CFD Simulations for the Selected Geometry (P/H = 10)

#### 3.2.1. Local Quantities

_{b}/c

_{w}(bottom row), it should be observed that the distributions of all quantities on the upper and lower walls are identical due to symmetry reasons and to the boundary conditions adopted. The comparison of the θ maps relevant to the three angles investigated shows that the case of γ = 0° provides a fairly uniform distribution of the wall salt concentration while, for γ = 90°, the wall concentration is strongly nonuniform in the lateral (spanwise) direction and becomes lower than the bulk concentration in two curved regions, symmetrically located about the flow direction, despite the net overall salt flux being into the channel. The case of γ = 30° exhibits a nonuniform distribution, with one S-shaped central region characterized by large values of θ. The significant nonuniformity of the polarization coefficient maps exhibited by the cases with flow attack angles of γ = 30° and γ = 90° are associated to inhomogeneous distributions of the fluid velocity. The occurrence of marked stagnant regions and restricted zones with a high fluid velocity results in uneven θ distributions.

_{τ}= 2.5 and γ = 90° (flow parallel to the shorter diagonal of the unit rhombus) are shown, as evidenced in the inset. Only the undeformed (middle column, Re = 5.1), the most compressed (TMP = +0.4 bar, left column, Re = 2.4), and the most expanded (TMP = −0.4 bar, right column, Re = 8.9) configurations are examined.

_{b}/c

_{w}on either of the walls of the fluid-filled channel (see the above discussion on the symmetry between the upper and lower walls). The corresponding values of the normalized Darcy friction coefficient (F ratio) and of the Sherwood number are also indicated.

#### 3.2.2. Global Quantities

_{τ}are aligned along a curved row since they correspond to different values of Re.

#### 3.2.3. Comparison with OCF Profiled Membranes

_{τ}= 2.5. The definition of γ for the two geometries is reported in Figure 1 of the present work and of Reference [20]. Note that, for each configuration, the P/H ratio considered is the largest admissible, i.e., withstanding the TMP value of 0.8 bar without collapsing.

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Round pillar (RP) profiled membranes: The repetitive unit (periodic cell) of a stack is highlighted on the left and is shown enlarged in the central inset. The geometric parameters H (channel thickness), P (pitch), α (intrinsic angle), and γ (flow attack angle) are indicated.

**Figure 2.**Computational domain: A detailed image of the Finite Element computational mesh is also shown.

**Figure 4.**Deformations for different P/H ratios under TMP = +0.8 bar. The quantity shown is the displacement in the direction orthogonal to the undeformed membranes. Top: external view; bottom: view after sectioning by a diagonal plane AA.

**Figure 5.**Deformations for P/H = 10 in both 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. The corresponding deformed configuration of the fluid volume is shown in the insets on the right.

**Figure 6.**Volume (normalized by the undeformed value) as a function of the transmembrane pressure for P/H=10.

**Figure 7.**Illustrating the flow direction (top row), 3-D streamlines (middle row), and maps of the concentration polarization coefficient θ = c

_{b}/c

_{w}on either of the walls (bottom row) for the P/H = 10 undeformed configuration and a friction velocity Reynolds number of 2.5 (Re = 5.1). c

_{b}= 500 mol/m

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

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

**Figure 8.**Influence of deformation on flow and mass transfer for P/H = 10, γ = 90°, and Re

_{τ}= 2.5. Left column: compressed (TMP = +0.4 bar, Re = 2.4); middle column: undeformed (Re = 5.1); right column: expanded (TMP = −0.4 bar, Re = 8.9). Top row: distribution of the streamwise velocity component in the central cross section of the channel (for symmetry reasons, only half map is shown); bottom row: distribution of the polarization coefficient on either of the walls. 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). The F ratio and Sherwood number are also reported.

**Figure 9.**Darcy friction coefficients (F ratio) for P/H = 10 as a function of the Reynolds number for different values of transmembrane pressure and three values of flow attack angle γ. (

**a**) γ = 0°; (

**b**) γ = 30°; (

**c**) γ = 90°.

**Figure 10.**Sherwood numbers on either of the walls as a function of the Reynolds number for P/H = 10 at different transmembrane pressures and flow attack angles. (

**a**) γ = 0°; (

**b**) γ = 30°; (

**c**) γ = 90°.

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

Membrane Young modulus, E | 150 | MPa |

Membrane Poisson ratio, ν | 0.4 | - |

Channel thickness, H | 200 | μm |

Membrane thickness, δ | 120 | μm |

Pitch-to-height ratio, P/H | 9–11 | - |

Pillar diameter, d | 1 | mm |

Intrinsic angle of pillar lattice, α | 60 | deg |

**Table 2.**Grid dependence results for the mechanical simulations (transmembrane pressure (TMP) = +0.8 bar, P/H = 10).

FE Mesh | No. Elements | Maximum Displacement (μm) |
---|---|---|

RP-I | 21 × 10^{3} | 67.69 |

RP-II | 60 × 10^{3} | 67.97 |

RP-III | 180 × 10^{3} | 68.06 |

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, Sc = µ/(ρD) | 622 | - |

FV Mesh | No. Elements | f | Sh |
---|---|---|---|

RP-A | 2.576 × 10^{6} | 6.941 | 13.769 |

RP-B | 4.142 × 10^{6} | 6.965 | 13.719 |

RP-C | 6.091 × 10^{6} | 6.976 | 13.689 |

**Table 5.**Approach velocity, Darcy friction factor, and mass transfer coefficients for the undeformed, +0.4 bar compressed and −0.4 bar expanded configurations for an RP with P/H = 10 and OCF with P/H = 8 [20] geometries with γ = 90° and Re

_{τ}= 2.5.

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

RP | OCF | RP | OCF | RP | OCF | |

U (cm/s) | 0.5 | 0.4 | 1.1 | 1.0 | 2.0 | 2.0 |

F | 129.0 | 260.5 | 28.9 | 43.2 | 9.5 | 10.3 |

k (m/s), upper wall | 3.2 × 10^{−5} | 2.6 × 10^{−5} | 1.9 × 10^{−5} | 1.9 × 10^{−5} | 1.9 × 10^{−5} | 2.2 × 10^{−5} |

k (m/s), lower wall | 3.2 × 10^{−5} | 2.6 × 10^{−5} | 1.9 × 10^{−5} | 1.4 × 10^{−5} | 1.9 × 10^{−5} | 1.1 × 10^{−5} |

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## Share and Cite

**MDPI and ACS Style**

Battaglia, G.; Gurreri, L.; Airò Farulla, G.; Cipollina, A.; Pirrotta, A.; Micale, G.; Ciofalo, M.
Pressure-Induced Deformation of Pillar-Type Profiled Membranes and Its Effects on Flow and Mass Transfer. *Computation* **2019**, *7*, 32.
https://doi.org/10.3390/computation7020032

**AMA Style**

Battaglia G, Gurreri L, Airò Farulla G, Cipollina A, Pirrotta A, Micale G, Ciofalo M.
Pressure-Induced Deformation of Pillar-Type Profiled Membranes and Its Effects on Flow and Mass Transfer. *Computation*. 2019; 7(2):32.
https://doi.org/10.3390/computation7020032

**Chicago/Turabian Style**

Battaglia, Giuseppe, Luigi Gurreri, Girolama Airò Farulla, Andrea Cipollina, Antonina Pirrotta, Giorgio Micale, and Michele Ciofalo.
2019. "Pressure-Induced Deformation of Pillar-Type Profiled Membranes and Its Effects on Flow and Mass Transfer" *Computation* 7, no. 2: 32.
https://doi.org/10.3390/computation7020032