# Fluid–Structure Interaction and Flow Redistribution in Membrane-Bounded Channels

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

#### 1.1. Hydrodynamics in Undeformed Membrane-Bounded Systems

#### 1.2. Motivation and Strategy of the Present Study

- In the ideal case of a uniform (along the whole channel) transmembrane pressure (TMP), a relatively simple chain of effects and consequences would be triggered: (a) Membrane/channel deformation; (b) changes in the friction coefficients in both fluid compartments with respect to those in the undeformed configuration; and, finally, (c) consequent changes in the solutions flow rates for any given inlet-to-outlet imposed pressure drop. Such TMP effects in ED and RED processes were documented in our previous studies [22,23], which revealed a significant influence of membrane deformation on the hydrodynamics and mass transfer in channels equipped with different profiled membranes.
- In the realistic case of a non-uniform TMP, the above chain becomes a closed loop (a two-way fluid–structure interaction, FSI) made up of: (a) Space-dependent membrane/channel deformation; (b) space-dependent changes in the friction coefficients, f, in both fluid compartments; (c) consequent space-dependent changes in the flow rates for any given inlet-to-outlet imposed pressure drop; (d) space-dependent changes in the frictional pressure losses and, thus, in the pressure distribution in both channels; and, finally, (e) changes in the spatial distribution of TMP (leading us back to point a). Once an equilibrium configuration is attained by the system, the flow distribution will generally turn out to be uneven (redistribution), also affecting the concentrations, solute mass transfer rates, and electric current densities.

^{®}or Ansys

^{®}. However, 3-D simulations of an entire channel equipped with spacers or profiled membranes would require an inordinate computational effort. Therefore, a hybrid 3-D/2-D modelling approach was chosen, following a strategy similar to that proposed by Kostoglou and Karabelas for undeformed channels in pressure-driven membrane processes [10,11].

- At the small scale of the unit cell (periodic domain identified by the membrane profiles), fully three-dimensional structural mechanics simulations were conducted by using the Ansys-Mechanical
^{®}code in order to compute the deformation of membranes/channels under different values of the transmembrane pressure, TMP, as discussed in a previous paper [22]. - Still at the unit cell scale, fully three-dimensional CFD simulations were conducted for each deformed configuration using the Ansys-CFX
^{®}code; these simulations provided the relation between the flow rate and driving pressure gradient (hydraulic characteristic) as a function of the amount of deformation, and thus of the applied TMP, as discussed in the same paper [22]. - In the present study, focused on the larger cell pair scale, the above results were summarized in the form of a correlation between the apparent hydraulic permeability (itself a function of the flow rate) and the transmembrane pressure, as will be discussed in Section 2.2.
- Finally, the above information was fed to a 2-D simplified model of the cell pair as will be described in Section 2.4 and Section 2.5.

## 2. Materials and Methods

#### 2.1. From a Small- to Large-Scale Description of Membrane-Bounded Channels

^{2}for a square-planform unit cell). Note that in an undeformed channel, h is <H due to the volume occupied by the membrane profiles; in a compressed channel, h is always <H; in an expanded channel, h may be either <H or even >H. Values of h under different load conditions are reported in Table A1 of Appendix A.

_{inter,s}, along the same generic direction, s, is defined as the volume average of the s-component of the actual fluid velocity in the unit cell, and can be computed as U

_{s}H/h. It takes into account both the presence of membrane profiles and the possible deformation (compression or expansion) of the channel.

_{s}, and the hydraulic diameter of the void and undeformed plane channel, 2H:

_{s}:

_{s}and H were defined previously.

_{s}on the flow attack angle was systematically investigated both for profiled membrane configurations and for spacer-filled channels. The former results (see, for example, Figure 7 in [22]) showed that for OCF profiles with P/H = 8, up to Re ≈ 100, the equivalent friction coefficient varied in a negligible way with the flow attack angle, γ, thus exhibiting an isotropic behavior, and that no pressure gradient orthogonal to the main flow direction arose.

_{s}, was proportional to Re

^{−1}(i.e., the relation between the velocity and pressure drop was linear) only up to Re ≈ 10, indicating a regime of creeping flow characterized by the self-similarity of the flow field. At higher Re, the dependence of the pressure drop on the velocity became nonlinear due to inertial effects.

_{app}is a (scalar) apparent permeability, independent of the flow direction, s. At low Reynolds numbers (i.e., <10), K

_{app}attains a constant value, K, independent of the velocity, U

_{s}, and Equation (3) takes the form of the classical Darcy law. At higher Reynolds numbers, when inertial effects occur, the Darcy law is replaced by the Darcy–Forchheimer equation:

_{app}= K/(1 + ρKFU

_{s}/μ).

_{s}, by:

_{app}deduced from the friction coefficient using Equation (5) can then be employed in Equation (3) to estimate the superficial velocity associated with a given pressure gradient in the fluid domain. A straightforward solution of Equation (3) is not possible if the channel permeability varies in space as a function of the local transmembrane pressure and the closed-loop interactions mentioned in Section 1.2 occur. The following sections present a mathematical model based on an iterative algorithm dealing with this issue, thus computing the steady-state fluid distribution in the channels in the presence of significant membrane deformation.

#### 2.2. Computational Domain and Modelling Assumptions

_{app}, of the concentrate channel is reported in Figure 3 as a function of the superficial velocity, U

_{s}, for different TMP values. It can be observed that the assumption of Darcyan flow (i.e., of a linear dependence of the velocity on the pressure gradient, with K constant and independent of U

_{s}) is well satisfied only for velocities of the order of a few cm/s (Re ≈ 10 for H = 200 μm), which are common in RED but rather low in ED.

- (a)
- The flow field is steady.
- (b)
- The fluid properties are constant and are the same in both the DIL and CON channels (the values ρ = 997 kg/m
^{3}and μ = 8.89·10^{−4}Pa·s were used in the examples discussed in this paper). - (c)
- AEM and CEM membranes share the same mechanical properties and profiles geometry, so that the same correlation for the channel apparent permeability applies to both channels.
- (d)
- Transmembrane water transport (due to osmotic flow and electro-osmotic drag) is neglected. Therefore, the inlet flow rate coincides with the outlet flow rate. This assumption is justified by the fact that the transmembrane water flow rate is much less than the main water flow rate along the channels. For example, even in the unrealistically extreme case of ED with a large concentration gradient (seawater–freshwater), a high current density (100 A/m
^{2}), large channel length/thickness ratio (3000, e.g., L = 0.6 m, H = 200 μm), low superficial velocity (1 cm/s) in both channels, large membrane osmotic permeability (10 mL/(m^{2}h bar)), and hydration number of 7 (water molecules/ion), the total transmembrane water flow rate estimated by elementary balances is less than 8% of the axial flow rate of each solution.

- Flow rates exiting a computational block are assumed to be positive, while flow rates entering a block are assumed to be negative.
- The TMP is calculated as the difference between the local pressures in the DIL and CON compartments ($TM{P}_{i,j}={P}_{i,j}^{DIL}-{P}_{i,j}^{CONC}$), so that, as mentioned above, it is positive when DIL is expanded and CON compressed. By definition, if the CON compartment locally experiences a given value of TMP, at the corresponding location, the DIL compartment is subjected to −TMP.

^{®}environment.

#### 2.3. Discretized Continuity Equation

_{x}× L

_{y}, is divided into N

_{x}× N

_{y}rectangular blocks of size Δx = L

_{x}/N

_{x}, Δy = L

_{y}/N

_{y}. Each channel is treated as a continuous, porous, two-dimensional medium, and the block size, being an arbitrary computational construct, is unrelated to the size of the individual unit cell as defined by the membrane profiles.

#### 2.4. Discretized Darcy Equation for the Case of Low Velocity

_{s}, the apparent permeability in Figure 3 becomes a function of transmembrane pressure only and is shown in Figure 4.

^{2}and TMP is in kPa.

_{app}depends not only on TMP but also on U

_{s}, requires a slightly more complex treatment and will be separately discussed in Section 2.5.

^{−9}kPa). Once convergence is attained, superficial velocities are computed from Equations (8) and (9).

#### 2.5. Model Adjustment for Non-Darcyan Flow Regime

^{2}, the linear relation between the velocity and pressure gradient expressed by the Darcy law is no longer valid, i.e., K

_{app}depends on U

_{s}. For the sake of clarity, the relation between the velocity and pressure gradient obtained from 3-D CFD simulations [22] is reported in Figure 6 for the undeformed case; in the presence of deformation, a similar graph applies for each value of TMP. The blue solid line represents the U

_{s}vs. ∂p/∂s relation computed by CFD while the dotted red line represents the linear extrapolation of the behavior observed in the limit of low velocity. A significant departure from the linear behavior is observed for velocities larger than ~10 cm/s. Therefore, in order to investigate higher flow regimes without excessively modifying the proposed model, a piecewise linear fitting (represented in Figure 6 by the red line segments separated by symbols) was applied to capture the dependence of the apparent channel permeability on the velocity. Each linear segment is identified by its slope, K’/μ, and its intercept, a, as shown in Figure 6, with K’ = K and a = 0 in the first segment.

#### 2.6. Flow Arrangement and Boundary Conditions

_{in}, at the inlet boundaries. In the cross flow case, values of 3.12, 16.29, and 34.3 kPa were set for P

_{in}, yielding superficial velocities in the undeformed channels of 1, 5, and 10 cm/s, respectively. Only the value of P

_{in}= 34.3 kPa was investigated in the case of the counter flow configuration, where it yielded a superficial velocity in the undeformed channels of ~8.5 cm/s.

^{®}code. Details of these comparisons are reported in Appendix B.

## 3. Results and Discussion

#### 3.1. Cross Flow Arrangement

#### 3.1.1. Low Velocity Case (P_{in} − P_{out} = 3.12 kPa, Yielding U_{s} ≈ 1 cm/s)

_{s}= 1 cm/s, being s either x for the CON channel or y for the DIL channel) are shown in Figure 9 for the case in which membrane deformation effects are not taken into account. In each channel, pressure decreases linearly along the flow direction. The resulting TMP exhibits an anti-symmetric behavior about the descending diagonal of the channel, attaining its lowest value at the bottom left corner and the highest value at the top right corner. Velocity maps are not shown since a flat distribution exists.

_{x}), which is shown in map (e) for the concentrate channel. This quantity exhibits a stratification in the direction orthogonal to the main flow direction, with a mean value 〈U

_{s}〉 of ~0.976 cm/s (almost the same as in the undeformed case) and a variation of about ±3%.

_{x}, and exhibits a symmetric distribution about the channel’s descending diagonal.

_{s}, and channel equivalent height, h.

#### 3.1.2. Higher Velocity Case (P_{in} − P_{out} = 34.3 kPa, Yielding U_{s} ≈ 10 cm/s)

_{s}≈ 10 cm/s along the main flow direction.

_{s}, with a mean value 〈U

_{s}〉 of 9.82 cm/s (1.8% reduction with respect to the undeformed case) and a maximum variation of +39% in the lower region of the stack (close to y = 0.6 m) and −27% in the upper region (close to y = 0).

_{y}, in the concentrate channel). This quantity is now two to three orders of magnitude lower than the streamwise component, U

_{s}, and exhibits a more complex distribution than in the lower velocity case of Figure 10f.

_{s}= 1 cm/s case but presents much larger relative variations (about ±35%) and a much more non-uniform gradient.

_{s}≈ 10 cm/s in the presence of membrane deformation), a better understanding of the behavior of the different quantities in Figure 12 can be achieved by considering the profiles of different variables along the main flow direction (x for the concentrate channel). Figure 13 reports such profiles as obtained at two spanwise locations, namely, y = 0.05 m (upper region of the maps in Figure 12) and y = 0.55 m (lower region of the same maps).

_{s}, graph (e), can be calculated as the product of the pressure gradient by the channel’s apparent permeability. As a result, U

_{s}is lower at y = 0.05 m than at y = 0.55 m, since the pressure gradient is almost the same at the two locations, while the permeability is lower in the upper-compressed region of the channel than in the lower-expanded one.

_{s}H/h. This quantity increases along the flow direction x at both y locations because the superficial velocity, graph (e), varies much less than the channel equivalent height, graph (b). Moreover, the interstitial velocity is lower at y = 0.05 m (upper, compressed region of the channel) than at y = 0.55 m (lower, expanded region) because the superficial velocity varies along y more than the equivalent channel height.

#### 3.1.3. Pressure Profiles for All Cross Flow Cases

_{in}− P

_{out}) of 3.12, 16.29, and 34.3 kPa, respectively, yielding 〈U

_{s}〉 values of approximately 1, 5, and 10 cm/s. Both the undeformed and the deformed conditions are considered.

_{s}〉 under deformed conditions is small (1.8% for P

_{in}− P

_{out}= 34.3 kPa, as mentioned in Section 3.1.2, and practically negligible for lower pressure drops).

_{s}〉 ≈ 1 cm/s, membrane deformation is small, as already discussed in Section 3.1.1, and does not significantly affect the pressure profile. As the velocity, and thus the TMP, increases, pressure profiles depart from the linear trend. Cubic polynomial functions are found to fit the pressure profiles well when membrane deformation becomes considerable.

#### 3.2. Counter Flow Arrangement

_{s}≈ 8.5 cm/s (with s corresponding to x). Note that pressure changes due to abrupt area variations, present in this configuration, are not taken into account by the model, but this is not a severe limitation because, at the present low velocity, these terms (proportional to the kinetic pressure. $\rho {U}_{s}^{2}$/2) are negligible compared to the distributed losses (order of 0.01 kPa, compared to the imposed inlet–outlet pressure variation of 34.3 kPa). This conclusion is also supported by the comparison with the CFD results in Appendix B.2.

_{s}〉 is now ~7.5 cm/s, ~12% lower than that computed in the undeformed case (~8.5 cm/s). This result can be attributed to the asymmetric behavior of the hydraulic permeability, which varies less under compression than under expansion.

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Dependence of Equivalent Channel Height and Channel Hydraulic Permeability on TMP

#### Appendix A.1. Equivalent Channel Height

^{2}= 2.56 mm

^{2}) for TMP varying between −40 and +40 kPa.

^{®}program (h is in μm, TMP in kPa):

TMP [kPa] | V [mm^{3}] | h [μm] |
---|---|---|

40 | 0.349 | 136 |

30 | 0.376 | 147 |

20 | 0.404 | 158 |

10 | 0.432 | 169 |

0 | 0.462 | 180 |

−10 | 0.491 | 192 |

−20 | 0.521 | 203 |

−30 | 0.550 | 215 |

−40 | 0.579 | 226 |

#### Appendix A.2. Channel Permeability for Non-Darcyan Flow

0 < U < 3 cm/s | 3 < U < 7 cm/s | |||
---|---|---|---|---|

TMP [kPa] | K [m^{2}] | a [cm/s] | K′ [m^{2}] | a [cm/s] |

40 | 6.754 · 10^{−10} | 0 | 6.289 · 10^{−10} | 0.20677 |

30 | 8.651 · 10^{−10} | 0 | 8.052 · 10^{−10} | 0.20784 |

20 | 1.088 · 10^{−9} | 0 | 1.012 · 10^{−9} | 0.20885 |

10 | 1.356 · 10^{−9} | 0 | 1.261 · 10^{−9} | 0.20938 |

0 | 1.670 · 10^{−9} | 0 | 1.553 · 10^{−9} | 0.20934 |

−10 | 2.034 · 10^{−9} | 0 | 1.891 · 10^{−9} | 0.20973 |

−20 | 2.439 · 10^{−9} | 0 | 2.271 · 10^{−9} | 0.20753 |

−30 | 2.893 · 10^{−9} | 0 | 2.693 · 10^{−9} | 0.20727 |

−40 | 3.397 · 10^{−9} | 0 | 3.163 · 10^{−9} | 0.20678 |

7 < U < 12 cm/s | 12 < U < 20 cm/s | |||
---|---|---|---|---|

TMP [kPa] | K′ [m^{2}] | a [cm/s] | K′ [m^{2}] | a [cm/s] |

40 | 5.777 · 10^{−10} | 0.7598 | 5.169 · 10^{−10} | 1.9422 |

30 | 7.393 · 10^{−10} | 0.7634 | 6.612 · 10^{−10} | 1.9506 |

20 | 9.287 · 10^{−10} | 0.7668 | 8.302 · 10^{−10} | 1.9584 |

10 | 1.158 · 10^{−9} | 0.7686 | 1.034 · 10^{−9} | 1.9625 |

0 | 1.425 · 10^{−9} | 0.7684 | 1.274 · 10^{−9} | 1.9622 |

−10 | 1.735 · 10^{−9} | 0.7697 | 1.551 · 10^{−9} | 1.9651 |

−20 | 2.085 · 10^{−9} | 0.7623 | 1.865 · 10^{−9} | 1.9483 |

−30 | 2.473 · 10^{−9} | 0.7615 | 2.213 · 10^{−9} | 1.9462 |

−40 | 2.905 · 10^{−9} | 0.7598 | 2.600 · 10^{−9} | 1.9425 |

## Appendix B. Grid Dependence and Validation Against CFD Results

#### Appendix B.1. Grid Dependence

_{x}and N

_{y}) on the model’s results was systematically addressed by comparing five grids characterized by: (A) 30 × 30; (B) 60 × 60; (C) 120 × 120; (D) 240 × 240; and (E) 480 × 480 blocks, with a total number of 900, 3600, 14,400, 57,600, and 230,400 blocks, respectively.

_{in}= 34.3 kPa, Figure A1 reports profiles of the x-velocity component along a line orthogonal to the main flow direction and located 2 cm away from the inlet side; see the left broken line in Figure 8a. The inset shows an enlarged portion of the same plot. It can be observed that maximum differences (grids A vs. E) are less than 0.25 cm/s, and grids C–E yield practically identical results. The discrepancies between consecutive grids decrease with the number of blocks, indicating an asymptotic convergence.

**Figure A1.**Cross flow configuration at P

_{in}= 34.3 kPa: profiles of the x-velocity component along a line orthogonal to the main flow direction and located 2 cm away from the inlet, see Figure 8a. Results obtained with five grids of increasing resolution are reported. The inset shows a magnified portion of the same plot.

**Figure A2.**Counter flow configuration at P

_{in}= 34.3 kPa: pressure profiles along a line orthogonal to the main flow direction and located 2 cm upstream of the outlet, see Figure 8b. Results obtained with five grids of increasing resolution are reported.

#### Appendix B.2. Comparison with CFD Results

^{®}18.1 was employed. Both cross flow and countercurrent configurations were simulated. The CFD settings were as follows:

- The porous media model was assumed, and steady state simulations were performed.
- Values of the permeability and of the resistance loss coefficient were determined by means of a quadratic regression of the undeformed channel characteristics (Figure 6). The permeability was set to 1.65·10
^{−9}m^{2}and the resistance loss coefficient to 989 m^{−1}. - Free slip wall boundary conditions were set at the upper and lower walls of the channel (representative of membrane surfaces). The latter condition was imposed to avoid viscous fluid–wall interaction, which would lead to erroneous results; in fact, the friction characteristics of the membrane surfaces and profiles are already taken into account by the permeability and resistance coefficients.
- Symmetry or no slip boundary conditions were imposed at the lateral edges of the domain.

_{in}= 18.64 kPa. Only the three finest grids, C, D, and E, were considered. It can be observed that the CFD predictions are practically coincident with the results provided by the finest grid (E, 480 × 480 blocks) but agree fairly well also with those obtained by coarser (120 × 120 or 240 × 240) grids. A similar agreement was also obtained for other locations, the other configuration (i.e., cross flow), and various flow velocities, thus confirming the reliability of the present model.

**Figure A3.**Single undeformed channel in the counter flow configuration at P

_{in}= 18.64 kPa: pressure profiles along a line orthogonal to the main flow direction and located 2 cm away from the inlet, see Figure 8b. Results obtained with three grids of increasing resolution are compared with CFD predictions.

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**Figure 1.**A channel delimited by profiled membranes of the overlapped crossed filaments (OCFs) type. A repetitive periodic unit (“unit cell”) is shown, enlarged, in the inset. The geometric parameters H (channel thickness), P (pitch), and γ (flow attack angle) are indicated. Adapted from [22].

**Figure 2.**Sketch of a cell pair with diluate (DIL) and concentrate (CON) channels. Membranes are shown only for the sake of clarity but are not part of the computational domain. The bottom row reports a representation of the computational “molecules” adopted for the formulation of discrete balance equations in both channels.

**Figure 3.**Apparent permeability, K

_{app}, of the concentrate channel as a function of the equivalent velocity, ${U}_{s}$, at different values of the transmembrane pressure for OCF-profiled membranes with P/H = 8, computed by 3-D mechanical-CFD simulations [22].

**Figure 4.**Permeability, K, at U

_{s}→ 0 as a function of transmembrane pressure for OCF-profiled membranes with P/H = 8, computed by 3-D mechanical CFD simulations [22].

**Figure 6.**Dependence of the superficial velocity on the pressure gradient for the undeformed channel case (solid blue line), computed by 3-D CFD simulations [22]. The dotted red line represents the linear relation valid at low velocities. The straight red segments separated by symbols represent a piecewise linear fitting. As an example, the intercept, a, and the slope, K’/μ, of the fourth segment are indicated.

**Figure 8.**The flow arrangements analyzed in the present work. (

**a**) cross flow; (

**b**) counter flow. The insets in the top row show the actual geometry of inlets and outlets. The second and third rows report the approximated geometry simulated here that apply to the two solutions (e.g., concentrate and diluate). The broken lines in the middle graphs are those along which the velocity and pressure profiles obtained with different grids were compared (Appendix B).

**Figure 9.**Results for a cross flow stack of 0.6 m sides at P

_{in}− P

_{out}= 3.12 kPa (U

_{s}= 1 cm/s) in the absence of membrane deformation: relative pressure maps for the CON (

**a**) and DIL (

**b**) channels, and TMP (

**c**).

**Figure 10.**Results for a cross flow stack of 0.6 m sides at P

_{in}− P

_{out}= 3.12 kPa (U

_{s}≈ 1 cm/s) in the presence of membrane deformation: relative pressure maps for the CON (

**a**) and DIL (

**b**) channels, TMP (

**c**), and concentrate channel equivalent height (

**d**), superficial velocity components along the x (

**e**) and y (

**f**) directions, and interstitial velocity along the x direction (

**g**).

**Figure 11.**Results for a cross flow stack of 0.6 m sides at P

_{in}− P

_{out}= 34.3 kPa (U

_{s}= 10 cm/s) in the absence of membrane deformation: relative pressure maps for the CON (

**a**) and DIL (

**b**) channels, and TMP (

**c**).

**Figure 12.**Results for a cross flow stack of 0.6 m sides at P

_{in}− P

_{out}= 34.3 kPa (U

_{s}≈ 10 cm/s) in the presence of membrane deformation: relative pressure maps for the CON (

**a**) and DIL (

**b**) channels, TMP (

**c**), and concentrate channel equivalent height (

**d**), superficial velocity components along the x (

**e**) and y (

**f**) directions, and interstitial velocity along the x direction (

**g**).

**Figure 13.**Results for a cross flow stack of 0.6 m sides at P

_{in}− P

_{out}= 34.3 kPa (U

_{s}≈ 10 cm/s) in the presence of membrane deformation: profiles of TMP (

**a**), and concentrate channel equivalent height (

**b**), apparent permeability (

**c**), pressure gradient (

**d**), superficial velocity (

**e**) and interstitial velocity (

**f**) along the x direction at y = 0.05 and 0.55 m. Corresponding results would be obtained for the diluate channel.

**Figure 14.**Pressure profiles along the x-direction near the lower edge (y = 0.55 m) of the concentrate channel for the cross flow configuration and 〈U

_{s}〉 ≈ 1, 5, and 10 cm/s. Profiles are reported both in the presence and absence of membrane deformation.

**Figure 15.**Results for a cross flow stack of 0.6-m sides at P

_{in}− P

_{out}= 34.3 kPa (〈U

_{s}〉 = 8.5 cm/s) in the absence of membrane deformation: maps of relative pressure in the CON (

**a**) and DIL (

**b**) channels, TMP (

**c**), and superficial velocity along the x direction in the concentrate channel (

**d**).

**Figure 16.**Results for a cross flow stack of 0.6-m sides at 〈U

_{s}〉 = 7.5 cm/s in the presence of membrane deformation: maps of relative pressure in the CON (

**a**) and DIL (

**b**) channels; TMP (

**c**); and channel equivalent height (

**d**), superficial velocity (

**e**), and interstitial velocity (

**f**) along the x direction in the concentrate channel.

© 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.; Cipollina, A.; Pirrotta, A.; Velizarov, S.; Ciofalo, M.; Micale, G.
Fluid–Structure Interaction and Flow Redistribution in Membrane-Bounded Channels. *Energies* **2019**, *12*, 4259.
https://doi.org/10.3390/en12224259

**AMA Style**

Battaglia G, Gurreri L, Cipollina A, Pirrotta A, Velizarov S, Ciofalo M, Micale G.
Fluid–Structure Interaction and Flow Redistribution in Membrane-Bounded Channels. *Energies*. 2019; 12(22):4259.
https://doi.org/10.3390/en12224259

**Chicago/Turabian Style**

Battaglia, Giuseppe, Luigi Gurreri, Andrea Cipollina, Antonina Pirrotta, Svetlozar Velizarov, Michele Ciofalo, and Giorgio Micale.
2019. "Fluid–Structure Interaction and Flow Redistribution in Membrane-Bounded Channels" *Energies* 12, no. 22: 4259.
https://doi.org/10.3390/en12224259