Scholarly article on topic 'Theoretical power density from salinity gradients using reverse electrodialysis'

Theoretical power density from salinity gradients using reverse electrodialysis Academic research paper on "Chemical engineering"

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Abstract of research paper on Chemical engineering, author of scientific article — David A. Vermaas, Enver Guler, Michel Saakes, Kitty Nijmeijer

Abstract Reverse electrodialysis (RED) is a technology to generate power from mixing waters with different salinity. The net power density (i.e. power per membrane area) is determined by 1) the membrane potential, 2) the ohmic resistance, 3) the resistance due to changing bulk concentrations, 4) the boundary layer resistance and 5) the power required to pump the feed water. Previous power density estimations often neglected the latter three terms. This paper provides a set of analytical equations to estimate the net power density obtainable from RED stacks with spacers and RED stacks with profiled membranes. With the current technology, the obtained maximum net power density is calculated at 2.7W/m2. Higher power densities could be obtained by changing the cell design, in particular the membrane resistance and the cell length. Changing these parameters one and two orders of magnitude respectively, the calculated net power density is close to 20W/m2.

Academic research paper on topic "Theoretical power density from salinity gradients using reverse electrodialysis"

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SciVerse ScienceDirect

Energy Procedia 20 (2012) 170- 184

Technoport RERC Research 2012

Theoretical power density from salinity gradients using

reverse electrodialysis

David A. Vermaasa,b', Enver Gulera'D, Michel Saakes0, Kitty Nijmeijera

a Membrane Science & Technology, University of Twente, MESA+ Institute for Nanotechnology, P.O. Box 217, 7500 AE Enschede,

The Netherlands

b Wetsus, Centre of Excellence for Sustainable Water Technology, P.O. Box 1113, 8900 CC, Leeuwarden, The Netherlands

Abstract

Reverse electrodialysis (RED) is a technology to generate power from mixing waters with different salinity. The net power density (i.e. power per membrane area) is determined by 1) the membrane potential, 2) the ohmic resistance, 3) the resistance due to changing bulk concentrations, 4) the boundary layer resistance and 5) the power required to pump the feed water. Previous power density estimations often neglected the latter three terms. This paper provides a set of analytical equations to estimate the net power density obtainable from RED stacks with spacers and RED stacks with profiled membranes. With the current technology, the obtained maximum net power density is calculated at 2.7 W/m2. Higher power densities could be obtained by changing the cell design, in particular the membrane resistance and the cell length. Changing these parameters one and two orders of magnitude respectively, the calculated net power density is close to 20 W/m2.

© 2012 Published by Elsevier Ltd. Selection and/or pe er-review under resp onsibility of the Centre for Cenewable Enere;y.

Keywords: ion exchange membranes; boundary layer; profiled membranes; spacers, reverse electrodialysis; salinity gradient energy

1. Introduction

Reverse electrodialysis (RED) is a technology to generate electricity from the salinity difference between two solutions, e.g. seawater and river water. The principle of RED is illustrated in Fig. 1. A RED system is composed of ion exchange membranes and compartments for seawater and river water (in alternating order). The ion exchange membranes are selective for either cations or anions. The salinity difference

* Corresponding author. Tel.: +31 58 284 3182 ; Fax: +31 58 284 3001. E-mail address: david.vermaas@wetsus.nl

1876-6102 © 2011 Published by Elsevier Ltd. Selection and/or peer-review under responsibility of the Centre for Renewable Energy. doi:10.1016/j.egypro.2012.03.018

between seawater on one side and river water on the other side of the membrane creates a potential difference. Multiple cells, each comprising a cation exchange membrane (CEM), a seawater compartment, an anion exchange membrane (AEM) and a river water compartment, can be piled up to increase the voltage. Electrodes at both ends of the pile facilitate a redox reaction, which generates an electrical current to power an external device.

Fresh water Salt water-

Fe(CN)6 P

Fe(CN)63-

Fe(CN)64-

Fe(CN)6

Brackish * water

Figure 1. Principle of reverse electrodialysis (RED)

The global potential for salinity gradient power is large. Each cubic meter of river water can generate 1.4 MJ of energy when mixed with equal amounts of seawater (assuming a salinity of 30 %o), and over 2 MJ when mixed with an excess of seawater [1]. The global runoff of river water into the sea has a potential to generate more than the prospected global electricity demand for 2012 [2]. Moreover, the power output from RED could be controlled by regulating the water flow, especially when a lake is available for fresh water storage. As such, salinity gradient energy can be stored and used when the power production from sun and wind is at a low level.

Recent developments improved the experimentally obtained power density (i.e. power per membrane area) for representative seawater and river water, to a maximum value of 2.2 W/m2. When taking into account the energy spent for pumping the water, a maximum net power density of 1.2 W/m2 was found [2]. The increase in practical net power output in RED to a value of 1.2 W/m2 was obtained by optimizing the intermembrane distance, imposed by spacers. Thinner spacers improve the power output, but also increase the power consumption for pumping the feed waters through the thin compartments between the CEM and AEM. An optimum intermembrane distance of 100 nm was found experimentally [2]. The non-conductive spacers obstruct the transport of feed water and reduce the ion transport and consequently the power output. To solve this issue, we proposed a spacer-free design that uses profiled membranes, supplied with straight ion conductive ridges, to integrate the membrane and spacer functionality. With these profiled membranes, the pumping losses were reduced by a factor of 4 and the absence of the non-conductive spacers reduced the ohmic resistance significantly [3]. On the other hand, the boundary layer resistance increased when using profiled membranes. Nevertheless, the net power density was approximately 10% higher than when a stack with spacers was used [3]. A design with profiled membranes not only increases the maximum value of the net power density, but also shifts the maximum to higher flow rates [3] and enables lower intermembrane distances.

The present work aims to estimate the optimum intermembrane distance and flow rate to reach the maximum net power density obtainable in RED, for designs with spacers and designs with profiled

membranes. The power is estimated using theoretical equations for the produced stack voltage, stack resistance and pumping power. Previous attempts [4-6] to estimate the maximum power density did not include estimations for the boundary layer resistance. However, experiments indicate that the maximum net power density can be found for relatively small intermembrane distances and using profiled membranes [2, 3]. Under these conditions, the boundary layer resistance cannot be neglected. This research demonstrates how the boundary layer resistance can be estimated based on the practical parameters residence time and intermembrane distance.

The remaining of this paper will describe the individual components (electromotive force, ohmic resistance, boundary layer resistance and pumping power) that are required to calculate the net power density. The calculated boundary layer resistance is calibrated using several data sets from previous research [2, 3]. Using these components, the optimum conditions for a maximized net power density and the sensitivity of the individual design parameters are shown.

Nomenclature

b width between profiled ridges (m) Ssp/Vsp ratio between the surface and volume

c concentration of feed water (mol/liter) of the spacer filaments (1/m)

dh hydraulic diameter (m) T temperature (K)

E electromotive force (V) tres residence time (s)

F faraday constant (96485 C/mol) v velocity (m/s)

h intermembrane distance (m) z valence of ions (-)

j current density (A/m2) a membrane permselectivity (-)

Nm number of membranes (-) 0 mask factor (-)

Ap pressure difference over feed water 7 activity coefficient (-)

compartment (Pa) £ porosity (-)

P J net net power density (W/m2) 1 (non-ohmic) overpotential

R universal gas constant (8.31 J/(molK)) K feed water conductivity (S/m)

Rohmic o^mc area resistance (Q-m2) viscosity of water (Pas)

Rbl bom^^ layer area resistance (Q-m2)

rAC area resistance due to bulk concenfration changes (Q-m2)

2. Theory

2.1. Voltage

The salinity difference over each membrane creates a potential difference which is given by the Nernst equation. The electromotive force over a series of Nm membranes, each with an apparent permselectivity a, is given by:

In which E is the electromotive force (V), R is the universal gas constant (8.31 J/(molK)), T is the absolute temperature (K), z is the valence of the ions (-), F is the Faraday constant (96485 C/mol), y is the activity coefficient (-) and c is the concentration at the membrane-solution interface (mol/liter). The subscripts sea and river indicate the solution on either side of the membrane.

The produced voltage drives an electrical current when an external circuit is connected. Due to the ohmic resistance of the stack itself, the voltage over the electrodes will decrease. At a current density j (A/m2), the voltage U (V) is given by:

In which Rohmic is the ohmic area resistance (iî-m2). When no current is applied, the electromotive force E can be estimated using the inflow concentrations of the feed waters in eq. 1. This voltage is referred to as the open circuit voltage EOCV. When a current is applied, ions are transported from the salt water side through the membranes to the fresh water side, and the concentration within each compartment changes. The concentration difference at the membrane-solution interfaces will be smaller than the concentration difference between the seawater and river water at the inflow. As a consequence, the electromotive force will be lower than EOCV. This decrease in potential can be subdivided into a contribution due to the concentration change in the boundary layers, r/BL, and a contribution due to the concentration change in the bulk of the solution, rjAC. In fact, r\BL considers the concentration gradient perpendicular to the membrane surface within each compartment (assuming developed boundary layers), while r/jc considers the concentration gradient parallel to the membranes. Including these two potential losses, the voltage over the stack is given by:

In which EOCV, t]BL and r\C are in Volt. The losses due to boundary layer effects and concentration changes in the bulk can be compared to the ohmic loss when r\BL and r\C are divided by the current density, i.e. interpreting both as a (non-ohmic) resistance:

U = E-Rohmic' j

U — EOCV Vac VbL Rohmic' J

U = EOCV~ (Rohmic+ RAC+ RBl) ' j

The next part gives a mathematical description on how to estimate the contribution of each of the individual resistance factors Rohmic, R^c and RBL.

2.2. Resistance

When spacer effects are neglected, the ohmic resistance is determined by the membrane resistance, the water conductivity and the intermembrane distance [7, 8]. However, the non-conductive spacer does have a significant effect on the ohmic resistance because it blocks part of the membrane area (spacer shadow effect) [9] and imposes a tortuous ionic flow in the water compartments [1, 10]. A more accurate way to estimate Rohmic is to include the porosity e, and a mask fraction /? that accounts for the spacer shadow effect [1]. Including these effects, the ohmic resistance can be estimated by [1]:

R = Nm

ohmic ~

RArns RCEM | hsea | h-

1_ « 1_ fl „2 „2

\ ' ' ° sea ° river /

+ ReIectrodes (5)

In which Raem and Rcem are the area resistances of the anion and cation exchange membrane (ii-m2), respectively, h is the intermembrane distance (m), k is the electrolyte conductivity (S/m) and Relectrodes is the (ohmic) resistance of both electrodes and their compartments (Q-m2). The spacer porosity s, and the mask fraction fi are dimensionless. Their values vary between 0 and 1; a completely open compartment would give s = 1 and fi = 0, whereas a solid spacer would be represented by s = 0 and /? = 1. The porosity is squared in eq. 5, because two effects play a role: 1) the spacer filaments block a certain fraction of the compartment, so the current density is intensified in the pores and 2) the spacer filaments force a longer, tortuous path for the electrical current. The geometry of the spacer filaments influences the exact relation between the electrical resistance and the porosity. As a first approximation, s is squared in eq. 5, in accordance with previous research [1].

R¿c can be estimated from the concentration change due to charge transport. Assuming a linear decrease of the electromotive force between feed water inlet and outlet, and neglecting changes in activity coefficients due to ion exchange, Rac can be estimated from [2]:

2 z-F-J

^ river

\ sea J

In which A„w = 1h--J_!_res-and Ase= 1--j tres-, in which tres is the residence time

river t^ t sea t^ t >

H.F-h 'c F • F•h • c

^ river river ^ sea sea

of the feed water in the stack (s). Fig. 2 shows the calculated Rc versus tres/h using the combined experimental data of Vermaas et al. [2, 3]. Both data sets were obtained from a stack with 5 RED cells, with an electrode dimension of 10 cm by 10 cm and using artificial seawater (0.510 M) and river water (0.017M). The first data set contained 4 different intermembrane distances (60, 100, 200 and 485 nm) using spacers [2], whereas the other data set compared the use of profiled membranes and spacers, both with an intermembr^e &s^ce of 240 ^m [3]. For all data, the current density was chosen such that the power density was maximized.

°k 0.4 q:

0 20000 40000 60000

tres/h (slm)

Figure 2. RC as function of the residence time of the feed water in a RED stack (tres) divided by the intermembrane distance h. Data adopted from [2, 3]. The symbols with 'sp' are obtained for stacks with spacers, whereas 'pr' stands for the stack with profiled membranes.

Fig. 2 shows that the data from different stacks are close to a unified line when scaled to trJh. The slight deviations are due to differences in current density, which are a combined effect of the electromotive force and the ohmic resistance. Even without eq. 6, the dependency on trJh could be expected. The cumulative ion transport from a volume of seawater to a volume of river water increases with the residence time, while the effect on the concentration is inversely proportional to the water volume, thus to the intermembrane distance. The experimental data in Fig. 2 show a linear fit with R2 = 0.97. This linearity cannot be derived from eq. 6 a priori, due to the fractions and the logarithm. Apparently, instead of eq. 6, a linear approach would estimate RAC closely. This work uses the more complex, but more physical, formulation from eq. 6.

The boundary layer resistance RBL is dependent on the change in concentration at the middle of a compartment and the concentration at the membrane-solution interface [7]. RBL was not estimated theoretically before for applications in RED using input parameters only. Previous research showed that Rbl reduces when the velocity of the feed water increases, since higher velocities improve the mixing rate [2, 3, 8, 9]. Ramon et al. [5] suggested a relation with the Reynolds number, Schmidt number, diffusion coefficient and hydraulic diameter. However, when the experimental data [2, 3] are combined, these do not correlate well (R2<0.5), neither with the feed water velocity, nor with the flow rate, Reynolds number, Sherwood number or the suggested relation of Ramon et al. [5]. A new, physically based approach is proposed in this research. The mixing in the boundary layers can be assumed proportional to the momentum exchange toward the membrane, which is proportional to the velocity shear at the membrane-solution interface [11] (i.e. the velocity gradient perpendicular to the membrane). Therefore, RBL can be expected to be inversely proportional to the velocity shear at the membrane-solution interface:

r2 = 0.97 ..''

% O □ A

o" □

$ O. & □ sp 60 urn

O .' Asp100|im ■

* A o sp 200 |jm

/ O sp 240 (im v sp 485 |im ft pr 240 urn

f i v1 d \dv\ rbl dt

v * J membr.sol.interrface

In which v is the local velocity magnitude (m/s) and y is the coordinate perpendicular to the membrane (m). For laminar uniform flow (Poiseuille flow), this yields [11]:

average

~tresL

In which va

. is the average velocity (m/s) and L is the cell length (m), i.e. the average path length of

the feed water in each compartment.

Experimental data from [2, 3] are used to show the relation between RBL and tresh/L. For simplicity, the compartments with spacers are considered as straight flow channels, disregarding the volume of the spacers. Fig. 3 shows the calculated values for RBL for a) stacks with several spacer thicknesses and b) stacks with profiled membranes.

0.8 0.7 0.6 0.5

g 0.4 ф

0.2 0.1 0

□ 60 дт Spacers

. a 100 [im □

О 200 [im

■ 0 240 цт

v 485 |лт о

R = 0.72

□ V'

□ О ■•'

О .-''«к 0

wA'0 А

■ ' А

0 240 цт Profiled membranes

R2 = 0,98

0.4 t -h/L (s)

res v '

t h/L (s)

res ^ '

Figure 3. Rbl as function of tresh/L for RED stacks with a) different intermembrane spacer thicknesses and b) profiled membranes [2, 3].

Although the data start to deviate from the linear fit for larger values of tres h/L, most values for RBL are on a reasonable linear line, especially considering that the difference between the thinnest and thickest spacer is a factor 8. Scattering is explained by non-ideal uniform flow (near profiled ridges and spacer filaments) and the relatively large error of RBL in the measurements. RBL is derived from the difference (Rtotai - Rohmic - Rac), each contribution with a certain error. The values for Rtota¡ and R¿c are largest for high tres, so the error in RBL increases when tres h/L increases. In addition, the different spacers have minor changes in spacer mesh size, mesh angle and porosity, which may also influence the mixing rate [12].

Fig. 3 also shows that RBL is clearly higher when profiled membranes are used compared to a design with spacers. For spacers, RBL can be approximated by:

0.62^ + 0.05

while for profiled membranes the data best fit:

rbl =-

0.96tres-^ + 0.35

These equations show that the linear fit for the stack with profiled membranes does not intersect the origin, whereas the trendline for the stack with spacers does approach the origin. This unexpected behavior for profiled membranes might be caused by preferential flow paths. Small irregularities on a profiled membrane may prevent flow through that complete channel, whereas in the case of spacers water can flow around such an obstruction.

2.3. Pumping losses

A part of the obtained power is required to pump the feed waters through the stacks. This pumping power can be calculated from the pressure drop over the inlet and outlet of the feed waters and the flow rate of the feed waters. In the most ideal case, for a laminar, fully developed flow in an infinite wide uniform channel, the pressure drop can be estimated using the Darcy-Weisbach equation [11]:

h 2 tres -1/4- dh 2

In which dh is the hydraulic diameter of the channel (m). In the case of an infinite wide channel this hydraulic diameter equals 2h [13].

The experimental pressure drops are significantly higher than this idealized case. The experimentally determined pressure drop in a stack with profiled membranes was approximately 20 times higher than the idealized equivalent values, whereas the spacers showed pumping losses of more than 80 times the values calculated from eq. 11. The excess in pressure drop for profiled membranes is partly due to the finite width of the channels and partly due to a non-optimal design. The flow was non-uniform especially at inflow and outflow, where the flow was forced to make sharp corners. In the case of spacers, the pumping power is additionally increased due to the spacer filaments that obstruct the flow. The spacer filaments make the effective hydraulic diameter smaller than that of a non-filled channel. To anticipate on that effect, the hydraulic diameter for spacer filled channels can be derived from [13, 14]:

dh = 2/ h + (1-e) -Ssp/Vsp (12)

In which Ssp/Vsp is the ratio between the surface and volume of the spacer filaments.

For profiled membranes, the hydraulic diameter can be derived from [13]:

d jwj^ (13)

h 2b + 2h

In which b is the width of each channel between the profiled ridges (m).

Assuming both compartments to be equally thick, the pumping power Ppump (W/m2), for both feed waters together can be calculated from [3]:

^ A ~tres2-1/4.dh2 ( '

In which <P is the volumetric flow rate (m3/s) and A is the total membrane area (m2). This pumping power should be representative for a large scale operation. Small scale experiments still show a higher pressure drop [2-4, 15] due to relatively high losses at the in- and outlet of the feed water or parameters as the spacer mesh angle. A more complex approach to calculate the pressure drop in spacers is available [14], but is beyond the scope of this paper.

2.4. Net power density

The gross power density generated with a RED stack can be calculated by multiplying the stack voltage U (eq. 4) by the current density j. To calculate the net power density, the power spent on pumping (eq. 14) needs to be subtracted from this generated power. This yields for the net power density of a RED stack:

P _ EOCV • J ~ jRohmic+ RAC + RBl\ J 2 _ p 15)

net pump ' /

Combined with the set of previous equations for EOcv, Rohmic, rac, rbl and dh (eq. 1, 5-14), the net power density can be estimated using design parameters only as input.

3. Results and discussion

The parameters that determine the net power density are membrane and spacer properties, electrode resistance, feed water concentrations, temperature, cell dimensions and residence time. Most parameters cannot be tuned in a wide range. For example, the feed water concentrations are limited to the availability. Three parameters that can be tuned in a wide range and are expected (deduced from previous experiments) to have a major impact on the net power density are the residence time, the intermembrane distance, the current density and the cell length.

The cell length has no (finite) optimum. A smaller cell length reduces the pumping power significantly, as indicated by eq. 14. A reduced pumping power allows smaller intermembrane distances and smaller residence times, which would reduce RBL and Rohmic, and consequently a smaller cell length would always lead to a higher net power density. The benefit of small cell length was already recognized in previous research [4]. Practical limitations determine the cell length. As a first estimate, a value of 0.1m is chosen.

The intermembrane distance, the residence time and the current density can be varied to find the optimum net power output. The residence time and intermembrane distance of the seawater was set equal to that for river water. Table 1 shows representative values for the other relevant parameters for a large scale operation that serve as input parameters for the calculations to estimate the maximum net power

output obtainable in RED. Although stacks with these specifications have not been tested experimentally, a combination of previous research indicates that such stacks can be manufactured with the current technology [2, 3, 16, 17].

Table 1. Typical specifications for a RED stack using spacers and a RED stack using profiled membranes.

Parameter

Spacers

Profiled membranes

h ( hriver hsea) tres ( tres, river tres, sea)

Raem = Rcem

Reiectrodes

csea criver

Ss]/Vsp

Varied between 1 - 200 цт Varied between 0.5 - 200 s Varied between 1 - 100 A/m2 0.1 m

1 Пет2 a

2 iixm2 0.97 b 100

0.510 M NaCl 0.017 M NaCl 298 K 0.50 c 0.70 b

Varied between 1 - 200 цт Varied between 0.5 - 200 s Varied between 1 - 100 A/m2 0.1 m

1 Пет2 a

2 Пет2 0.97 a 100

0.510 M NaCl

0.017 M NaCl

The net power density, as estimated using the equations presented in this work and by characteristic values summarized in Table 1, is shown in Fig. 4 as function of tres and h, for a) stacks with spacers and b) stacks with profiled membranes. The current density j is chosen for each value of tres and h such that the net power is maximized.

b Based on Fumatech FKS / FAS membranes [2] c Based on open area and porosity of Sefar woven spacers [2]

d Assuming 10% of the membrane area occupied by profiled ridges and neglecting ion-conduction through the profiled ridges. If this ion-conduction is not neglected, p is even lower. e Based on 4/dfilament [14]

Figure 4. Net power density for stacks with spacers (a) and stacks with profiled membranes (b), as function of the intermembrane distance and residence time. The residence time is plotted on a logarithmic scale.

Fig. 4b shows that the maximum net power density obtainable is 2.72 W/m2, using a stack with profiled membranes, an intermembrane distance of 52 nm and a residence time of 2.4 s. The net power density for a stack with spacers is only 1.34 W/m2, for an intermembrane distance of 70 nm and a residence time of 7.2 s (Fig. 4a). This is close to the experimentally derived maximum of 1.2 W/m2, obtained for an intermembrane distance of 100 nm and a residence time of 8 sec. Fig. 4 also shows that the optimum residence time is rather independent of the intermembrane distance for h >50 nm; the highest net power densities, for all intermembrane distances between 50 and 200 nm, are obtained for a residence time of approximately 2.5 sec. The larger residence time for stacks with spacers (approximately 7 sec) compared to stacks with profiled membranes is caused by the larger contribution of the pumping power loss in the case of stacks with spacers.

To improve the net power density, different values of the parameters listed in Table 1 can be considered. A sensitivity analysis is performed to investigate what parameter has the largest influence on the net power density. The temperature and feed water concentrations are left out of consideration, because these parameters can not be influenced. The sensitivity analysis was done by changing each of the variables to a 1% lower or higher value and calculating the maximum net power density, not necessarily at the same tres, h and j. For every run, the optimum values for tres, h and j were determined. The result of this sensitivity analysis is shown in Table 2.

Table 2. Sensitivity analysis for parameters determining the net power density. Positive values indicate that a higher parameter value would imply an increase in net power density and vice versa

Relative increase in P„,t per relative increase in parameter value (-)

Parameter

Spacers Profiled membranes

a 1.862 1.383

Raem = Rcem -0.240 -0.088

I^kctrodd -0.002 -0.002

Nm 0.002 0.002

L -0.308 -0.141

ß -0.240 -0.010

e 1.473 0.344

b - 0.014

Ss]/Vsp -0.168 -

Table 2 shows that the net power density is most sensitive for the permselectivity (a) and the porosity (e). These parameters can be improved only to a limited extend, which would slightly increase the net power density. The membrane resistance and cell length have a smaller influence on the net power density, but have relatively much larger possibilities for improvement. Theoretically, these parameters have no minimum value, whereas the permselectivity is limited to a value of 100%. Therefore, reducing the membrane resistance and cell length are promising for improving the net power density than improving the membrane permselectivity. Fig. 5 shows the net power density as a function of the cell length, for different membrane resistances.

10"3 10~2 10"1 io~3 io~2 10"1

Cell length L (m) Cell length L (m)

Figure 5. Net power density as function of the cell length, for membrane resistances of 0.1, 0.3, 0.5 and 1.0 fi-cm2, for stacks with spacers (a) and stacks with profiled membranes (b). The cell length is plotted on a logarithmic scale.

Fig. 5 shows that the net power density increases rapidly as the cell length decreases, reaching a value of almost 20 W/m2 for profiled membranes with a cell length of 0.001m and a membrane resistance of 0.1 Qcm2. Naturally, a 1 mm cell length and such a low membrane resistance are not realistic at the present state of technology. Moreover, the optimum current density to obtain 20 W/m2 at these conditions can get up to 500 A/m2, which requires attention to suit the electrode system [18]. Nevertheless, the tremendous increase in net power density for small cell lengths emphasizes the sensitivity of the net power density on the cell length. The straight or even convex shaped graphs using logarithmic x-axes in Fig. 5 suggest an asymptotic increase to an infinite net power density as the cell length approaches zero. However, the finite membrane resistance will prevent an infinite net power density when the cell length approaches

Fig. 5 also shows that decreasing the membrane resistance is more effective when a small cell length is chosen. For example, reducing the membrane resistance from 1 fi-cm2 to 0.5 fi-cm2 leads to a 19% increase in net power density for a cell length of 0.1m, whereas the same reduction leads to a 47% increase in net power density for a cell length of 0.001m. Reducing the cell length is accompanied with decreasing optimum intermembrane distances, and therefore a larger influence of the membrane resistance.

A design with very short cell lengths requires an intelligent feed water distribution system for an operation at large scale. An example is given by Veerman et al. [4], proposing a fractal design for distributing feed waters. Such a design involves profiled membranes with (deep) channels carved out to supply the feed water to (shallow) cells with a cell length in the order of 1 mm. A RED-design with such small cell lengths was not tested experimentally before, but can be manufactured with the current technology. This calculation shows that the net power density can be improved significantly in this way.

4. Conclusions

The power density obtained from reverse electrodialysis can be estimated based on a set of analytical equations. This estimation of the net power density better reflects the reality in comparison to previous attempts, where the boundary layer resistances and pumping power were often left out of consideration. This research shows that the boundary layer resistance can be estimated based on input parameters, in this case toesh/L. The highest net power density, using parameters that are typical for the current state of technology, is 2.7 W/m2. This value is predicted for a stack with profiled membranes, with an intermembrane distance of 52 |xm and a residence time of 2.4 s. Higher net power densities can be obtained by improving the membrane properties (permselectivity, resistance), increasing the (spacer) porosity and using shorter cell lengths. The combination of decreasing cell length and decreasing membrane resistance is an effective strategy to improve the net power density. A net power density close to 20 W/m2 is envisaged when the membrane resistance is reduced to 0.1 Qcm2 using a cell length of 1 mm. A design with such a small cell length is not tested yet and such a small membrane resistance is not obtained yet. This research demonstrates that the strategy to reduce both the cell length and membrane resistance is very effective to improve the net power density in reverse electrodialysis.

Acknowledgements

This research is performed at Wetsus, Technological Top Institute for Water technology. Wetsus is funded by the Dutch Ministry of Economic Affairs, the European Union Regional Development Fund, the Province of Fryslän, the City of Leeuwarden and the EZ/Kompas program of the 'Samenwerkingsverband Noord-Nederland'. The authors are thankful for the support of Alliander, Eneco, Frisia Zout, Fujifilm, Landustrie, Magneto Special Anodes, MAST Carbon and A. Hak.

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