Journal of Membrane Science 478 (2015) 37–48
Contents lists available at ScienceDirect
Journal of Membrane Science journal homepage: www.elsevier.com/locate/memsci
Efficiency of electrochemical gas compression, pumping and power generation in membranes Jacopo Catalano a, Anders Bentien a,n, David Nicolas Østedgaard-Munck a, Signe Kjelstrup b a b
Department of Engineering, Aarhus University, Hangoevej 2, 8200 Aarhus N, Denmark Department of Chemistry, Norwegian University of Science and Technology, 7491 Trondheim, Norway
art ic l e i nf o
a b s t r a c t
Article history: Received 22 August 2014 Received in revised form 17 December 2014 Accepted 22 December 2014 Available online 1 January 2015
The electrokinetic effects in membranes and porous materials can be used for direct conversion of pressure into electricity or conversion of electricity into pumping power, and they have potential applications within actuators, small scale pumping and energy harvesting devices. Still, in the literature only electrokinetic effects with liquid reservoirs on each side of the membrane are considered. In the present paper, isothermal electrokinetic effects are investigated using non-equilibrium thermodynamics in the case with gas phase reservoirs on each side of the membrane. For comparison the case with liquid reservoirs is included. We describe how the figure-of-merit, energy conversion efficiency, power density, compressor and generator curves depend on observable transport properties. The derived equations are used to analyse two examples with gas phase reservoirs. The first being related to electrochemical hydrogen compression/liquefaction, and the second being the electrochemical gas compression for cooling cycles. In the latter case, experimental transport data predict high efficiency and a power density which is promising with respect to membrane-based electrochemical cooling applications. & 2015 Published by Elsevier B.V.
Keywords: Proton exchange membrane Electrochemical gas compression Power generation Electrokinetic
1. Introduction Electrokinetic effects in porous materials and membranes arise due to the coupling between the movement of ions and solvent. In e.g. water such a coupling can be explained and modelled by electrostatic interactions between charged ions and polar water molecules. Electrokinetic effects can be utilised for direct conversion of potential or kinetic energy (e.g. Pressurised liquids and gases) into electrochemical energy or vice versa [1]. This is potentially attractive for many applications [2] such as microfluidic pumps, gas compressors for cooling cycles, small generators in e.g. domestic compressed air energy storage [3] or heat engines e.g. organic Rankine cycles [4]. The Saxèn relations have been known since long, but specific studies of electrokinetic energy conversion were done first in the 1960s by Osterle and co-workers [1,5] and Burgreen and Nakache [6,7]. They predicted a maximum first law efficiency (ηEK ) of the order 1–3% and up to 17%, respectively. During the past decade renewed interest in this topic has emerged; and most reports focus on the transport properties of straight nanochannels with well-defined dimensions [8–23]. Recently [2,24], experimental
n
Corresponding author. E-mail address:
[email protected] (A. Bentien).
http://dx.doi.org/10.1016/j.memsci.2014.12.042 0376-7388/& 2015 Published by Elsevier B.V.
values of the elecktrokinetic figure-of-merit [1,2,24] have shown that efficiency of about 20% can be obtained in commercial Nafion membranes and it is increasing with temperature. Up to now all theoretical studies of electrokinetic energy conversion have only considered incompressible fluids and in particular aqueous solutions. However, gas phase electrokinetic energy conversion is possible too and the potential applications are electrochemical gas compression or power generation (expander). Some experimental works have been reported on electrochemical compression of H2 through a membrane [25–29], with focus on high pressure liquefaction of H2. Electrochemical gas compression in membranes for cooling applications is described in the patent literature, e.g. Refs. [30–32]. None of the patents disclose technical details or theoretical considerations with respect to e.g. efficiency and power density. Fig. 1 gives a schematic illustration of electro-osmosis with liquid (left) and gas reservoirs (right). In the first case, an ion, say Li þ , migrates through the membrane due to an external electric potential (ϕ) difference. Each Li þ couples with a number of water molecules, that are quantified by the water transport number (t w ). In popular terms one may say that Li þ is dragging or pumping water through the membrane as it is moving in the electric field. Alternatively, a gas phase can supply the conducting ion by gas oxidation and subsequent reduction on the two membrane sides.
38
J. Catalano et al. / Journal of Membrane Science 478 (2015) 37–48
be included emperically as overpotentials as will be done later in the discussion.
2. Electrokinetic energy conversion with liquid phase reservoirs 2.1. Transport equations
Fig. 1. Schematic representation of electrokinetic pumping (left) and gas compression (right) across a membrane (M) having thickness Δx ¼ x2 −x1 . Electrodes (E) generate an external electrical potential difference (Δϕ ¼ ϕ2 ϕ1 ) across the membrane that results in an ion current density (j) through the membrane. In the liquid phase a volume flux (J v ) and in the vapour phase a gas flux (J n ) arises due the coupling between the ion and the solvent. During operation a pressure difference (Δp ¼ p2 p1 ) between the two sides is present. In a practical system the pump could be designed similar to that of a redox flow battery where faradaic processes takes place in the whole volume of the porous (e.g. carbon felt) electrode. The compressor could be designed similar to that of a fuel cell with a membrane electrode assembly. Here the faradaic processes only take place in the catalyst layer (C) between the membrane surface and the electrode (electrically conductive gas diffusion layer).
The membrane surface is then covered with an electrode/catalyst. Examples of such gas reactions are 1=2 H2 ⇆H þ þe , 1=2 Br2 þ e ⇆Br or 1=2 Cl2 þ e ⇆Cl . Other gas molecules, e.g. water, methanol or ammonia, in the gas phase can dissolve in the membrane and be transported along with the ion. This is quantified by the gas transport number (t n ) and in cooling cycles, this transported fluid acts as a refrigerant. Since electrokinetic pumping/compression with liquid or gas reservoirs are reversible processes an external pressure difference can be converted into an electrical power. The basic theoretical description [1,5] of electrokinetic energy conversion is obtained from non-equilibrium thermodynamics. The theory has been used to find a general expression for maximum efficiency that increases with the electrokinetic figure-of-merit (β ¼ ðν2 σ Þ=κ H ), where ν is the streaming potential coefficient, σ is the ion conductivity and κ H is the hydraulic permeability of the membrane [1,5]. One of the aims of the present article is to explore this theory in more details. In particular, we want to define pump and generator curves, and find requirements for maximum efficiency, power density, maximum pressure (pump head) etc. These parameters are very important for assessing electrokinetic energy conversion with respect to various applications. Still, the major aim of the current article is to formulate the transport equations for the gas phase in a practically useful way. The theory of non-equilibrium thermodynamics is indispensable when it comes to correctly define the coupling phenomena, which are central in electrokinetic energy conversion. This includes the derivation of a general figure-of-merit and maximum efficiency along with other important parameters e.g. compressor and generator (expander) curves, requirements for maximum efficiency, power density, maximum pressure etc. Despite applications within the field of electrokinetic H2 compression and the many patent applications for electrochemical cooling cycles, a theoretical basis has not yet been reported in the literature. The following two theoretical sections cover electrokinetic energy conversion with liquid and gas phase reservoirs, respectively, and can be read independently. They are followed by a section that discusses the significance of the results along with estimates of efficiency of electrochemical gas compression based on transport data from the literature. The theory does not include effects from mass transport and catalytic activity in electrodes and electrode membrane interface. It can however
Fig. 2 shows a circuit diagram of electrokinetic energy conversion. The central unit is a membrane, characterised by the ion conductivity (σ ) and hydraulic permeability (κ H ). Electric power can be generated (Fig. 2 left) when a pressure difference (Δp ¼ p2 p1 ) is applied across the membrane, resulting in a volume flux (J v ) through the membrane. Due to the coupling between the movement of solvent and ions an electric current density (j) and a potential difference (Δϕ ¼ ϕ2 ϕ1 ) are induced in the electrical circuit, where electrical power can do work in an variable external load (RL ). In the pumping operation (Fig. 2 right), an electric potential difference across the membrane induces a current density along with a volume flow and a pressure difference arises across the membrane. The hydraulic work can be dissipated within a variable hydraulic load/resistance (RL ). An isothermal system, at a fixed temperature T, exposed to an electric and a mechanical driving force obeys the following phenomenological transport Eqs. [1,5,33,34]. 1 Δp 1 Δϕ þLvϕ ð1Þ J v ¼ Lvv T Δx T Δx 1 Δp 1 Δϕ þ Lϕϕ j ¼ L ϕv T Δx T Δx
ð2Þ
The direction and magnitude of the volume flux (J v ) and current density (j) is determined by the transport coefficients (Lij ) where the off-diagonal coefficients (written with appropriate units) obey the Onsager relation Lvϕ ¼ Lϕv . In the following, the phenomenological transport coefficients are related to observable transport coefficients. The electric
Fig. 2. Diagram of electrokinetic power generation/expander (left) and compression (right) across a membrane. The system is charachterised by an ion conductivity (σ), electrical current density (j), gas permeability (κ n ) and gas flux (J n ). During power generation (left) the pressure difference (Δp ¼ p2 p1 ) is negative and the potential difference (Δϕ ¼ ϕ2 ϕ1 ) is positive. For compression the signs are reversed. With this convention j and J n are always positive. The streaming potential coefficient (not shown in figure) is negative for cation conductive membranes and postive for anion conductive membranes. The arrows at the top and bottom indicate wheter electric power (P el ) or compression/expansion work (P comp ) is inserted to or extracted from the system. The case for liquids reservoirs on each side is fully equivalent, here κ n and J n are replaced with the hydraulic permeability (κ H ) and volume flux (J v ), respectively. While P comp is replaced with the hydraulic power (P hyd ¼ J v ðΔp=ΔxÞ).
J. Catalano et al. / Journal of Membrane Science 478 (2015) 37–48
conductivity is defined for uniform pressure, as Lϕϕ jΔx σ¼ ¼ T Δϕ Δp ¼ 0
ð3Þ
The hydraulic permeability (κ H ) is defined for zero current density (j ¼ 0), and is: ! Lvϕ Lϕv J v Δx Lvv 1 κH ¼ ¼ ð4Þ T Lvv Lϕϕ Δp j ¼ 0 The streaming potential is related to the transport coefficients through: L ϕv Δϕ ν¼ ¼ ð5Þ Δp j ¼ 0 Lϕϕ and can be related to the salt (t s ) and water (t w ) transport numbers by ν ¼ t s V s þ t w V w þ ΔV =F [33–35], where V s and V w are the molar volume of the salt and water, respectively, and ΔV is the volume transferred by the electrode reaction. The value of ΔV is often known, and can be added to the sum t s V s þ t w V w . The transport Eqs. (1) and (2) can now be rewritten in terms of observable transport coefficients (Eqs. (3)–(5)): Δp Δϕ Jv ¼ κH 1 þ β þ σν Δx Δx
ð6Þ
Δp Δϕ j ¼ σν σ ; Δx Δx
ð7Þ
where β ¼ ðν2 σ Þ=κ H is the electrokinetic figure-of-merit in the case with liquid reservoirs on each side. 2.2. Pump and generator curves
39
In the power generation mode, a constant Δp=Δx is used as driving force, alternatively this can be a constant J v , and again this does not change the generality of the analysis. In this case Δϕ=Δx decreases linearly with current density and Δϕ=Δx ¼ 0 at jgen ¼ 1, where the external electric load is zero. With respect to J v , it seen that it increases linearly with current density and can be explained by the lower hydraulic resistance due to a change in the electroviscous effect (electrolyte drag effects) when the electrical load is lowered. The equations for the pump and generator curves along with the maximum and minimum values of the potentials and fluxes are collected in Table 1. Electro-viscous effects can be observed experimentally as changes in the flow or current whenever the external electrical or hydraulic load is changed, respectively. An interesting point is that the ratios Δϕ=Δx J ¼ 0 = Δϕ=Δx Δp ¼ 0 ¼ v J v;Δϕ ¼ 0 = J v;j ¼ 0 ¼ 1 þ β, that define the electro-viscous effects, depend on the elecktrokinetic figure-of-merit only, and can be used for a direct experimental determination of β . For instance measurements of J v;Δϕ ¼ 0 and J v;j ¼ 0 correspond to measurement of κ H with and without electro-viscous effects, respectively. In a practical setup this corresponds to measurement of κ H where the two sides of the membrane are electrically short- and open circuited, respectively. Similarly, Δϕ=Δx J ¼ 0 and Δϕ=Δx Δp ¼ 0 v correspond to measurements of the ion conductivity of the membrane with infinite and zero hydraulic resistance, respectively. A figure-of-merit above 1 has been found experimentally in Nafion [2,24], and significant differences between e.g. Δϕ=Δx J v ¼ 0 and Δϕ=Δx Δp ¼ 0 are thus expected.
2.3. Efficiency and power density
For evaluating the performances and efficiency of generators or pumps, the corresponding generator or pump curves must be known. For generators the performance is characterised by the ðΔϕ=ΔxÞ j curve, while pumps are characterised by the ðΔp=ΔxÞ J v curve. These curves are equivalent to e.g. the Δϕ j curve of a fuel cell or battery. By rearrangement and mutual insertion of Eqs. (6) and (7), one can write:
Δp 1 1 β 1 Δϕ 1 ν J þ ¼ ¼ Jv j κH Δx κH 1 þ β ν Δx 1 þ β κH v
ð8Þ
1 Δϕ Δp j ν ¼ν ¼ J 1þβ j σ κH v Δx Δx σ
ð9Þ
This gives the pump (Eq. (8)) and generator (Eq. (9)) curve for electrokinetic energy conversion. The middle parts of Eqs. (8) and (9) express the curves in terms of potential/pressure difference along with a flux, while the right hand sides are expressed as functions of the fluxes only. The pump and generator curves are shown in Fig. 3, where the dimensionless volume flow, J pump ¼ ðJ v =jνÞ, and the dimensionless current density, jgen ¼ j=ðσνΔp=ΔxÞ are introduced. In an experimental setup the fluxes J v or j are varied by adjusting the external hydraulic or electrical resistances (RL in Fig. 2) in pump or generator operation, respectively. In a pumping mode, the electric input power can be either a constant current density or constant potential difference. Here a constant current density is used, however, the outcome of the following analysis with respect to efficiency, power density etc. is independent of this choice. From Fig. 3 it is seen that Δp=Δx decreases linearly with J pump up to J pump ¼ 1, where Δp=Δx ¼ 0. Since j is constant, Δϕ=Δx varies with the volume flux and reaches a minimum value at J pump ¼ 1 whenever the external hydraulic resistance (Fig. 2) is zero.
For both pumping and power generation the energy inserted into or work extracted from the membrane system is the area under the pump or generator curve. In terms of volume power density (P), power per unit volume, it is
P hyd ¼ J v
2
Δp Δp ¼ κH Δx Δx
Δp J 2 J jν ¼ v v κH κH Δx
jν
J jν Δϕ j2 Δp j2 ¼ þ jν ¼ 1þβ v σ Δx σ Δx κH
P el ¼ j
ð10aÞ
ð11aÞ
where the subscripts refer to hydraulic and electric power, respectively. A negative value of P corresponds to work that is done on the membrane system while a positive value corresponds to work that can be extracted from the membrane system. The area power density is found from P Δx. The lower part of Fig. 3 includes the power density curves, Eqs. (10b) and (11b), that explicitly are P hyd ¼
j2 ν2
κH
P el ¼ βκ H
2 Δp J pump J 2pump ¼ κ H 1 þ β jgen Δx
Δp Δx
2
i j2 h jgen j2gen ¼ 1 þ βð1 J pump Þ
σ
ð10bÞ
ð11bÞ
in terms of the dimensionless fluxes. For pumping it is seen that the electric power introduced into the system decreases with J pump . The hydraulic power output shows a maximum output for J pump ¼ 1=2. For power generation, similar curves are obtained, here P hyd increases with jgen and the output P el shows a maximum at jgen ¼ 1=2. The electrokinetic energy conversion efficiency (η) is given as the ratio between the amount of energy that is extracted from the
40
J. Catalano et al. / Journal of Membrane Science 478 (2015) 37–48
Fig. 3. (Top) Pump (top left) and generator (top right) curves for electrokinetic energy conversion as function of the dimensionless fluxes J pump ¼ ðJ v =jνÞ and jgen ¼ j=ðσνΔp=ΔxÞ. For pump operation it is assumed that j is constant whereby Δϕ=Δx is variable (right y-axis of the pump curve). For generator operation a constant Δp=Δx is assumed, whereby J v is variable (rigth y-axis of the generator curve). Equation numbers refer to equations in text. Specific values of Δp=Δx, Δϕ=Δx and J v have been included for J pump and jgen at values 0 and 1 have been included and can also be found in Table 1. (Bottom) Power density curves for pump (bottom left) and generator (bottom right) operation as function of the dimensionless fluxes. The electric power input in pumping and hydraulic power input in power generation is dependent on the dimensionless flux and is a consequence of the value of the external loads. The graphs also show the efficiency (η) (rigth y-axis). The conditions for maximum power density and maximum effiencieny are indicated by dotted lines and the conditions for this can be found in Table 1 along with the corresponding expressions for the power and efficiency. Values of Δp=Δx, Δϕ=Δx and J v at maximum power density and maximum effiency can be derived from the equations for the pump (Eq. (8)) and generator curves (Eq. (9)).
Table 1 Collection of equations that characterise electrokinetic energy conversion in the liquid phase based on measurements of ion conductivity (σ), hydraulic permeability (κ H ) and streaming potential coefficient (ν). Left side is with respect to pump operation, while right side is with respect to power generation. Generator operation with constant Δp.
Pump operation with constant j Pump curve
Δp Δx ¼
κ1H J v κνH j
J pump ¼ 0-J v ¼ 0 Δp jν Δx J v ¼ 0 ¼ κ H (pump head) Δϕ j Δx J ¼ 0 ¼ σ ð1 þ β Þ v
Generator curve J pump ¼ 1-Δp ¼ 0
Δp j Δϕ Δx ¼ νΔx σ
jgen ¼ 0 -j ¼ 0 Δϕ Δp Δx j ¼ 0 ¼ νΔx (open circuited potential)
J v;Δp ¼ 0 ¼ jν (max flow) Δϕ j Δx Δp ¼ 0 ¼ σ
J v;j ¼ 0 ¼
κ H Δp Δx
jgen ¼ 1 -Δϕ ¼ 0 jΔϕ ¼ 0 ¼ σνΔp Δx (max current) J v;Δϕ ¼ 0 ¼ κ H ð1 þ βÞΔp Δx
Figure-of-merit β ¼ νκHσ 2
Dimensionless volume flow J pump ¼ Jjνv
j Dimensionless current density jgen ¼ σνΔp=Δx
Maximum power density is obtained at J pump;MaxP ¼ 1=2
Maximum power density is obtained at jgen;MaxP ¼ 1=2 2 ηgen;MaxP ¼ 2ð2βþ βÞ P el;MaxP ¼ βκ4H Δp Δx pffiffiffiffiffiffiffiffi 1þβ1 Maximum efficiency is obtained at jgen; Max η ¼ β pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi Δp2 2 þ 2 1 þ β β þ β2 1þβ1 ηgen; Max η ¼ pffiffiffiffiffiffiffiffi P el; Max η ¼ κ H
ηpump;MaxP ¼ 2ð2βþ βÞ pffiffiffiffiffiffiffiffi 1þβ Maximum efficiency is obtained at J pump; Max η ¼ pffiffiffiffiffiffiffiffi 1þ 1þβ p ffiffiffiffiffiffiffi ffi pffiffiffiffiffiffiffiffi 2 β 1þβ 1þβ1 P hyd; Max η ¼ jσ pffiffiffiffiffiffiffiffi2 ηpump; Max η ¼ pffiffiffiffiffiffiffiffi 2
P hyd;MaxP ¼ βj 4σ
1þ
1þβ
Δx
1þβþ1
J 2pump þ J pump J pump þ 1 þ β =β
ð12Þ
while for power generation
ηgen ¼
P el ¼ P hyd
j2gen þ jgen
jgen þ ð1=β Þ
1þβþ1
and the condition for maximum efficiency is
system and the amount inserted and for pumping it is
P ηpump ¼ hyd ¼ P el
β
pffiffiffiffiffiffiffiffiffiffiffi 1þβ pffiffiffiffiffiffiffiffiffiffiffi J pump ¼ 1þ 1þβ and pffiffiffiffiffiffiffiffiffiffiffi 1þβ1 jgen ¼
β
ð13Þ
The efficiency curves are also plotted in the lower parts of Fig. 3. The maximum efficiency (ηmax ) is found by standard mathematical methods with differentiation of Eqs. (12) and (13),
ð14Þ
ð15Þ
for pumping and power generation, respectively. In both cases the maximum efficiency is given by pffiffiffiffiffiffiffiffiffiffiffi 1þβ1 ηmax ¼ pffiffiffiffiffiffiffiffiffiffiffi 1þβþ1
ð16Þ
J. Catalano et al. / Journal of Membrane Science 478 (2015) 37–48
These results are in accordance with literature [1,5,36], however, with the difference that the conditions for maximum efficiency (Eqs. (14) and (15)) are explicitly shown only here. The conditions for maximum efficiency, maximum power density along with the efficiencies and power densities are also included in Table 1.
3. Electrokinetic energy conversion with gas phase reservoirs
41
number: Δϕj ¼ 0 ¼
Lϕn p tn p RT ln 2 ¼ RT ln 2 Lϕϕ p1 F p1
ð18Þ
With t n ¼ 1=2 for H2 this gives Δϕj ¼ 0 13 mV ln p2 =p1 and is in agreement with experimental literature data [25]. Furthermore, the streaming potential coefficient (νn ) in the gas phase " # L ϕn Δϕ νn ¼ ¼ RT ð19Þ Lϕϕ ln p2 =p1 j¼0
3.1. Phenomenological transport equations This section is outlined in the same manner as the previous section where the liquid phase reservoirs were used. The basic principles of electrokinetic energy conversion in the presence of gas phases were shown in Figs. 1 and 2. As an example, we consider H2 and methanol (MeOH), in the gas phase on each side of the membrane, at different total and partial pressures. At the left hand side of the membrane/electrode surface 12 H2 dissociates into H þ ð12 H2 -H þ þ e Þ. The proton migrates through the membrane due to the externally applied electrical field. At the other side of the membrane the opposite reaction occurs (H þ þ e -12 H2 ) and H2 is formed. Due to electrostatic interactions, MeOH molecules are transported along with the proton through the membrane. Electrokinetic gas compression and power generation in the gas phase has, to the best of the authors' knowledge, not been described in terms of non-equilibrium thermodynamics. A rigorous derivation of simplified phenomenological transport equations is given in Appendix A. The result is R p 1 Δϕ J n ¼ Lnn ln 2 ðA6Þ þ Lnϕ Δx T Δx p1 R p 1 Eϕ ln 2 j ¼ Lϕn þ Lϕϕ Δx T Δx p1
ðA7Þ
where Lnϕ ¼ Lϕn through the Onsager relation and J n is the total molar gas flux. In the example shown in Fig. 1 J n ¼ J H2 þ J g , where J g is the molar flux of e.g. MeOH. It is assumed that the gases are ideal and that the compositions on both sides of the membrane are the same. In a practical application, the latter may not always be the case; as the concentrations of the transported ion and gas may change over time. This accompanying change in the chemical potential must then be included into the equations, but they are left out here for simplicity. In the following the phenomenological transport coefficients (Lij ) in Eqs. (A6) and (A7) will be related to observable transport coefficients using the definitions of the observable transport coefficients. Electroosmosis is defined as the number of molecules transported across the membrane along with the electric current ðJ n =jÞ in the case where there is no trans-membrane pressure difference ðΔp ¼ p2 p1 ¼ 0Þ. From Eqs. (A6) and (A7) this gives Lnϕ t n Jn ð17Þ ¼ ¼ j Δp ¼ 0 Lϕϕ F where F is the Faraday constant and t n is the gas transport number, which is the total number of gas molecules transported across the membrane per unit of charge. In comparison to the case with liquid reservoirs, t n also includes the H2 while t w is only the number of water/solvent molecules that are dragged through the membrane. If only H2 is present and it is assumed that H2 does not diffuse through the membrane, t n ¼ 1=2. The open-circuited potential (j ¼ 0), the streaming potential that arises whenever a pressure difference is present, is also found from Eqs. (A6) and (A7) and the definition of gas transport
is defined. The ion conductivity of the membrane is defined for uniform pressure, as Lϕϕ jΔx ð20Þ σ¼ ¼ T Δϕ Δp ¼ 0 The total gas permeability of a gas mixture of non-interacting gases is normally defined as being a linear combination of the single-gas permeabilities and the respective driving forces: i.e. P P J n ¼ J i ¼ κ gas;i ðΔpsurf =ΔxÞ in which Δpsurf represents the i i i i partial pressure difference of the species i between the membrane surfaces. Considering the case of a binary mixture, no concentration polarisation (psurf ¼ pi ¼ pyi ) and the same gas concentrations i in both sides (y ¼ y as: i;1 i;2 for i ¼ 1; 2), the gas flux simplifies J n ¼ y1 κ gas;1 þ 1 y1 κ gas;2 p2 p1 ¼ κ gas p2 p1 in which κ gas is a lumped permeability coefficient. However, for electrokinetic energy a more convenient definition is as conversion function of ln p2 =p1 . The gas permeation for zero current density (j ¼ 0) where both sides of the membranes are electrically isolated from each other, is found from Eqs. (A6) and (A7): ln p2 =p1 ð21Þ Jn ¼ κn Δx where the gas permeability coefficient is defined as ! " # Lnϕ Lϕn J n Δx κn ¼ ¼ Lnn 1 R Lnn Lϕϕ ln p2 =p1
ð22Þ
j¼0
If the ratio p2 =p1 is sufficiently small Eq. (21) can be Taylor expanded to give J n ¼ κ n ð1=p1 ÞðΔp=ΔxÞ. For practical purposes an experimentally determined/literature value of the gas permeability obtained from J n ¼ κ gas ðΔp=ΔxÞ can be translated into κ n from the expansion, κ n ¼ p1 κ gas , where the units of κ gas and κ n are [mol m 1 Pa 1 s 1] and [mol m 1 s 1], respectively. Alternatively, κ n can be estimated from Eq. (21) provided that J n , p2 and p1 are given, which often is the case in the literature. According to non-equilibrium thermodynamics theory, the phenomenological coefficients in the flux-force relations do not depend on the fluxes or forces themselves. They may depend on state variables, such as temperature or pressure. Any experimental deviation from a linear J n vs ln p2 =p1 relationship can be caused by such a dependence, which will appear as a consequence of the integration across the membrane (Eq. (A5)). We have assumed that the gasses are ideal in the integration, while say, a temperature dependence, may appear as a non-linearity after integration. This will also mean that compression and power generation efficiency becomes pressure dependent. By introducing the definitions of the observable transport coefficients the transport Eqs. (A6) and (A7) can be rewritten as ln p2 =p1 t n σ Δϕ Jn ¼ κn 1 þ β ð23Þ F Δx Δx t n σ RT ln p2 =p1 Δϕ j¼ σ ð24Þ F Δx Δx where β ¼ ðRT=F 2 Þðt 2n σ =κ n Þ is defined as the figure-of-merit for electrokinetic energy conversion with gas phase reservoirs.
42
J. Catalano et al. / Journal of Membrane Science 478 (2015) 37–48
3.2. Compressor and generator (expander) curves The performance and efficiency of the electrokinetic generator/ gas expander can be evaluated from the Δϕ=Δx j characteristics. For electrokinetic gas compression theperformance and efficiency is evaluated from the ðRT=ΔxÞln p2 =p1 J n curve. The area under this curve gives the (reversible) work per unit time and volume that is done on the gas. From rearrangement and introduction of Eqs. (23) and (24), the compressor and generator curves are found: ln p2 =p1 1 1 F β 1 Δϕ 1 tn J ¼ ¼ Jn þ j ð25Þ κn F κn Δx 1 þ β κ n n RT 1 þ β t n Δx Δϕ t n RT ln p2 =p1 j t n RT ð1 þ β Þ ¼ j ¼ J Δx F σ κn F n σ Δx
ð26Þ
The middle part of Eqs. (25) and (26) gives the pressure and potential difference as function of the fluxes and the potentials Δϕ=Δx and ðRT=ΔxÞln p2 =p1 , while the right hand side of the equations gives the curves as function of the fluxes J n and j only. With Eqs. (25) and (26) the compressor and generator curves for electrokinetic energy conversion in the gas phase can be constructed and they have the same functional behaviour as the ones for liquid reservoirs (Eqs. (8) and (9)) and are shown in Fig. 4, where they are plotted as function of the dimensionless gas flow (J comp ¼ J n F=ðjt n Þ) and the dimensionless current density (jgen ¼ −ðjFΔxÞ=ðσt n RT lnðp2 =p1 ÞÞ). The explicit dependence of Eqs. (25) and (26) on J comp and jgen can be found by inserting the expressions for into the equations. As for liquid reservoirs, and without loss of generality, it is assumed that the constant driving force for compression is a constant j while Δϕ=Δx is variable. For generation it is assumed that the constant driving force is ln p2 =p1 while J n is variable. The equations for the pump and generator curves are collected in Table 2. From an experimental point-of-view electro-viscous effects are observed as changes in the flow or current whenever the
external electrical or compression load is changed, respectively. The ratios Δϕ=Δx J ¼ 0 = Δϕ=Δx Δp ¼ 0 ¼ J n;Δϕ ¼ 0 = J n;j ¼ 0 ¼ 1 þ β, n that define the electro-viscous effects depend on the electrokinetic figure-of-merit only and can be used for a direct experimental measurement of β . I.e. J n;Δϕ ¼ 0 and J n;j ¼ 0 correspond to measurements of κ n where the two sides of the membrane are electrically short- and open-circuited, respectively. Similarly, Δϕ=Δx J n ¼0 and Δϕ=Δx Δp¼0 correspond to the measurements of the ion conductivity of the membrane with infinite and zero external compression load, respectively.
3.3. Efficiency and power density The volume power density with respect to the amount of work that is done on gas for compression and generation is the area under the ðRT=ΔxÞln p2 =p1 J n and ðΔϕ=ΔxÞ j curves, respectively. The area power density can be calculated from P Δx. Explicitly, the equations found are P comp ¼ J n RT
2 ln p2 =p1 ln p2 =p1 ln p2 =p1 tn t n RT RT ¼ jJ J 2 ¼ jRT κ n RT Δx F Δx Δx F κn n κn n
ð27aÞ
1þβ 2 Δϕ j2 t n RT ln p2 =p1 t n RT J j ¼ j ¼ j F σ κnF n σ Δx Δx
P el ¼ j
ð28aÞ
As for the case of liquid reservoirs (Eqs. (10a), (10b), (11a) and (11b)), a negative value corresponds to external work that is inserted into the membrane while a positive value corresponds to work that can be extracted from the membrane. Eqs. (27a) and (28a) are also shown in the lower part of Fig. 4 and the explicit dependence on J comp and jgen can be found:
P comp ¼
ln p2 =p1 β ¼ κ n RT J J 2 σ comp comp Δx
2
jgen þ 1
ð27bÞ
Fig. 4. (Top) Compressor (top left) and generator (top right) curves for electrokinetic energy conversion as function of the dimensionless fluxes J comp ¼ J n F=ðjt n Þ and jgen ¼ −ðjFΔxÞ=ðσt n RT lnðp2 =p1 ÞÞ. For pump operation it is assumed that j is constant whereby Δϕ=Δx is variable (right y-axis of the pump curve). For generator operation a constant ln p2 =p1 is assumed, whereby J n is variable (rigth y-axis of the generator curve). Equation numbers refer to equations in text. Specific values of ln p2 =p1 , Δϕ=Δx and J n have been included for J comp and jgen at values 0 and 1 have been included and can also be found in Table 2. (Bottom) Power density curves for compression (bottom left) and generator (bottom right) operation as function of the dimensionless fluxes. The electric power input in compression and expansion power input in power generation is dependent on the dimensionless flux and is a consequence of the value of the external loads. The graphs also show the efficiency (η) (rigth y-axis). The condition for maximum power density and maximum effiencieny are indicated by dotted lines and the conditions for this can be found in Table 2 along with the corresponding expressions for the power and efficiency. Values of ln p2 =p1 , Δϕ=Δx and J n at maximum power density and maximum effiency can be derived from the equations for the compressor (eq. 25) and generator curves (eq. 26).
J. Catalano et al. / Journal of Membrane Science 478 (2015) 37–48
43
Table 2 Collection of equations that characterise electrokinetic energy conversion in the gas phase based on measurements of ion conductivity (σ), gas permeability (κ n ) and gas transport number (t n ). Left side is with respect to compressor operation, while right side is with respect to power generation. Generator operation with constant Δp.
Compressor operation with constant j Compressor curve
lnðp2 =p1 Þ ¼ Δx
tn κ1n J n þ Fκ j E
J comp ¼ 0 -J n ¼ 0 lnðp2 =p1 Þ ¼ jFκtnn (max compression ratio) Δx Jn ¼ 0 Δϕ j Δx J ¼ 0 ¼ σ ð1þ β Þ n
Generator curve J comp ¼ 1 -Δp ¼ 0 J n;Δp ¼ 0 ¼ jtFn (max flow) Δϕ Δx
¼ σj Δp ¼ 0
Δϕ Δx ¼
lnðp2 =p1 Þ κ n Δx
Figure-of-merit β ¼
ηcomp;MaxP ¼ 2ð2βþ βÞ
2
pffiffiffiffiffiffiffiffi 1þβ Maximum efficiency is obtained at J comp; Max η ¼ pffiffiffiffiffiffiffiffi 1þ 1þβ p ffiffiffiffiffiffiffi ffi pffiffiffiffiffiffiffiffi 2 β 1þβ 1þβ1 P comp; Max η ¼ jσ pffiffiffiffiffiffiffiffi2 ηcomp; Max η ¼ pffiffiffiffiffiffiffiffi 1þ
P el ¼ βκ n RT
1þβ
1þβþ1
4
2 i ln p2 =p1 j2 h jgen j2gen ¼ 1 þ β 1 J comp σ Δx ð28bÞ
P comp ¼ P el
J 2comp þJ comp J comp þ 1 þ β =β
ð29Þ
For generation the efficiency is:
ηgen ¼
P el ¼ P comp
j2gen þ jgen
jgen þ ð1=β Þ
J n;Δϕ ¼ 0 ¼
Δx
(max current)
lnðp2 =p1 Þ κ n ð1 þ βÞ Δx
n RT
jFΔx lnðp2 =p1 Þ
Maximum power density is obtained at jgen;MaxP ¼ 1=2 2 lnðp2 =p1 Þ ηgen;MaxP ¼ 2ð2βþ βÞ P el;MaxP ¼ βκn4RT Δx pffiffiffiffiffiffiffiffi 1þβ1 Maximum efficiency is obtained at jgen; Max η ¼ β pffiffiffiffiffiffiffiffi ffi 2 2 þ 2pffiffiffiffiffiffiffi 1þβ1 1 þ β β þ β2 κ n RT lnðp2 =p1 Þ ηgen; Max η ¼ pffiffiffiffiffiffiffiffi P el; Max η ¼
The power density that can be extracted from the membrane shows a maximum whenever J comp ¼ 1=2 or jgen ¼ 1=2 for compression and generation, respectively. Due to the electro-viscous effect, the amount of work/power (density) inserted into the system depends on gas flux and current density. For instance with compression, P el decreases with J comp and is a consequence of a decreased external compression load (see Fig. 2) that goes to zero at J comp ¼ 1. For power generation P comp increases with jgen and is a consequence of decreasing external electrical load that goes to zero at jgen ¼ 1 The efficiency for compression is
ηcomp ¼
jgen ¼ 1 -Δϕ ¼ 0 lnðp2 =p1 Þ
jΔϕ ¼ 0 ¼ σ tnFRT
2 RT t n σ F 2 κn
Dimensionless current density jgen ¼ σt
Maximum power density is obtained at J comp;MaxP ¼ 1=2 P comp;MaxP ¼ βj 4σ
lnðp2 =p1 Þ σj Δx
jgen ¼ 0 -j ¼ 0 Δϕ t n RT lnðp2 =p1 Þ (open circuited potential) Δx j ¼ 0 ¼ F Δx J n;j ¼ 0 ¼
Dimensionless gas flow J comp ¼ JjtnnF
tnFRT
ð30Þ
These curves coincide with the ones for energy conversion in the liquid state (Eqs. (12) and (13)). Maximum efficiency can be found frompstandard mathematical pffiffiffiffiffiffiffiffiffiffiffi methods and is found to be ffiffiffiffiffiffiffiffiffiffiffi 1 þ β þ 1 where β ¼ ðRT=F 2 Þðt 2n σ =κ n Þ is the η¼ 1þβ1 = figure-of-merit for electrokinetic compression or power generation in p the gasphase. The condition for maximum p efficiency is pffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffi J comp ¼ 1 þ β= 1 þ 1 þ β and jgen ¼ 1 þ β 1 =β for compression and power generation, respectively. Table 2 summarises the results of the compressor, generator curves, power density and efficiency. Some of the results that are of particular relevance with respect to dimensioning and applications are the equations for the maximum compression ratio, power density and the condition for maximum efficiency.
4. Discussion The equations for efficiency and power density follow from standard non-equilibrium thermodynamics and are generally valid for reversible electrodes. In real electrokinetic energy conversion systems, electrode overpotentials and polarisation phenomena at the electrode and membrane surfaces may decrease the conversion efficiency and power density. The equations for the efficiency and
Δx
β
1þβþ1
power density are still valid, but the effective potential rather than the potential difference at reversible conditions will enter the equations. Alternatively, the transport properties should be measured in a way that includes the effect of overpotentials, whereby the equations for the efficiency and power density are fully valid. 4.1. Condition for maximum for efficiency in power generation From an experimental point of view it is often an advantage to have a theoretical prediction of the conditions for maximum efficiency. As an example an electrokinetic generator with liquid reservoirs (Fig. 2, left) is chosen. The generator operates with constant pressure difference and the maximum efficiency is obtained by varying the magnitude of the external electrical load (RL) inducing the question of the magnitude of RL which maximises the efficiency. From Fig. 2 it is seen that the current density 1 1 in the generator is j ¼ σ sys ðΔϕ=ΔxÞ, where σ sys ¼ σ internal þ σ L 1 is the combined reciprocal electrical conductivity from the membrane and the external load. The product between the streaming potential coefficient and the pressure difference is the internal voltage of the generator Δϕ=Δx ¼ νðΔp=ΔxÞ and hence j ¼ σ sys νðΔp=ΔxÞ. From Table1pitffiffiffiffiffiffiffiffiffiffiffi is seen that maximum efficiency is obtained at jgen; Max η ¼ 1 þ β 1 =β ¼ j= σ internal νΔp=Δx , where the subscript internal has been added to the conductivity in order to distinquish between the two conductivities. By combining these two expression it is found that pffiffiffiffiffiffiffiffiffiffi σ internal =σ L ¼ RL =Rinternal ¼ β=ð 1 þ β−1Þ−1, and in the case with e. g. β ¼ 1 the external resistance has to be about 1.4 times the internal resistance in order to obtain maximum efficiency. On the other hand it is well-known that maximum power output is obtained from RL ¼ Rinternal , which is stated as jgen;MaxP ¼ 1=2 in Table 1. The corresponding power densities and efficiencies can be found from the relevant equations in the table. 4.2. Energy conversion with liquid reservoirs As an example we consider the electrokinetic transport properties (σ , ν, κ H and β) of Nafions 117 (in the following referred to as Nafion), which has been thoroughly investigated [2,24]. A relatively high conversion efficiency of the order 20% was found in 0.03 M LiCl solutions. The transport properties of Nafion can be used to estimate the pump characteristics of electrokinetic pumping. Here it is desirable to have a large current density in order to maximise the power density. At a high electrolyte concentration (40.3 M) a current density of 100 mA cm 2 is reasonable without
44
J. Catalano et al. / Journal of Membrane Science 478 (2015) 37–48
very large concentration polarisation effects. From the equations for electrokinetic energy conversion with liquids in Table 1 some of the important pump characteristics like the area power density and pressures are calculated and given in Table 3 for a 180 μm Nafion 117 membrane. A current density (100 mA cm 2) can generate a maximum pressure difference of 226 bar, whereas the pressure differences at maximum power density and efficiency are 113 bar and 92 bar respectively. Experimentally, values up to about 25 bar with j 30 mA cm 2 to 180 mA cm 2 have been observed [37,38]. However, in these experiments the pressure difference was far from being fully developed, presumably due to mechanical limitations of the test cell. A maximum pressure difference of the order 200 bar is not unrealistic and is in agreement with predictions from the transport properties. In many applications, high power densities and efficiencies are key parameters. In the case of a 180 μm Nafion membrane high performance can only be obtained if the system is operated at high pressure differences (4100 bar). If a more moderate operating pressure difference is needed (e.g. 10 bar), it can be obtained by (i) shifting to a more extreme point at the pump curve at the expense of a an efficiency that drops below 5%, (ii) reduction of the current density to 10 mA cm 2, which is at the expense of a power density that is reduced by a factor of 100 or (iii) reduction of the membrane thickness, which maintains the efficiency and increases the power density. Due to their relatively high pressure difference at maximum efficiency and power density, Nafion membranes are potential targets for applications in HPLC (high pressure liquid chromatography). The pumps in HPLC operate at very high pressure and with relatively small volume flow. 4.3. Electrokinetic hydrogen gas compression One of the main results in the present article is the derivation of conditions for maximum efficiency and power density for electrokinetic energy conversion with gas phase reservoirs. It can be used for dimensioning of and search for efficient membrane materials for electrochemical gas compression. The figure-of-merit for gas phase electrokinetic energy conversion, β ¼ ðRT=F 2 Þðt 2n σ =κ n Þ, resembles that for conversion with liquid phases, with the squared streaming potential term (strength of the electrokinetic coupling) and the conductivity and permeability term in the nominator and denominator, respectively. The figure-of-merit increases with the gas transport number (t n ) squared and this can be used for screening membrane materials with high efficiency. t n can be determined from t n ¼ νn ðF=RTÞ (Eqs. (17) and (19)) by using the experimental values of the streaming potential coefficient (νn ). In the case of pure H2 on both sides of the membrane t n ¼ 1=2. The efficiency and power density for energy conversion in Nafion membranes is discussed as practical examples for the case with pure H2 and mixed gas phases on both sides of the membrane. Nafion is well characterised with respect to both H2 permeation and proton conductivity for a range of different humidity levels. Table 4 gives a compilation of literature transport
data for Nafion surrounded by H2 in the gas phase [39]. Estimates of κ n are made using the definition of κ n (Eq. (22)) and with values of J n from literature permeation rates [39] assuming p2 ¼ 10 bar and p1 ¼ 1 bar. This most likely leads to overestimated values of κ n . The interesting result here is that for Nafion in pure H2 β ¼ 74 corresponding to η ¼ 79%. Even if κ n is a factor 10 larger, which results in β ¼ 7:4, the maximum efficiency still has a high value of 49% and experimental data unambiguously show that there is a large coupling in the case of Nafion and pure H2. This fact is not surprising, in view of the electrode reaction, which fully couples to proton transport to the electric current density. It is only the ohmic loss of ion transport and the impact of H2 diffusion on the reservoir pressures, which can decrease the efficiency. Despite the very high efficiency, there are two major drawbacks, the first being a very low power density and the second is that overpotentials in the dissociation/recombination of H2 at the electrodes have not been included in the calculation of the efficiency. With respect to the power density two different examples of gas compression with varying membrane thickness (180 mm and 50 mm corresponding to commercial Nafion 117 and 212 respectively) and with a compression ratio of 210 between the low and high pressure side have been included. It is seen that highest obtainable power density is about 11 mW cm 2, for a 50 mm membrane with a compression ratio of 210. Although the current density is relatively high (180 mA cm 2) the low power density comes from the relatively low potential drop of 78 mV across the membrane. If this technology is to be used for large scale H2 compression/liquefaction, large membrane areas are needed, adding significantly to the total system costs. In the literature [29], experimental value of a steady state compression ratio of 150 was found for j ¼ 200 mA cm 2, in quite good agreement with the estimated values in Table 4. Furthermore Grigoriev et al. also made an experimental plot of the cell voltage as function of compression (Fig. 6 in ref. [29]) and found an approximate logarithmic relationship. This is expected from ln p2 =p1 =Δx J ¼ 0 ¼ jt n =ðFκ n Þ (Table 2) if the ratio κn F=ðt n ΔxÞ is n independent of pressure indicating that κ n too is somewhat independent of pressure. Still, in the efficiency calculation, overpotentials in the electrode materials have not been included. Depending on the definition, the overpotential accounts for both activation losses and reaction kinetic effects at the electrode. In the present study a crude model for the overpotential is applied. A constant overpotential of 50 mV per electrode that is independent of current density is assumed, this is most likely a low and optimistic value [25,29]. Nonetheless, the main point is that the overpotential has a large impact and the maximum efficiency decreases from 79% to 35% (last column Table 4) because the overpotential is of the same magnitude as the potential difference across the membrane. 4.4. Electrokinetic gas compression – cooling applications Alternatively, electrokinetic effects with gas phase reservoirs on each side of the membrane can be used in cooling cycles or power generation from heat sources. In the present discussion
Table 3 Calculation of pump characteristics for Nafion 117 membrane with a thickness of 180 μm based on literature data of the transport data [2,24] in the left column. The power density is calculated from the relevant equations in Table 1 and assuming a current density of 100 mA cm 2. Transport coefficient
Value
Current density: j¼100 mA cm 2
Power density (mW cm 2)
Pressure (bar)
κ H (m2 s 1 Pa-1) ν (μV bar 1) σ (S m 1) β η (%)
3.1 10 17 390 2.36 1.2 20
Pump head (zero flow) Maximum power density Maximum efficiency
– 2.2 2.1
226 113 92
J. Catalano et al. / Journal of Membrane Science 478 (2015) 37–48
45
the power density and current density were calculated for 50 mm membranes where the compression ratio is 10. The results are shown in Table 4. The main result is that lower current densities are needed whenever t n increases in order to maintain a compression ratio of 10. At the same time the potential difference across the membrane increases because the amount of work that is done per unit charge increases with t n and the combined effect is a constant power density. Despite the higher compression ratio for pure H2 it can be seen from Table 4 that the power density is 1–2 orders of magnitude larger in the example with MeOH. In fact, it can be shown from the equations in Table 2 that P comp p κ n for a constant β and constant compression ratio. For practical applications this is very important since this shows that high power densities could be found in membrane/solvents/gas systems with relatively large gas permeability. An equally important point is that the total system efficiency is larger whenever a refrigerant is used along with the gas e.g. MeOH together with H2. As already noted, the potential difference across the membranes is larger in the case where refrigerants are used and it increases with t n , as can be seen from Table 4. When electrode overpotentials are included in the calculation of the efficiency it is seen that the effect is relatively small and that the system efficiency still is of the order 60%, much larger than that with pure H2. This is explained by the larger electrical potential difference across the membrane that is a consequence of the larger amount of work that is done per unit charge whenever proton couple to MeOH. Domestic refrigerators, about 100 W cooling effect, typically have a coefficient of performance (COP) of about 1.5, depending on the cooling temperature. If we assume that most losses are related to the mechanical compressor, the compressor efficiency is below 30%. As seen from Table 4, compressor efficiencies with Nafion above 50% are realistic. Furthermore, a cooling capacity of 100 W can be obtained by a membrane area of approximately 20 cm 20 cm. This is a relatively small area and it further underlines the feasibility of this technology for domestic cooling applications. Furthermore, the cooling capacity is simply regulated by the magnitude of the DC current in an electrokinetic gas compressor and can be kept at the maximum efficiency of the compressor curve at all times. Mechanical compressors need frequency regulation and permanent magnet motors to regulate the capacity, which adds significantly to the costs. The weaknesses of the above calculations are of course a lack of comprehensive experimental data. Nevertheless, the main result is that very high efficiencies and power densities can be predicted
only gas compression for cooling applications is considered and it is implicitly assumed that it is a condensation cycle type similar to that in mechanical refrigeration systems. The use of pure H2 in cooling cycles is not realistic since it would require extremely high compression ratios in order to liquefy H2 and the power density is at the same time quite low. Alternatively, a system containing H2 along with a refrigerant will be analysed and it will be shown that higher system efficiencies and power densities can be obtained and with compression ratios that are much more adequate for cooling applications. From the literature it is well established that the hydration shell increases with K þ , Na þ , Li þ , (H þ ) as observed by the largest water transport number in e.g Nafion [40]. Water transport numbers t w 2–3 that are found experimentally for H þ in Nafion [41] are significantly lower than those for any of the other monovalent ions [40]. This discrepancy is explained by the extra transport tunnelling mechanism of protons in water, that reduces the drag of the water molecules, thereby also t w . In electrokinetic energy conversion the tunnelling mechanism decreases the efficiency and is thus not a wanted feature. Alternatively, in a solvent such as methanol (MeOH) protons cannot tunnel, and MeOH may therefore be a suitable choice. High values of the coupling between Li þ and MeOH have been observed experimentally with liquid reservoirs [42], corresponding to t MeOH 21. To the best of our knowledge there are no data for H þ and MeOH in Nafion. However, the transport number is expected to be of the same magnitude. This is supported by the large decrease of σ H þ ;water 10 S m 1 /σ H þ ;MeOH 1 S m 1 compared to σ Li þ ;water 1:3 S m 1 /σ Li þ ;MeOH 0:6 S m 1 in Nafion going from 100% water to 100% MeOH [43,44]. The large magnitude of σ H þ ;water =σ H þ ;MeOH indicates that tunnelling does not contribute in MeOH while the similar magnitude of σ H þ ;MeOH and σ Li þ ;MeOH suggests that the dragging effect of the solvent is similar for both ions. There are no experimental data available for t n in a system with Nafion and gas phase MeOH and H2. However, assuming that the coupling is similar to that with liquid reservoirs a value of t n from 5–20 is reasonable and is used in the following. The gas permeability of H2O and MeOH is several orders of magnitude larger than for H2 [45] and κ n is estimated to be 5 10 6 mol m 1 s 1. At the same time σ will decrease in MeOH and with increasing tn/solvent drag. Experimental values of σ H þ ;MeOH are found to be of the order 1 S m-1 [44]. However, in the calculations the values are chosen to be in the range 0.3–4.8 S m 1 under the constraint that β ¼ 32, in order to make the different cases comparable. With these transport properties
Table 4 Calculation of efficiency, current density, potential difference and power density for different membrane thicknesses and for different compression ratios. Calculations are based on the equations in Table 2. Gas transport properties of Nafion with pure H2 have been estimated from Ref. [39]. In the case of Nafion with H2 and MeOH the transport properties have been estimated based on a fixed gas permeability that is approximately 550 times larger [45] than in the case for pure H2 and under the constraint that the figure-of-merit is 32. Gas permeability (κ n ) mol m 1 s 1 Nafion with pure H2
9 10 9
5 10 6 Nafion with H2 and MeOH
Compression ratio at maximum efficiency
μm
Efficiency including 100 mV electrode overpotential %
78 78
180 50
3.1 11.2
210 210
35 35
360 720 1440
50
152 152 152
10
55 62 66
mA cm 2
79
50 180
70
600 300 150
Current density (j)
S m1
1/2
10
74
5 10 20
4.8 1.2 0.3
32
Ion conductivity (σ)
Area power density at maximum efficiency mW cm 2
Potential difference across membrane (Δϕ) mV
Figure- Efficiency (η) ofmerit (β) %
Gas transport number (t n )
Membrane thickness (Δx)
46
J. Catalano et al. / Journal of Membrane Science 478 (2015) 37–48
for gas compression in cooling cycles if a refrigerant is included in the system. This does in particular hold when overpotentials are included in the model. The calculations should be regarded as a guide to obtain high efficiencies and power densities. In order to identify compressor systems that have practical relevance, aimed research in membrane properties along with a suitable refrigerant must be addressed. In the above examples MeOH and Nafion has been chosen only because it is well characterised in the literature and this data can be used for estimating the performance based on the theory developed in the present article. Nonetheless, enhanced electrokinetic properties is most likely not found in Nafion membranes and similar. These membranes are so dense that the solvent (water) uptake per unit charge is about 20–40 water molecules per SO3 group depending on the solvent. This ultimately limits the gas/water transport number and thereby the streaming potential coefficient. Larger streaming potential coefficients can be found for membranes with larger solvent uptake and larger pore diameters [2]. With respect to the refrigerant it must be a polar solvent, otherwise the coupling to the ions will be diminished; it must otherwise be chemically compatible with the membrane and it should have a boiling point in the range around room temperature. Again MeOH has been chosen in the example because of the available literature data. Alternatives may be organic polar solvent/gases and an even more obvious choice may be ammonia that is a known refrigerant with excellent properties.
5. Summary and conclusions The main goal of the present study was to formulate a theoretical framework based on non-equilibrium thermodynamics for describing the electrokinetic effects in membranes with both gas phase and liquid reservoirs. In particular for gas phase reservoirs this has not yet be done in the literature. The theoretical description can have large impact on practical applications of membrane electrokinetic effects, e.g. electrochemical gas compression/cooling, micro pumps and power generation. Based on experimental transport data for dense ion conducting membranes (Nafion) we predict high efficiency for electrochemical gas compression/cooling in practical systems if mixtures of a transport gas (e.g. H2) and a refrigerant (e.g. MeOH) are used.
Nomenclature Faraday constant, C mol-1 phenomenological transport coefficients in appropriate units j electrical current density, A m-2 Jv volume flux, m s 1 Jw molar water flux, mol s 1 m 2 J n ; J g ; J H2 Molar gas flux, mol s 1 m 2 J pump dimensionless volume flow, dimensionless jgen dimensionless current density, dimensionless J comp dimensionless gas flow, dimensionless p; Δp pressure (difference), Pa P hyd ; P comp ; P el Hydraulic, compressor and electric power density, W m3 R gas constant, J K 1 mol 1 RL external load (electric or hydraulic), Ω or Pa s m 3 t s , t w , t n transference coefficient of salt, water and gas, dimensionless T, ΔT temperature (difference), K Vi molar volume of species i, m3 mol 1 x, Δx distance parallel to fluxes, membrane thickness, m β electrokinetic figure-of-merit, dimensionless
F Lij
η κH , κn μi ν; νn σ σe ϕ; Δϕ
first law efficiency, dimensionless Hydraulic permeability and gas permeability coefficient, m2 s 1 Pa 1 and mol s 1 m 1 chemical potential of component i, J mol 1 Streaming potential with liquid and gas phase reservoirs, V Pa 1 and V electrical/ionic conductance, S m 1 volumetric entropy production rate, J m 3 s 1 K 1 electric potential (difference), V
Acknowledgements AB wishes to thank the Villum Foundation (grant number: VKR022356) and The Aarhus University Research Foundation for funding. SK is grateful to the RENERGI project no. 164466, and the FRIENERGI project no. 197598/V30, both of the Norwegian Research Council.
Appendix A Consider a membrane surrounded by two gas reservoirs (Fig. 1). The reservoirs contain a mixture of two gases. One of them is H2; the other may be water or methanol. The membrane is saturated with solvent (e.g. H2O or MeOH), and electrodes (Pt) are deposited on to its two external surfaces. The catalyst is connected via an external electrical circuit. The arrangement allows H2 to dissociate on the membrane surface (left side). Hydrogen is transported through the membrane as H þ , and it recombines at the right hand side to form gas molecules again. The other gas in the reservoir can be transported through the membrane alone and along with the H þ . This scenario is likely if the gas is well dissolved in the membrane and is polarisable, like MeOH is. It is assumed that the system is isothermal. The entropy production (σ e ) in such a system, showing various electrokinetic effects has in the outset three independent terms: 1 ∂μH2 ;T 1 ∂μg;T 1 ∂ϕ þ Jg þj ðA1Þ σ e ¼ J H2 T ∂x T ∂x T ∂x Where J H2 and J g are molar gas fluxes, and μH2 ;T and μg;T are the chemical potentials at constant temperature of the H2 and gas, respectively, and ϕ is the electrical potential. Effects from electrode activation and reaction kinetics at the electrodes are not specifically included here, however, in the discussion section the effects are included empirically in some examples. The chemical potential depends on the concentration and the pressure through the relation 1 ∂μi;T 1 ∂ μ i ðc Þ ∂p þV i ðA2Þ ¼ T ∂x T ∂x ∂x where V i is the molar volume of species i. In the following it is assumed that there are no electrolyte concentration gradients, and only pressure gradients are considered and eq. (A1) can be rewritten as: 1 ∂p 1 ∂p 1 ∂ϕ þ Jg V g þj ðA3Þ σ e ¼ J H2 V H2 T ∂x T ∂x T ∂x If ideal gasses are considered, the molar volumes are the same V g ¼ V H2 ¼ RT p and 1 RT ∂p 1 ∂ϕ þj ; ðA4Þ σe ¼ Jn T p ∂x T ∂x where J n ¼ J H2 þJ g is the total molar gas flux. The entropy production in an infinitesimal area section is σ e dx. The total entropy production is obtained by integrating between the states that are
J. Catalano et al. / Journal of Membrane Science 478 (2015) 37–48
bounding the membrane. This assumes that we can anywhere find a state in equilibrium with the real membrane. The integration is carried out over the appropriate pressure and potential and the outcome is: Z Δϕ Z Δx Z p2 1 RT 1 11 dp þ dϕ σ e; tot ¼ σ e dx ¼ Jn j T p ∂x T ∂x 0 p1 0 1 RT p 1 Δϕ ln 2 ðA5Þ ¼ Jn þj T Δx T Δx p1 Where integration has been along a path of solutions in equilibrium with the membrane at any location. This equation is valid under isothermal conditions and it assumes that the gas compression/expansion is reversible. In this case the total internal energy is constant and W ¼ -q, where W is the work that the gas performs on the surroundings and q is the heat involved in the process. In order to maintain isothermal conditions under the compression/expansion process, it must be coupled to a heat reservoir. In the present case it will be the membrane and its coupling to the electrochemical cell and surroundings. Thus, the fulfilment of the isothermal condition depends strongly on the design of the mechanical set-up. The transport equations that follow from Eq. (A5) and describe electrokinetic gas compression and expansion are: 1 RT p 1 Δϕ J n ¼ Lnn ln 2 ðA6Þ þ Lnϕ T Δx T Δx p1 1 RT p 1 Δϕ ln 2 þLϕϕ j ¼ Lϕn T Δx T Δx p1
ðA7Þ
where Lij are phenomenological transport coefficients (with appropriate units) and the cross-term coupling coefficients are linked through the Onsager relation (Lnϕ ¼ Lϕn ). The coupling coefficient in these equations measures the number of vapour molecules transported with the protons through the membrane. Eqs. (A6) and (A7) are derived under ideal gas assumption and that the composition of H2 and the second gas is the same in both reservoirs. If at the same time it is assumed that J H2 is fully coupled to the electrical current: J H2 ¼
1j 2F
ðA8Þ
With the definition of J n , the total entropy production (Eq. (A5)) can be rewritten as: 1 RT p 1 Δϕ 1 RT p ln 2 ln 2 þj ðA9Þ σ e; tot ¼ J g T Δx T ∂x T 2F Δx p1 p1 For which the two following transport equations hold: 1 RT p 1 Δϕ 1 RT p ln 2 ln 2 þ Lgϕ J g ¼ Lgg T Δx T Δx T 2F Δx p1 p1 1 RT p 1 Δϕ 1 RT p ln 2 ln 2 þ Lϕϕ j ¼ Lϕg T Δx T Δx T 2F Δx p1 p1
ðA10Þ
ðA11Þ
These equations hold under the assumption in Eq. (A8), that is the membrane is impermeable to H2 and H2 does not diffuse through the membrane. The last term T1 2FRTΔx ln pp2 can be interpreted 1 as a correction to the electric potential due to volume transfer by the electrode reaction. In the present article we analyse examples with H2O/MeOH, that have gas permeabilities about three orders of magnitude higher than H2 in Nafion membranes [39,45]. In this case H2 permeation is negligible and Eqs. (A10) and (A11) should hold. Still in the main part of the article we use Eqs. (A6) and (A7) since these equations also apply for the case with pure H2 in both reservoirs. Eqs. (A10) and (A11) leads to an infinite electrokinetic figure-of-merit (100 % conversion efficiency). By comparison of Eqs. (A6) and (A7) to Eqs. (A10)
47
and (A11), we see that they have the same form and will result in slightly different definitions of the observable transport properties (Eqs. (17)–(22)).
References [1] F.A. Morrison, J.F. Osterle, Electrokinetic energy conversion in ultrafine capillaries, J. Chem. Phys. 43 (1965) 2111–2115. [2] B.S. Kilsgaard, S. Haldrup, J. Catalano, A. Bentien, High figure of merit for electrokinetic energy conversion in Nafion membranes, J. Power Sources 247 (2014) 235–242. [3] H.S. Chen, T.N. Cong, W. Yang, C.Q. Tan, Y.L. Li, Y.L. Ding, Progress in electrical energy storage system: a critical review, Prog. Nat. Sci. 19 (2009) 291–312. [4] B. Saleh, G. Koglbauer, M. Wendland, J. Fischer, Working fluids for lowtemperature organic Rankine cycles, Energy 32 (2007) 1210–1221. [5] R.J. Gross, J.F. Osterle, Membrane transport charachteristic of ultrafine capillaries, J. Chem. Phys. 49 (1968) 228–234. [6] D. Burgreen, F.R. Nakache, Electrokientic flow in ultrafine capillary slits, J. Phys. Chem. 68 (1964) 1084–1091. [7] D. Burgreen, F.R. Nakache, Efficiency of pumping and power generation in ultrafine electrokinetic systems, J. Appl. Mech. 32 (1965) 675–679. [8] J. Yang, F.Z. Lu, L.W. Kostiuk, D.Y. Kwok, Electrokinetic microchannel battery by means of electrokinetic and microfluidic phenomena, J. Micromech. Microeng. 13 (2003) 963–970. [9] H. Daiguji, P.D. Yang, A.J. Szeri, A. Majumdar, Electrochemomechanical energy conversion in nanofluidic channels, Nano Lett. 4 (2004) 2315–2321. [10] F.H.J. van der Heyden, D.J. Bonthuis, D. Stein, C. Meyer, C. Dekker, Electrokinetic energy conversion efficiency in nanofluidic channels, Nano Lett. 6 (2006) 2232–2237. [11] X.C. Xuan, D.Q. Li, Thermodynamic analysis of electrokinetic energy conversion, J. Power Sources 156 (2006) 677–684. [12] Y. Ren, D. Stein, Slip-enhanced electrokinetic energy conversion in nanofluidic channels, Nanotechnology 19 (2008) 195707. [13] C. Davidson, X. Xuan, Effects of stern layer conductance on electrokinetic energy conversion in nanofluidic channels, Electrophoresis 29 (2008) 1125–1130. [14] C. Davidson, X. Xuan, Electrokinetic energy conversion in slip nanochannels, J. Power Sources 179 (2008) 297–300. [15] R. Chein, H. Chen, C. Liao, Analysis of electro-kinetic pumping efficiency through finite-length nano-scale surface-charged capillaries, J. Electroanal. Chem. 630 (2009) 1–9. [16] R. Chein, C. Liao, H. Chen, Electrokinetic energy conversion efficiency analysis using nanoscale finite-length surface-charged capillaries, J. Power Sources 187 (2009) 461–470. [17] R. Chein, K. Tsai, L. Yeh, Analysis of effect of electrolyte types on electrokinetic energy conversion in nanoscale capillaries, Electrophoresis 31 (2010) 535–545. [18] C.-C. Chang, R.-J. Yang, Electrokinetic energy conversion in micrometer-length nanofluidic channels, Microfluidics Nanofluidics 9 (2010) 225–241. [19] R. Chein, C. Liao, H. Chen, Thermodynamic reversibility analysis of electrokinetic energy conversion in nanofluidic channels, Nanoscale Microscale Thermophys. Eng. 14 (2010) 75–94. [20] M. Wang, Q. Kang, Electrochemomechanical energy conversion efficiency in silica nanochannels, Microfluidics Nanofluidics 9 (2010) 181–190. [21] C.-C. Chang, R.-J. Yang, Electrokinetic energy conversion efficiency in ionselective nanopores, Appl. Phys. Lett. 99 (2011) 083102. [22] D. Gillespie, High energy conversion efficiency in nanofluidic channels, Nano Lett. 12 (2012) 1410–1416. [23] J. Hoffmann, D. Gillespie, Ion correlations in nanofluidic channels: effects of ion size, valence, and concentration on voltage- and pressure-driven currents, Langmuir 29 (2013) 1303–1317. [24] J. Catalano, A. Bentien, Influence of temperature on the electrokinetic properties and power generation efficiency of Nafions 117 membranes, J. Power Sources 262 (2014) 192–200. [25] R. Strobel, M. Oszcipok, M. Fasil, B. Rohland, L. Jorissen, J. Garche, The compression of hydrogen in an electrochemical cell based on a PE fuel cell design, J. Power Sources 105 (2002) 208–215. [26] K. Onda, K. Ichihara, M. Nagahama, Y. Minamoto, T. Araki, Separation and compression characteristics of hydrogen by use of proton exchange membrane, J. Power Sources 164 (2007) 1–8. [27] C. Casati, P. Longhi, L. Zanderighi, F. Bianchi, Some fundamental aspects in electrochemical hydrogen purification/compression, J. Power Sources 180 (2008) 103–113. [28] K.A. Perry, G.A. Eisman, B.C. Benicewicz, Electrochemical hydrogen pumping using a high-temperature polybenzimidazole (PBI) membrane, J. Power Sources 177 (2008) 478–484. [29] S.A. Grigoriev, I.G. Shtatniy, P. Millet, V.I. Porembsky, V.N. Fateev, Description and characterization of an electrochemical hydrogen compressor/concentrator based on solid polymer electrolyte technology, Int. J. Hydrog. Energy 36 (2011) 4148–4155. [30] B. Bahar, W. Grot, W. Parmelee, Electrochemical compression system. WO2013106718-A1, 2013 Jul 18. [31] D.P. Bloomfield, Electrochemically driven heat pump. US4593534-A, 1986 10 Jun.
48
J. Catalano et al. / Journal of Membrane Science 478 (2015) 37–48
[32] R.B. Trusch, Joule-Thomson refrigeration cycle employing a reversible drive electrochemical compressor. S5024060-A; EP463985-A; EP463985-A3; EP463985-B1; DE69100584-E 463985-B463981, 1991 18 Jun. [33] A. Bentien, T. Okada, S. Kjelstrup, Evaluation of nanoporous polymer membranes for electrokinetic energy conversion in power applications, J. Phys. Chem. C 117 (2013) 1582–1588. [34] K.D. Sandbakk, A. Bentien, S. Kjelstrup, Thermoelectric effects in ion conducting membranes and perspectives for thermoelectric energy conversion, J. Membr. Sci. 434 (2013) 10–17. [35] S. Kjelstrup, D. Bedeaux, Non-equilibrium Thermodynamics of Heterogeneous Systems, World Scientific, Singapore, 2008. [36] J.F. Osterle, Unified Treatment of Thermodynamics of Steady-State Energy Conversion, Appl. Sci. Res. Sect. A—Mechan. Heat Chem. Eng. Math. Methods 12 (1964) 425–434. [37] C.E. Evans, R.D. Noble, C.A. Koval, A nonmechanical, membrane-based liquid pressurization system, Ind. Eng. Chem. Res. 45 (2006) 472–475. [38] K.A. Jeerage, R.D. Noble, C.A. Koval, Investigation of an aqueous lithium iodide/ triiodide electrolyte for dual-chamber electrochemical actuators, Sens. Actuators B—Chem. 125 (2007) 180–188.
[39] T. Sakai, H. Takenaka, N. Wakabayashi, Y. Kawami, E. Torikai, Gas permeation properties of solid polymer electrolyte (SPE) membranes, J. Electrochem. Soc. 132 (1985) 1328–1332. [40] T. Okada, G. Xie, O. Gorseth, S. Kjelstrup, N. Nakamura, T. Arimura, Ion and water transport characteristics of Nafion membranes as electrolytes, Electrochim. Acta 43 (1998) 3741–3747. [41] S.K. Ratkje, M. Ottøy, R. Halseid, M. Strømgard, Thermoeletric power relevant for the solid-polymer-electrolyte fuel-cell, J. Membr. Sci. 107 (1995) 219–228. [42] V.M. Barragan, J.P.G. Villaluenga, M.R. Godino, M.A. Izquierdo-Gil, C. RuizBauza, B. Seoane, Swelling and electro-osmotic properties of cation-exchange membranes with different structures in methanol-water media, J. Power Sources 185 (2008) 822–827. [43] L. Chaabane, L. Dammak, V.V. Nikonenko, G. Bulvestre, B. Auclair, The influence of absorbed methanol on the conductivity and on the microstructure of ion-exchange membranes, J. Membr. Sci. 298 (2007) 126–135. [44] C.A. Edmondson, P.E. Stallworth, M.C. Wintersgill, J.J. Fontanella, Y. Dai, S.G. Greenbaum, Electrical conductivity and NMR studies of methanol/water mixtures in Nafion membranes, Electrochim. Acta 43 (1998) 1295–1299. [45] R. Struis, S. Stucki, M. Wiedorn, A membrane reactor for methanol synthesis, J. Membr. Sci. 113 (1996) 93–100.
