Energy from CO2 using capacitive electrodes – Theoretical outline and calculation of open circuit voltage

Journal of Colloid and Interface Science 418 (2014) 200–207

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Energy from CO2 using capacitive electrodes – Theoretical outline and calculation of open circuit voltage J.M. Paz-Garcia, O. Schaetzle, P.M. Biesheuvel, H.V.M. Hamelers ⇑ Wetsus, Centre of Excellence for Sustainable Water Technology, Agora 1, 8934 CJ Leeuwarden, The Netherlands

a r t i c l e

i n f o

Article history: Received 4 November 2013 Accepted 29 November 2013 Available online 8 December 2013 Keywords: Mixing energy Membrane potential Carbonic acid

a b s t r a c t Recently, a new technology has been proposed for the utilization of energy from CO2 emissions (Hamelers et al., 2014). The principle consists of controlling the dilution process of CO2–concentrated gas (e.g., exhaust gas) into CO2–dilute gas (e.g., air) thereby extracting a fraction of the released mixing energy. In this paper, we describe the theoretical fundamentals of this technology when using a pair of charge–selective capacitive electrodes. We focus on the behavior of the chemical system consisting of CO2 gas dissolved in water or monoethanolamine solution. The maximum voltage given for the capacitive cell is theoretically calculated, based on the membrane potential. The different aspects that affect this theoretical maximum value are discussed. Ó 2013 Elsevier Inc. All rights reserved.

1. Introduction Recently, a new technology has been conceived for harvesting energy from CO2 emissions [1]. The technology aims to generate electricity from the energy that is released when a gas with high CO2 concentration, such as exhaust gas from power plants, dilutes into a gas with low CO2 concentration, such as the atmosphere. In Ref. [1], it was estimated that an amount of 850 TWh of additional electricity would be available yearly in the flue gases of power plants without additional CO2 emissions. Therefore, the development of this new technology would increase the energetic efficiency of fossil-fuel and biomass combustion power stations. It is possible to extract mixing energy from two solutions with different concentrations using capacitive electrode cells. This principle has been explored for producing electricity from salinity gradients systems, such as in mixing river and seawater [2]. In Hamelers et al. [1], the viability of the generation of energy from CO2 concentration gradient systems was demonstrated using, as first approach, an experimental setup based on carbonated aqueous solutions. In that work, electrical energy was obtained by mixing two electrolyte solutions with different CO2 concentrations: one was saturated with pure CO2 gas and the other one with air. While salinity gradient systems are generally based on NaCl solutions (assumed chemically inert), CO2 concentration gradient systems involve additional physicochemical aspects that have to be taken into account for understanding the technology. If the system is operated with an aqueous solution, it will be limited by the

⇑ Corresponding author. E-mail address: [email protected] (H.V.M. Hamelers). 0021-9797/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jcis.2013.11.081

solubility of the gaseous carbon dioxide in the liquid phase. Also, the reactivity of the carbon dioxide in water (which will form carbonic acid, carbonates, bicarbonates and protons) might play a determining role [3]. Capacitive cells are manufactured using a pair of porous carbon electrodes (PCEs) sandwiching a spacer channel, through which an electrolyte can flow. PCEs have the ability to accumulate ionic charge within their porous structure because of the formation of electrical double layers (EDLs). Therefore, PCEs are also used to create EDL capacitors for energy storage [4–6] and for capacitive deionization [7,8]. If an external electric potential develops between the electrodes, they behave as an anode–cathode pair, attracting and accumulating anions and cations respectively. The ionic charge is balanced by electrons, which travel through the external circuit and the electric conductors from one electrode to the other. When a set of anion- and cation-exchange membranes is placed between the spacer channel and the porous carbon electrodes, it is possible to produce electrical energy from the spontaneous ionic current induced by the membrane potential, when solutions of different ionic composition are alternatingly pumped into the cell. The set of an ion-exchange membrane and a porous electrode is denoted in this work as charge-selective porous electrode. The membrane-enhanced capacitive system converts the chemical energy released in the mixing or dilution process to useful electrical energy. The capacitive cell behaves as an electric capacitor, producing electrical energy in consecutive charging and discharging steps, which occur by alternatively injecting in the cell the concentrate and the dilute CO2 solutions. The maximum concentration that can be obtained for the concentrated (CO2-flushed) solution is limited by the solubility of

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the carbon dioxide gas. At 25 °C, the saturation concentration of aqueous CO2 in water is 33.6 mM at 1 bar CO2 pressure, while the concentration of the main ions (protons and bicarbonate ions) is in the order only of 0.1 mM. In the experiments shown in Ref. [1], the possibility was explored to increase the energy extracted in the capacitive cell by increasing the CO2 absorption into the aqueous solutions using monoethanolamine (MEA). Ethanolamines are commonly used to capture acid gases in aqueous solution due to their alkaline buffer properties. The higher solubility of carbon dioxide in amine solutions is due to both the alkalinization of the media, and due to the formation of carbamate ions [9–14]. Capacitive cells operated with 0.25 M MEA solutions instead of deionized water resulted in higher conversion efficiencies and shorter (faster) charging/discharging cycles [1]. The capacitive cell discussed in this paper, depicted in Fig. 1, is based on that used in the experiments reported in Ref. [1]. It consists of a pair of ion-selective electrodes sandwiching a laminar flow channel (25 cm length, 2 cm height, 0.2 mm width). A porous spacer is placed between electrodes to assure enough separation for the flow channel. Between the spacer and the porous electrodes, ion-exchange membranes are placed. One of the electrodes is covered with an anion-exchange membrane (AEM) and the other one with a cation-exchange membrane (CEM). The cell operates cyclically. The terms ‘‘charging step’’ and ‘‘discharging step’’ are used to refer to the operation of the cell using CO2 concentrated (CO2-flushed) or CO2 dilute (air-flushed) solutions respectively, which are equilibrated in separated tanks. To further develop the technology, it is necessary to have a better understanding of the physicochemical processes involved, such as the reactive transport of chemical species in the porous electrode and through the ion-exchange membranes. In this work, we propose a physicochemical and mathematical model for the generation of electrical energy from mixing CO2 saturated solutions using capacitive cells. This paper is organized as follows. In Section 2, we describe the chemical system under consideration. Section 3 is devoted to describe a mathematical model for ionic reactive-transport. In

eAnode

Cathode RNHCOOCO32 HCO3 H+ RNH3+

-

OH

H2CO3

AEM

Flow channel

2. The H2CO3/MEA chemical system The chemical system considered in this research consists of that resulting from the dissolution of gaseous carbon dioxide, CO2 (g), in water or in an aqueous solution of MEA. The carbonate solutions are obtained by continuous flushing of gas (namely air, or pure CO2 gas) through the liquid assuring gas/liquid equilibrium. Table 1 collects a set of equilibrium stoichiometric equations that describes the carbonic acid chemical system in pure water and in MEA aqueous solution. Assuming ideal thermodynamics, the K-values in Table 1 describe the ratio of products over reactants concentration, when all concentrations are expressed in M. The species water is then omitted from this definition. CO2 (g) dissolves in water in the molecular form of CO2 (aq), denoted as ‘‘free hydrated CO2.’’ Henry’s law can be used to estimate the equilibrium concentration of CO2 (aq), as follows:

½CO2 ðaqÞ ¼

pCO2 ðgÞ KH

ð1Þ

where the term in square brackets means aqueous molar concentration, pCO2 ðgÞ stands for the partial pressure in atm and KH is the Henry’s constant for carbon dioxide, that at 298 K is approximately 29.41 (L atm mol1) [16]. Just as some other gases, the solubility of CO2 (g) decreases when increasing the temperature. Carbonic acid, H2CO3, is a diprotic and moderately weak acid. An aqueous electrolyte in equilibrium with the atmosphere has a pH  5.7, and a bicarbonate concentration of ½HCO 3   2 lM. In the case of equilibrium with pure CO2 gas, the pH decreases to a value of pH  4, and the bicarbonate ions’ concentration increases to around ½HCO 3   120 lM. Even at moderate alkaline conditions, the concentration of carbonate ions, CO2 3 , is lower than that of the bicarbonate ions. However, in very alkaline water (pH greater than 10) carbonate will be the predominant ion. The capacitive cell operated with carbonated water will have a slightly acidic pH. Nevertheless, alkaline conditions are expected inside the anion-exchange membrane at the anode. In equilibrium, slightly less than 1% of the dissolved CO2 hydrates to form non-dissociated molecules of carbonic acid [15]. For convenience, the total amount of carbon dioxide dissolved in water, H2 CO3 ¼ CO2 þ H2 CO3 , is frequently used [18]. The term carbonic acid is used in this work for the combined species H2 CO3 ; which replaces CO2(aq) in Eq. (1) while retaining the value of the Henry’s constant. Fig. 2 shows results for the pH of pure water in equilibrium with gas of different CO2 partial pressures, between the range of air Table 1 Chemical system.a

CO2-flushed

PCE

Section 4, the basics of porous electrode theory is described. In Section 5, the ionic transport through ion-exchange membranes is discussed. In Section 6, we describe the equivalent electrical circuit of the capacitive cell. Finally, in Sections 7 and 8, calculations and simulations are presented, focusing on the determination of the open circuit voltage of the cell.

CEM

PCE

Fig. 1. Schematic representation of the capacitive cell operating during the charging step. The spontaneous tendency of the ionic flow is shown. In the discharging step, ionic and electric currents will reverse.

Reaction

log 10(Keq)

H2 CO3 ¢ Hþ þ HCO 3 2 þ HCO 3 ¢ H þ CO3 H2 O ¢ Hþ þ OH þ RNHþ 3 ¢ RNH2 þ H RNHCOO þ H2 O ¢ RNH2

6.35 10.33

þ HCO 3

14 9.35 1.54

a Equilibrium constants at room temperature (from Refs. [15]  and [17]). RNH2, RNHþ 3 and RNHCOO stand, respectively, for the ethanolamine, the ethanolammonium ion, and the carbamate species, with R = C2H4OH.

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(pCO2 ¼ 3:5 Pa) and pure CO2 (1 bar). At these conditions, the pH of the solution is acidic and the concentration of hydroxide ions and carbonate ions is negligible. Accordingly, the electrolyte is almost a binary and equimolar solution of protons and bicarbonate ions. Their concentrations can be obtained from the pH of the solution. CO2 (g) dissolves better in aqueous amine solutions than in pure water. MEA, as other amines, is a base in aqueous solution [19], and it increases the pH of the solution forcing the dissociation of the carbonic acid by acid–base neutralization. In addition, by the formation of carbamate ions, bicarbonate ions are consumed and the equilibrium system is also displaced toward the dissociation of more carbonic acid. Several mechanisms have been proposed for the formation of carbamate ions. Nevertheless, in equilibrium, it is irrelevant which mechanism is chosen for determining the carbamate concentration. In the present work local equilibrium conditions are assumed for the set of aqueous phase reversible reactions. The local chemical equilibrium assumption is common for homogeneous aqueous phase reactions in the case of ionic species through membranes [3]. The reversible aqueous phase chemical reactions describing the carbonate chemical system in water and MEA solution are fast enough (in both direct and reverse directions) to accept the local chemical equilibrium conditions [14,20]. Fig. 3 shows equilibrium concentration for a volume of 0.25 M of MEA aqueous solution flushed with a gas with different CO2 concentrations, ranging from air to pure CO2. According to the calculations, a 0.25 M aqueous solution of MEA will have a pH  9.4 in equilibrium with air and pH  7 in equilibrium with pure CO2. In MEA solution, the ionic strength is considerably higher as compared to pure water. The main cation is RNHþ 3 , while the concentration of protons is very low due to the alkaline conditions. For the solution equilibrated with the pure CO2 gas, the main ions are ethanol ammonium and bicarbonate, in a concentration significantly higher than the concentration of all the other ions.

3. Mathematical model for reactive transport In the present study, the Poisson–Nernst–Planck (PNP) system of equations will be used to mathematically describe the reactive transport of chemical species taking place in the process. The PNP system of equations fully describes the coupled electro-diffusion transport of ions and non-ionic species by gradients of chemical and/or electric potentials. The Gouy–Chapman and Donnan models, used to describe the electric potential and the ionic concentration in the porous electrodes, can be derived from the PNP equations when applied to equilibrium between the nano- and the microscopic scale. Transport through ion-exchange membranes can also be effectively modeled using the PNP system. We propose here a generalized mathematical description for the reactive transport of chemicals in the direction between the electrodes (see Fig. 1). The PNP system consists of the combination of a conservation of mass equation, Eq. (2), for each of the N chemical species in the

Fig. 3. Equilibrium calculations for a 0.25 M MEA solution flushed with a CO2 containing gas at different partial pressures.

system, and the Poisson’s equation of electrostatics, Eq. (3), which couples between the concentration and the electrical potential.

@ðpci Þ ¼ r  Ji þ Gi ; @t

er2 w ¼ F

i ¼ 1; 2; . . . ; N

X c i zi þ r

ð2Þ ð3Þ

i

In Eqs. (2) and (3), ci (mol m3) is the concentration of the chemical species i, p (–) is the porosity, Ji (mol m2 s1) is the flux term, Gi (mol m3 s1) is the generation term, w (V) is the electric potential, e (F m1) is the permittivity in water, F (C mol1) is the Faraday constant, zi (mol mol1) is the ionic charge, and r (C m3) is the fixed charge density referred to the volume of electrolyte. When applied to systems on the macroscopic scale, the electrostatic term in the Poisson’s equation, i.e., the left-hand term in Eq. (3), is negligible and the equation becomes equivalent to the electroneutrality condition:

F

X c i zi þ r ¼ 0

ð4Þ

i

The flux term, Ji, is defined using the Nernst–Planck equation [21], assuming no advective contribution in the longitudinal direction between the electrodes:

F c i rw Ji ¼  Di rci  zi Di RT diffusion |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl}

ð5Þ

electromigration

where R (J K1 mol1) is the universal gas constant and T (K) is the absolute temperature. Defining the dimensionless voltage as / = FR1T1w, the Nernst–Planck equation can be rewritten as follows:

Ji ¼ Di ðrci þ zi ci r/Þ 2

ð6Þ

1

where Di (m s ) is the effective diffusivity, which is estimated 2 1 from the diffusivity at infinite dilution, DH ), and materiali (m s dependent parameters such as the porosity, p, and the tortuosity, s (both dimensionless):

p Di ¼ DH i

s

Fig. 2. pH value of carbonated water in equilibrium with a gas at different CO2 partial pressure.

ð7Þ

The chemical reaction term, Gi, is related to the reversible reactions expressed in Table 1, considered fast enough to accept the local chemical equilibrium condition [22].

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ci;mi ¼ ci;mA expðzi D/D Þ

4. Porous electrode theory In the proposed technique, the conversion between the ionic and the electric currents in the porous carbon electrodes takes place by means of electrostatic charge accumulation (in the form of EDLs) [23]. In a cathodic porous electrode, for example, the carbon surface charges negatively (with electrons coming from the anode through the external circuit and the conductors). This negative charge in the solid material is balanced by an excess of positive ionic charge in the vicinity of the contacted electrolyte (as e.g., protons). An anionic electrode has a shortage of electrons (positive charged surface), which is compensated by an excess of anions (as e.g., bicarbonate ions). The porous electrodes for capacitive cells are manufactured by casting fine particles of activated carbon on a graphite film, forming a thin layer. Therefore, in porous electrodes, two different ranges of porosity can be identified. The term macroporosity refers to the inter-particle space, i.e., the pores between the particles that remain charge-neutral. On the other hand, the term microporosity refers to the narrow pores within the carbon particles. Macropores have a typical size above 1 lm while micropores may have a few nanometers pore size. Referred to the total volume of porous matrix, the volume fractions of the micropores may be equal to or even higher than that for the macropores [23]. Herein, the terms ‘‘mA’’ and ‘‘mi’’ will be used for the macropores and micropores respectively. The ionic charge accumulated in the vicinity of the surface of porous electrodes can be situated in two different regions: The diffuse layer at the external part of the particle and that inside the micropores (see Fig. 4). Porous electrodes have high specific surface area (about 1000 m2 g1 or higher). As most of the active surface of the carbon particles corresponds to the micropore internal surface, it is commonly accepted that the accumulation of charge in porous electrodes is mainly due to that placed within the micropores, and thus the charge located on the outer particle–electrolyte interface is negligible in most practical models. In porous electrode theory, the accumulation of charge inside micropores is described by means of the Donnan model [24], which derives from the Gouy–Chapman (GC) model [25]. The Donnan model states that, due to the small pore diameter in the micropores, the diffuse layer inside them is strongly overlapped and, therefore, the electric potential can be considered constant (i.e., space independent) and equal to the surface potential, /mi(x)  /0. Consequently, the potential drop between the regions of this overlapped electric double layer and the electroneutral macropore solution, denoted as Donnan potential drop D/D = /0  /mA, does not depend on position within the pore, and the concentration within the micropores is given by:

Diffuse layer

where the terms ci,mi and ci,mA stand for the concentration of the species i inside the micropores and macropores. Some modifications of the Donnan model have been proposed to take into account the affinity that the activated carbon may have for the physicochemical adsorption of specific ions and the voltage drop in the Stern layer [23–26]. This extended or modified Donnan model will not be discussed here. The most accepted models for the one-dimensional transport of chemical species in porous electrode capacitive cells combine the Nernst–Planck model described before with the porous electrode theory described here [23,27]. In these models, the effective concentration in the electrode is the combination of the coexisting macro- and microporosity volumes. The conservation of mass equation, Eq. (2), in a porous electrode is therefore given by the following equation:

@ @J ðci;mA pmA þ ci;mi pmi Þ ¼  i þ Gi ; @t @x

+

- +

i ¼ 1; 2; . . . ; N

ð9Þ

In these models, the macro- and micropore electrolyte solution in the porous electrodes, for a given position in the one-dimensional domain, is assumed to remain in electrostatic dynamic equilibrium, described by Eq. (8). It means that the macroporosity solution is assumed electroneutral, and the excess of charge is immediately absorbed in or desorbed from the micropore solution, as schematically depicted in Fig. 5. 5. Membrane potential In the conceived technology, porous electrodes are designed as charge-selective by placing ion-exchange membranes between the electrode and the flow solution. Ion-exchange membranes are organic polymer layers containing a fixed amount of charge of one  sign. A CEM contains negative groups, such as e.g., —SO 3 , –COO 2 or —PO3 , fixed to the membrane backbone, while an AEM contains positively charged groups, such as —NHþ 3 . The membranes allow the passage of mobile charge of the opposite sign (counterions) but strongly reject ions with the same charge (coions). Typical membranes have a thickness between 0.01 and 0.5 mm, a fixed charge concentration between 3 and 5 M and permselectivities higher than 95% for CEMs and 90% for AEMs [28,29]. A simple but effective way to model the transport of ionic species through membranes is by considering the membrane as a

e

e

Microporosity

ð8Þ

_

e

_

_

e e

_

_

e

_

+ + - -+ + + - - + + - + + + - + + - + Macroporosity Fig. 4. Scheme for the macro- and microporosity in an anodic porous carbon electrode.

Fig. 5. Schematic representation of the one-dimensional model combining macroand microporosity for the accumulation of ions in an anodic charge-selective porous electrode. The macropore solution is assumed electrically neutral, and all the ionic charge entering the electrode electrolyte accumulates inside the micropores.

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homogeneous porous layer with a fixed amount of equally distributed charge [30]. The term r (C m3) already described in Eqs. (3) and (4) can be used to define the fixed charge density inside the membrane, and is related to the charge density xX (in M) commonly used in the membrane literature. The membrane fixed charge hinders the transport of the coions. Thus, as only ions of one type can pass (either cations or anions), it is possible to have a net ionic flux through the membrane. The driving force for this transport is the difference of chemical potential of the mobile ions (accounting also for the electrochemical contribution) between the boundaries of the membrane. At the interface between the charged membrane and each of the assumed-electroneutral electrolytes surrounding it, there is a narrow diffuse layer in which the concentration and electric potential experience sharp changes (the Donnan layer). These diffuse layers have a thickness of a few nanometers (see Fig. 6). Thus, on the macroscale, the concentration and electric potential drop between the membrane and the surrounding electrolytes have an apparent jump discontinuity. The fixed charge inside ion-exchange membranes is normally very stable and the electric potential between the external electrolyte and the membrane can also be described using the Donnan model. Accordingly, the potential drop between the membrane and each electrolyte is normally denoted as Donnan potential D/D, just as in the case of the porous electrodes [29,31]. The membrane potential, D/m, is described as the potential drop between the two electrolytes surrounding the membrane (see Fig. 6). The maximum membrane potential corresponds to that obtained in open-circuit conditions. In this case, the system would be in dynamic equilibrium, meaning that there may be ionic flux through the membrane, but the net flux of ionic current is zero [32,33]. This is expressed as:

 X X z2 D i F zi Ji ¼ zi Di rci þ i ci rw ¼ 0 RT i i

RT F

PN

i ðzi Di rci Þ PN 2 i ðzi Di c i Þ

! ð11Þ

diffuse layers (nanoscale) electric potential

φ

Δφm

Donnan potential drop

ci (mol m −3)

mobile cations

Fixed charge

mobile anions a

CEM

  RT ci;b ln Fzi ci;a

ð12Þ

Despite the high permselectivities of the ion-exchange membranes, the charge-selectivity for the case of carbonated systems is limited. It is known that, when ion exchange membranes separate two carbonate/bicarbonate containing solutions with different concentration, the pH of the depleted solution decreases and that of the concentrated solution increases [3,34–37]. This fact has been experimentally observed even when using open-circuit conditions, in which case the electroneutrality condition should hinder the ionic transport through the membrane. The reason is that non-dissociated carbonic acid and non-hydrated carbon dioxide are both non-ionic and they are not affected by the coion repulsion of the membrane, meaning that they are not affected by the membrane charge and they can diffuse through the membranes by concentration gradients. The diffusion transport of H2CO3/CO2 through the membranes is also influenced by its reactivity. As the pH in the AEM is more basic than in the external solutions, molecules of carbonic acid diffusing through the AEM will dissociate into protons and bicarbonate/carbonate ions. With the process, those ions will exchange most of the hydroxide ions in the membrane, which will become less alkaline and more carbonated. An AEM will hinder the transport of carbonic acid until most of the fixed charge of the membrane is balanced with carbonated ions. In this case, carbonate ions are favored inside the membrane structure due to both the high pH and the preference of the membrane to polyvalent ions. This process would take place only in the first cycles of the treatment, and may be prevented with a good pretreatment of the membranes.

6. The capacitive cell

Eq. (11) is nonlinear and its analytical integration between the limits of the membrane (a and b) is not straightforward. In a system in which there is a mobile ion whose concentration is much higher

membrane potential drop

Dw ¼ 

ð10Þ

The gradient of electric potential along the membrane can then be worked out, giving:

rw ¼ 

than of all the other mobile ions, the integration between the boundaries of the membrane leads to the simplified expression:

b

x (m)

Fig. 6. Schematic representation of ion transport and electrical potentials in a cation-exchange membrane (CEM) [28–33].

In the previous sections, the transport equations, the modeled chemical system, the porous electrodes and the membrane potential have all been described. In the present section, all the previous theory will be combined to describe the physicochemical phenomena that take place in a capacitive cell for harvesting energy from carbonated solutions. As mentioned before, the cell is operated cyclically, fed by the CO2-concentrate solution (charging step) or the CO2-dilute solution (discharging step), which are equilibrated in two different tanks prior injection in the cell. In the following, we will consider that the electrolyte in the pores of the electrodes has a concentration in between the dilute and the concentrate CO2-solutions. This is expected after a few cycles irrespective of the electrolyte used to pretreat the electrodes, due to the diffusion of non-ionic carbonic acid through the membranes. During the charging cycle, the concentrate (CO2-flushed) solution is fed into the cell through the flow channel. As the concentrations in the flow channel and in the electrolyte filling the pores of the electrodes are different, there is a spontaneous tendency for the ions to diffuse toward the region with lower concentration (see Fig. 1). In the half-cell covered by the CEM, cations (H+ protons and, if present, RHþ 3 ) have a tendency to migrate from the flow channel electrolyte toward the cathodic electrolyte. In the other half-cell, consisting of the electrode covered by the AEM, anions 2  (HCO 3 , CO3 , and, if present, RNHCOO ) will spontaneously diffuse into the anodic electrolyte. Since the fed solution (CO2-flushed) has a pH value equal to or lower than the solution filling the electrode pores, a flow of OH is expected in the opposite direction (from the electrode solution toward the flow channel).

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Ions pass through the membranes and reach the solutions in the electrodes, where they accumulate near the carbon surface, attracting or repelling electrons. As mentioned before, the accumulation of charge takes place only inside the micropores while the macropore solution remains electroneutral. The accumulation of charge is therefore related to the development in each porous electrode of a potential drop between the microporous and macroporous solutions (namely the electrode potential defined by the Donnan model). As the electrodes are accumulating ionic charge, the electric potential recorded in the cell decreases. Assuming that the electrodes are electrically connected, there will be a net ionic current and, simultaneously, an electric current flowing through the conductors, i.e., electrons going from one electrode to the other. The capacitive cell electrically behaves as a capacitor, and the system can be explained using a RC equivalent circuit, as illustrated in Fig. 7. The membranes are modeled considering a voltage source (or power supply) and an internal resistance. The magnitude of the voltage source depends on the concentration gradient, and the internal resistance depends on the conductivity of the membrane. The porous electrodes are modeled considering also an internal resistance in series with a capacitor. Among the different contributions for the internal resistance, we assume in our present model that those terms related to the conductivity through the membranes will be the most significant. Similar to a RC circuit, the electrical voltage and the ionic current registered between electrodes (through the external load) are both monotonically decreasing functions. It means that, as the electrodes get charged, the cell potential and the ionic and electric current decrease. After a certain time, when the current has vanished, the system is switched to the discharging step, where dilute electrolyte (air-flushed solution) is fed to the cell. In the discharging step, the air-flushed solution fed to the system is less concentrated than the solution in the porous electrode, which implies that the ionic flux is reversed compared to the charging step. The membrane potential, therefore, changes its sign. The porous electrodes release the accumulated charge in the micropores while the surface potential decreases. Electrons are therefore transferred from cathode to anode and electric current is again produced. The difference of concentration between the dilute and concentrate solutions with respect to the solution inside the electrodes may and will probably be different, and it is also not constant in time due to, for example, the diffusion of the carbonic acid. Consequently, the charging and discharging cycles are not symmetric.

7. Maximum potential – open-circuit conditions The open-circuit voltage (OCV) is the difference of electrical potential when an infinitely large resistance is placed in the external

Fig. 7. Equivalent electrical circuit of the membrane-enhanced capacitive cell.

205

circuit between the two electrodes. Since the electrical current is zero, there is no conversion between the electrical and the ionic current, and therefore ions cannot freely migrate across the membranes. Without ionic current, the OCV is approximately the sum of the two membrane potentials. The maximum OCV would be obtained in the ideal case of having the membrane separating the dilute (air-flushed) and the concentrate (CO2-flushed) solutions, ignoring the diffusion of carbonic acid and assuming permselectivities of 100% for the membranes. In a real case, all these effects will reduce the maximum value of the OCV, and will make it change in time. In the case of using pure water as electrolyte, we can assume that the bicarbonate ions and the protons are in a concentration much higher than hydroxide and carbonate ions, so the membrane potential can be obtained assuming a single mobile ion migrating through the membrane. According to the results in Fig. 2, the proton and bicarbonate ions concentration is 124 lM in the concentrate solution and 2.3 lM in the dilute solution (approximately the same for both ions). Using Eq. (12), each membrane would develop a potential of approximately 100 mV, resulting in a total membrane potential of 200 mV. In Ref. [1], the maximum experimental OCV obtained was about 90 mV, which is markedly lower. In the case of MEA solution, and according to the equilibrium concentrations shown in Fig. 3, in the concentrate solution (CO2flushed) there is also a predominant concentration of one of the species (ethanolammonium and bicarbonate). Accordingly, the simplified equation for the membrane potential assuming the diffusion of only one mobile counterion can also be used. For the case of the AEM, using bicarbonate concentrations of 19.6 mM and 233 mM, the theoretical membrane potential obtained from Eq. (12) is 62 mV. For the case of the CEM, based on ethanolammonium concentrations of 94.3 mM and 242 mM, the membrane potential obtained is 24 mV. Accordingly, the total OCV for the cell would be about 86 mV, in line with the experimental results for MEA reported in Ref. [1]. It should be noted that the CO2 (g)/CO2 (aq) gas–liquid equilibrium is a slow reversible process with respect to the rest of the equilibrium reactions. In the capacitive cell discussed in this research, the CO2 saturated solutions are obtained prior to being pumped into the cell. Nevertheless, it is possible that the airflushed MEA solution used in the experiments reported in Ref. [1] was not in equilibrium (as assumed for the calculations presented above). Based on the results shown in Fig. 3, the equilibrium pH of an air-flushed MEA solution (0.25 M) should be around 9.4, while in Ref. [1] the pH of that solution was about 10.6. MEA solutions have a great tendency to absorb CO2, but in the case of using a gas with low CO2 concentration, the equilibrium may be limited by the kinetics of the absorption process. To test this idea, in Fig. 8, results for the cell OCV are shown using Eq. (12) and considering the dilute solution in a range from equilibrium with air (pH  9.4) and almost infinite dilution (i.e. MEA solution that has not been flushed with air) (pH  11.4). For pH 10.6, the theoretical OCV would be around 200 mV. This value is similar to

Fig. 8. Theoretical OCV as a function of the pH of the dilute solution (in the range of MEA in equilibrium with air and pure MEA) in the case of using MEA as electrolyte.

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For a very dilute solution, the OCV increases until a value of approximately 450 mV. In the light of this result, it can be concluded that when using an alkaline substance such as MEA to increase the CO2 absorption, it may be convenient to use a dilute solution that is far from the equilibrium point for the CO2 absorption from air, i.e., pure MEA solution without any initial CO2 content. However, as mentioned before, the transport of the non-ionic carbonic acid (or non-hydrated CO2) through the membranes will limit this potential. The concentration of carbonic acid, and its constitutive ions, in the electrode pores will change during the process. It is expected that the concentration of the electrode pore solution will be somewhere in between the air-flushed and the CO2-flushed solutions, depending on the duration of the cycles.

(a)

(b)

8. Simulation results

Fig. 9. Electrical potential in the cell in open circuit conditions during a charging step of 5 min in pure water. Color lines (online version) stand for intermediate profiles and the solid black line stands for the 5th minute profile. (a) Case with membranes permeable to H2CO3 and (b) case with membranes impermeable to H2CO3.

that obtained when deionized water, and it is therefore consistent with the fact that in both cases the maximum OCV experimentally observed was in both cases about 90 mV.

In this section, simulation results are shown for the open circuit voltage computation of the modeled system. A one-dimensional Galerkin finite element method [38] was implemented and used for the numerical solution of the coupled system of algebraic and differential equations in Eqs. (2) and (3). The membranes are modeled as thin porous regions (p = 0.6, s = 10, 40 lm thick) with a fixed charge density of xX = 1 M (referred to the volume of electrolyte, with positive sign for the AEM and negative for the CEM). The transport resistance related to the flow channel and the macropores of the porous electrodes is considered negligible with respect to that in the membranes. Accordingly, the electrolyte region in the porous electrodes and in the flow channel are modeled using a tortuosity factor of s = 103, which results in a behavior similar to a stirred tank for these regions. Chemical reactions are solved assuming local chemical equilibrium. A sequential two-step method for equilibrium transport models is used, by combining the finite element solution with a Newton–Raphson method for multispecies chemical equilibrium [18] between each time step in the numerical integration. Dirichlet boundary conditions are fixed in

(a)

(b)

Fig. 10. Transient profiles in a 5 min charging step in pure water in open circuit conditions. Same simulation as Fig. 9a. (a) The bicarbonate ions and (b) the carbonic acid. Color lines stand for intermediate profiles and the solid black line stands for the 5th minute profile. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

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the middle point in the spacer channel of the one-dimensional domain. This is done to simulate that the constant flow of fresh solution into the spacer channel keeps the concentration of electrolyte at that position at a constant value. Fig. 9 shows the development of the OCV for the charging step, assuming the dilute (air-flushed) solution in both electrode electrolytes, and the concentrate (CO2-flushed) solution in the flow channel. Two cases are compared: membranes permeable and impermeable to the non-ionic carbonic acid. The maximum OCV obtained in the simulation is around 90 mV in both cases, which matches with the experimental results shown in Ref. [1]. However, in the more realistic case of having membranes permeable to carbonic acid, (case a), the membrane potentials related to both membranes (AEM and CEM) decrease faster than in the ideal case of having impermeable membranes. In both cases, the membrane potential associated with the AEM decreases due to the increasing pH in the electrode compartment, that produces a flux of hydroxide ions toward the flow channel which counteracts the flow of bicarbonates and gradually equilibrates the solutions. Moreover, in the AEM, the high pH in the membrane results in a substitution of hydroxide by bicarbonate ions, according to:

H2 CO3 þ OH ! HCO3 þ H2 O

ð13Þ

This process will take place only during the first cycles, and will lead to a highly carbonated membrane with lower pH, and lower conductivity. Fig. 10 shows the time-transient profiles of bicarbonate ions in the charging step for the case of having membranes permeable to the carbonic acid. In Fig. 10a, the formation of bicarbonates inside the AEM due to the process shown in Eq. (13) is shown. Fig. 10b, shows that the carbonic acid diffuses and equilibrates the solutions in both electrodes, but the diffusion rate through the membranes is different.

9. Conclusions In the present work, a theoretical discussion has been given for the process of harvesting energy from CO2 emissions using membrane-enhanced capacitive cells. In this study, we focused on the different physicochemical aspects that affect the open circuit voltage (OCV), which is the maximum voltage that the cell can supply, and it is directly related to the power density and, therefore, to the energy efficiency. In a membrane-enhanced capacitive cell, the OCV is related to the membrane potential, which is determined by the difference in ionic concentration across the membrane. In this work, we discussed theoretically, and showed with numerical simulations, that the membrane potential in the carbonated system is not constant in time. Two different transport mechanisms have been discussed here as the main ones responsible for the decrease of the membrane potential: (1) the counterion exchange between hydroxide and bicarbonate ions through the AEM, and (2) the non-ionic H2CO3 transport through both membranes. Simulation results are in good agreement with the experimental results obtained previously [1]. Moreover, we explain the experimentally obtained OCV-values for the case of 0.25 M MEA-solution as the electrolyte, which may help to optimize the technique of harvesting energy from CO2-emissions, and increase the efficiency of the process.

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