1. Introduction
Processes based on ion exchange membranes are increasingly being adopted in industrial applications, from water treatment [
1] to food processing [
2] and energy harvesting [
3], as both environmentally friendly and economically attractive. In electrodialysis (ED) [
4], ions are driven by an imposed electric field from a dilute electrolyte solution to a concentrate one. Conversely, reverse electrodialysis (RED) [
5] harvests electrical energy from the controlled mixing of two solutions at different salt concentration. ED and RED units are built by alternately stacking anion- and cation- exchange membranes, separated by net spacers or built-in profiles creating the fluid channels where the two solutions (concentrate and diluate) flow. The two membranes and the two solutions form the repeating unit, referred to as cell pair. Spacers cover part of the membrane surface, thus reducing the actual active area, and increase the electrical resistance, as they are electrically non-conductive.
Profiled membranes have recently been presented as an innovative solution to overcome net spacers drawbacks [
4,
6,
7]. Profiled membranes simplify the stack assembly avoiding the use of spacers, and may improve the process performance. Numerical simulations [
8,
9,
10] and experimental lab scale tests [
4,
11,
12,
13,
14,
15] have confirmed their potential benefits. However, the actual performance of profiled membranes stacks depends on the specific profile geometry. Simple geometries (e.g., pillar or ridges profiles) are characterised by reduced hydraulic friction, but may exhibit lesser mixing properties than spacers [
10,
12,
13,
14,
16]; on the other hand, improved profile shapes may provide better trade-off solutions among pressure drops, mixing and Ohmic resistance, thus improving the stack performance [
8,
15,
16].
In membrane-based processes, a trans-membrane pressure (TMP) between the different solutions flowing through a module may be a design feature or may arise for various reasons (e.g., flow arrangement or differences in geometry, flow rate or physical properties). This may lead to local deformations of membranes and membrane-bounded channels. As a result, the channel geometry (shape and average size) may be modified with respect to the nominal one, affecting fluid dynamics and transport mechanisms (of mass, heat, ions) and, thus, the process performance.
The effects of membrane/channel deformation have been studied in the context of different processes. She et al. [
17] tested pressure retarded osmosis (PRO) modules at pressures up to 16 bar. Experimental performance became worse than theoretical predictions as the hydrostatic pressure increased; this difference was attributed to a more severe membrane deformation at high pressures. Later, She et al. [
18] studied in detail the influence of spacer geometry on PRO efficiency under pressure loads up to 20 bar. The spacer with the largest mesh pitch gave the poorest performance in terms both of power density and of pressure drop.
Karabelas et al. [
19] investigated the influence of the compressive stresses that arise in reverse osmosis (RO) spiral wound membrane modules, provided with spacers, during the assembly stage. The stresses localized at the membrane-spacer contact regions were systematically addressed as functions of spacer compaction, channel gap, membrane indentations and pressure drop. Interestingly, mild applied pressures (1–2 bar) were sufficient to cause significant effects. Correlations for the frictional losses were obtained for various applied pressures and were implemented into a process model predicting the performance of RO units.
Huang [
20] simulated flow and heat transfer in deformed channels for liquid-to-air membrane energy exchanger (LAMEE) units. Membrane deformation was not actually computed, and the deformed membrane was modelled as a spherical surface. As membrane deformation increased, the friction coefficient was found to increase in the compressed (air) channel and to decrease in the expanded (liquid) channel. Heat transfer was affected by deformation in a complex way.
The influence of channel deformation on the performance of proton exchange membrane fuel cells (PEMFC) was assessed in several studies following similar approaches. Shi and Wang [
21] predicted the compression of the porous gas diffusion layer due to the clamping (assembly) force, and simulated fluid dynamics, mass transport and electrochemical phenomena in the deformed geometries. The authors considered a serpentine channel and found that the assembly compression of the units enhanced pressure drop in the fluid channels, and that the process performance was particularly affected by deformation at high current densities. Zhou et al. [
22] simulated a unit with a single straight channel including the membrane. As expected, most of the deformation was found to occur in the porous gas diffusion layer due to its lower mechanical stiffness. The spatial distributions of porosity and permeability were computed and the effects of assembly pressure, gas diffusion layer thickness and membrane features were assessed.
Hereijgers et al. [
23] measured membrane deflection and mass transfer coefficients in membrane microcontactors using round and diamond-shaped pillar spacers of different pitch. They found that trans-membrane pressure exhibited a minimum as the spacer pitch was made to vary, and that membrane deflection had a positive or negative impact on mass transfer depending on the diffusion coefficients in the two immiscible phases.
Time-dependent membrane deformation has recently been considered as a possible means to improve process performance. Moreno et al. [
24] introduced the concept of “breathing cell” for reverse electrodialysis systems. In the breathing cell, the channels thickness changes dynamically due to the intermittent (5–15 cycles per minute) closure of an outlet valve in the concentrate channels. As a result, the Ohmic resistance of the diluate compartment (which is the predominant one) decreases. Some effects on concentration polarization are also expected. This cyclic operation was shown to yield higher net power densities in a range of flow rates.
Some ED/RED practical applications are poorly affected by these issues (TMP ≈ 0). However, in prototype and industrial size stacks with non-parallel flow layouts (cross flow, counter flow) and/or with asymmetric channels (different geometries, fluid properties, flow rates), where the pressure distribution in the two compartments is different, appreciable values of TMP may arise. In particular, when some factors enhancing pressure drop are present, TMP values amounting to some tenths of a bar can be exhibited (higher TMP levels can cause severe risks of leakages [
25,
26,
27]).
For example, in the cross-flow RED prototype units (44 × 44 cm
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.
ED units operate with fluid velocities higher than those typical of RED (in order to increase the limiting current density) and, despite the usually higher channel thickness, exhibit large pressure drops [
1]. For example, Wright et al. [
30] performed ED tests in a bench-scale unit and in a commercial-scale unit with parallel flow, measuring pressure drops up to ~0.65 bar and ~1.30 bar, respectively, at fluid velocities up to ~9 cm/s. If such operating conditions were adopted in non-parallel flow arrangements, they would lead to significant levels of
TMP.
Recent studies showed that asymmetric channels are optimal for RED applications [
31,
32]. However, they can be affected by
TMP-related issues. For example, in ref [
32] it was shown that for the couple of NaCl solutions 15–500 mol/m
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.
It must also be added that ion exchange membranes may have very different mechanical features. The Young modulus (
E) may vary within a broad range from 10 MPa to 1 GPa [
33,
34,
35,
36,
37,
38,
39,
40,
41,
42,
43] or even to higher values in some cases [
44,
45], but decreases with ageing due to membrane usage [
34,
35,
36,
44]. Moreover, the new generation membranes are manufactured with low thickness, e.g., from 80 to 250 µm [
46,
47]; even lower values can be found among commercial membranes and experimental membranes prepared in laboratory [
48]. A theoretical study [
49] has recently found optimal thicknesses of 15–20 and 50–70 µm for ED and RED applications, respectively. Therefore, it is quite common that ion exchange membranes exhibit a low stiffness, due to the combined effects of a low
E and a low thickness. This feature makes the membranes susceptible to large deformations in stacks with a non-negligible TMP, depending also on the spacer features.
In particular, a fluid-membrane mechanical interaction will be triggered, which will find an equilibrium state characterized by some distribution of pressure, geometry, flow rate, hydraulic friction, mass transfer coefficient, current density, Ohmic and non-Ohmic resistances in both compartments. Compared to the nominal conditions, the values of any of the above quantities under deformed conditions may be: (i) either higher or lower in the whole channel (e.g., in asymmetric configurations); (ii) higher in some parts of the channel and lower in other ones (e.g., in non-parallel flow arrangements). In both cases, these deviations from the undeformed conditions may impair the process performance due to the lack of compensation of effects between compressed zones and expanded zones (in the same or in different channels). For instance, an increase in the thickness of the diluate (which often provides the predominant resistance), in the whole channel or in a part of it, especially where the solution is less conductive, causes an increase in the average Ohmic resistance. Imbalances may also affect hydraulic friction, increasing the overall pressure drop. An increment in non-Ohmic resistance is another well-known detrimental effect of uneven flow rate distributions [
13].
All the aspects of practical interest examined in this section have provided the motivation to the present work. In particular, this paper goes inside the unexplored field of the TMP effects, taking a first step concerning mechanical response (deformation), flow and mass transfer characteristics at the local scale of a periodic unit. For this purpose, simulation tools implementing well-established and validated physical models and numerical methods were developed. Profiled membranes of the Overlapped Crossed Filaments (OCF) type were simulated. They are made by an array of semi-cylinders on both membrane sides, placed at 90° each other, as shown in
Figure 1.
4. Conclusions
Integrated mechanical and fluid dynamics simulations were performed for profiled membranes of the OCF (“overlapped crossed filaments”) type. The membranes were treated as linearly elastic, homogeneous and isotropic, and values of the Young modulus and of the Poisson ratio representative of ion exchange membranes’ features (E = 150 MPa, ν = 0.4) were adopted.
Under these assumptions, the largest value of the pitch to height ratio withstanding a trans-membrane pressure of 0.8 bar without collapsing (i.e., without exhibiting a contact between opposite membranes) was found to be P/H = 8.
The influence of TMP (which was investigated here in the range −0.4–+0.4 bar) is an increase of friction under compression conditions and a reduction of it (although to a lesser extent) under expansion conditions. This imbalance of effects may produce an increase of total pressure drop in the stack. The influence of the flow attack angle is negligible, indicating a substantial hydrodynamic isotropy of the profiled membrane lattice at low Reynolds numbers.
The influence of TMP on the Sherwood number is more complex. On the whole, compression enhances mass transfer and expansion reduces it; the influence of channel deformation on mass transfer is less marked than on friction. Some anomalous behaviour of Sh is observed in the cases characterized by γ = 90° and Re > ~50, in which the highest Sh is obtained for the largest expansion.
This study shows that TMP values of practical interest for ED/RED units can produce significant effects on deformation, flow and concentration fields (and, thus, on hydraulic friction and mass transfer coefficient). In general, other important quantities, e.g., Ohmic resistance, non-Ohmic resistance and limiting current density, will also be affected.
As far as membrane manufacturers are concerned, the main suggestion arising from this study is probably that stiffer membranes should be preferred in order to reduce the undesirable effects of membrane deformation, which result in an impairment of the ED/RED equipment performance. On the other hand, the current trend is towards the reduction of membrane thickness, mainly sought in order to reduce Ohmic losses. Therefore, producing membranes with the highest possible Young modulus (provided the membrane’s electrochemical behaviour is not impaired) will probably become a priority.
At a larger scale, combinations of geometric/mechanical features and operating conditions expose ED/RED stacks to the risk of severe local deformations, providing a new set-up, different from the nominal undeformed one, where all parameters are distributed as a result of a fluid-membrane mechanical interaction. Therefore, the actual process performance may be heavily affected by an imbalance of effects between compressed and expanded zones. This implies that the design and the performance prediction of devices under realistic operating conditions should take into account the membranes’ mechanical features.
From this perspective, correlations describing (a) the dependence of deformation on trans-membrane pressure and (b) the dependence of friction coefficients and Sherwood numbers on deformation, derived from the results of the present modelling approach, will be implemented into higher-scale (stack-level) models in order to close the fluid-structure interaction loop and characterize the amount and effects of maldistribution phenomena. Moreover, the simulation method presented in this study can be applied to the more traditional configurations of flat membranes with net spacers by suitable adjustments.