## 1. Introduction

Thermoelectricity involves the direct coupling of fluxes of heat and electric charge. The coupling can refer to the way a temperature difference can produce electricity, or to the reverse, how an electric current can create a temperature difference. The first part of the thermoelectric effect, the conversion of heat to electricity, was discovered in 1821 by Thomas Seebeck [

1]. The second effect was explored in more detail by Jean Peltier, and is referred to as the Peltier effect. They are linked by the Onsager reciprocal relations (see below). Thermoelectric devices provide the only direct possibility to convert low-temperature heat sources into electric power. This property makes them potential candidates for industrial waste heat conversion. In contrast to heat engines, thermoelectric generators have no moving parts. Thermoelectric energy converters have in practice, since long been made from semiconductors [

2], and there are several ideas for their improvement, e.g., nanostructured materials [

3,

4,

5]. The ideas were proposed by Hicks and Dresselhaus already in 1993 [

6,

7]. Semiconductors are, however, often expensive or rare, and do not offer particularly large Seebeck coefficients, typically 200–300

$\mathsf{\mu}$V/K, even if colossal values have been found for peculiar conditions [

8].

It may therefore be interesting to examine the potential of other conductors. Materials such as solid state ionic conductors and electrolytes [

9], or ionic liquids [

10,

11], are relevant. Granular porous media [

12] and electrically conductive polymers [

13] also show thermoelectric effects. This review will, however, focus on a large class of materials which may be more accessible, namely ion-exchange materials. In the last several years, the possibility of using ion-exchange membranes in renewable energy technology is in the crosshairs [

14,

15,

16,

17,

18,

19].

The aim of this work is to review state-of-the-art knowledge on thermoelectric energy conversion in cells with ion-exchange membranes. The hope is to provide a basis for further explorations of their use, i.e., in reverse electrodialysis (RED) [

18,

20]. In a RED concentration cell, isothermal alternating compartments of sea water and brackish water separated by ion-selective membranes permit the production of electric power [

21]. Recent work demonstrates that a thermoelectric potential can be added to the RED concentration cell to increase the electromotive force by 10% per 20 K difference for given electrolyte conditions [

20]. The class of ion exchange materials may provide cheaper cell components than presently used and help make renewable technologies more competitive. We shall review experimental results from cells with ion-conducting membranes.

The energy conversion that takes place in these cells can be well described by non-equilibrium thermodynamics. This theory relates properties that are critical for energy conversion, and we shall decompose measurements as far as possible in terms of these properties.

The typical experimental cell for thermoelectric energy conversion, or thermocell for short, which is relevant for waste heat exploitation (below 100 °C), can be schematically written as

Salt solutions of MCl are separated by an ion-exchange membrane. The electrodes here are reversible to the chloride ion. Other electrodes are also relevant. The temperature difference is ideally across the membrane alone.

The study of such cells is not new. Already in 1956, Tyrrell [

22] described this type of cells. In 1957, Hill et al. [

23] described the potential difference in terms of irreversible thermodynamics. A year later, Ikeda and coworkers [

24,

25] reported a thermal membrane potential of 24.6

$\mathsf{\mu}$V/K with 0.1 M KCl solutions for collodion membranes. However, Tyrrell et al. [

22] had obtained a value 10 times higher with ion-exchangers.

Non-isothermal transport phenomena in charged membranes have been reported occasionally in the literature since then [

26,

27,

28,

29]. In the 1970s and 1980s, Tasaka and co-workers were central [

30,

31,

32,

33,

34,

35,

36,

37], while new researchers joined in the 1990s [

38,

39,

40,

41,

42,

43,

44,

45,

46,

47,

48,

49,

50,

51,

52]. Different aspects were studied, such as the influence of the membrane type [

40,

41,

42,

51,

52], the concentration and nature of the electrolyte solutions in contact with the membrane [

39,

42], and the membrane-transported entropy of ions [

42,

45]. Experimental techniques were refined, to resolve thermal polarization of membranes and quantify contributions from thermo-osmotic processes [

48]. Temperature effects were also an issue [

49].

Non-equilibrium thermodynamics [

28,

53,

54,

55,

56,

57] can be used to describe the conversion of thermal to electric energy. Two sets of variables are then relevant: the practical set according to Katchalsky and Curran [

53], which consists of measurable variables, and the set most often used of ionic variables. We consider it an advantage to have two equivalent paths for derivation of expressions to be used in the laboratory, but shall systematically use the practical set and compare it to the other in the end. The practical set is suited to make clear the relation between experiment and theory and give advice on experimental design.

The reader who is interested in the experimental results alone may go directly to the end of

Section 4 where we present the final equations, which enable us to explain and compare experimental results.

The main aim of this work is thus to bring out the potential of a new class of materials, the ion-exchange materials, for thermoelectric energy conversion purposes. In spite of good knowledge about thermoelectric generators in general, see, e.g., [

12,

13], properties of cells with ion-exchange membranes have scarcely been studied systematically. It is our hope that this work can provide a basis and pinpoint needs for further research.

## 2. The Cell

The typical cell membrane has transport of salt and water, see

Figure 1 for an illustration. We consider electrodes of silver and silver chloride, but other electrodes (calomel) have often been used. The membrane is cation- or anion-selective. We can imagine that waste heat sources are used to maintain a temperature difference across the membrane. In a saline power plant, say the reverse electrodialysis plant, there is already a concentration difference across the membrane [

21]. A pressure difference can also arise, but this possibility will be neglected for now.

The transport processes that we consider take place along the horizontal axis of the cell. This will be referred to as the x-axis. The system is normally stirred and therefore homogeneous in the y-z plane for any fixed x-coordinate. All membrane fluxes can then be given by the scalar x-component of the vectorial flux. The membrane surfaces can be assumed to be in local equilibrium. We shall mainly examine emf-experiments, which are carried out in the limit of very small current densities (open circuit potential measurements). The electrodes are connected to a potentiometer via Cu wires. The potentiometer is at room temperature ${T}_{0}$, while the electrodes have different temperatures, T or $T+\Delta T$, like their thermostated electrolyte solutions, respectively. We are seeking an expression for the emf of the cell in terms of properties that can be measured.

The

emf of the cell in

Figure 1,

$\Delta \varphi $, represents the ideal electric work done that can be done by the cell, and can be found by adding contributions along the circuit, from the leads that connect the electrodes to the potentiometer (two contributions giving

${\Delta}_{\mathrm{ext}}\varphi $, from the left and right electrodes (giving

${\Delta}_{\mathrm{el}}$) and from the membrane (

${\Delta}_{\mathrm{m}}\varphi $). The

emf is thus

Subscripts on the symbol $\Delta $ indicate the origin of the contributions; from the connecting leads (“ext”), from the electrodes a and c (“el”), and from the membrane (“m”). The electrolyte solutions are stirred, so they are isothermal and fully mixed. Therefore, they do not contribute to the emf, ${\Delta}_{\mathrm{aq}}\varphi =0$.

## 3. The Electromotive Force of the Ag|AgCl-Cell

We derive the expression for the measured electromotive force (emf), $\Delta \varphi $, of the cell, in terms of measurable properties of the cell. The cell has nine distinct phases—two connecting leads from electrodes to a potentiometer, anode- and cathode-surfaces, two bulk electrolyte solutions, a membrane, and two interfaces between the membrane and the solutions. Out of these nine phases, eight can be considered pairwise physically equivalent, differing only in parameter values. We consider aqueous solutions that are uniform and in local equilibrium with the membrane on each of the membrane sides.

#### 3.1. Connecting Leads

The wires connecting the Ag|AgCl electrodes to the potentiometer conduct heat as well as charge. The entropy production per unit volume,

${\sigma}_{\mathrm{ext}}$, is [

55]

Here

${J}_{q}^{\prime}$ is the measurable heat flux, and

j is the electric current density. The contribution to the overall

emf is measured under reversible conditions (when

${\sigma}_{\mathrm{ext}}=0$). The

emf-contribution depends on the temperature difference as

where we have applied the Onsager relation between transport coefficients, see [

56] for details. The definition is in accordance with Haase [

58] and with Goupil et al. [

59]. The heat transported reversibly with the electric current is the transported entropy of the charge carrier,

${S}_{{\mathrm{e}}^{-}}^{*}$. For all practical purposes, it is constant with temperature. We integrate Equation (

3) for the a-side and the c-side, and obtain the first contribution to Equation (

1).

#### 3.2. The Electrochemical Reaction at the Interface

The electrochemical reaction takes place at the electrode-solution interfaces. The general expression for the entropy production of the anode contains terms from heat and component fluxes into and out of the electrode surface, see [

55] for details. Under reversible conditions only two contributions are effective, the term due to reaction and the electric potential jump:

where

${r}^{\mathrm{s}}$ is the chemical reaction rate, and

${\Delta}_{\mathrm{n}}G$ is the reaction Gibbs energy of the neutral components of the chemical reaction. The expression for the right-hand side electrode interface (the cathode) are similar. The reaction rate is proportional to the electric current,

${r}^{\mathrm{s}}=j/F$. The electric potential drop can be alternatively expressed by the electrochemical potential difference of the chloride ion. The

emf contribution for the anode is therefore

There is a similar contribution for the other electrode, c. Subscript (a,aq) denote that the property belongs to the interface between the solid phases a and the aqueous solution. By adding the two electrode surface contributions, respectively,

${\Delta}_{\mathrm{a},\mathrm{aq}}\varphi $ and

${\Delta}_{\mathrm{aq},\mathrm{c}}\varphi $, we obtain the next contribution to the

emf in Equation (

1)

This expression gives the contribution from the electrode reactions to the Seebeck coefficient.

#### 3.3. The Membrane

The

emf-measurements take place under reversible conditions, and one can safely assume equilibrium at the membrane solution interfaces. This makes it convenient to deal with the membrane as a discrete system [

57]. The entropy production has contributions from the membrane transport of heat, mass, and charge. In the membrane frame of reference, we have

In order to arrive at this expression, we eliminated the measurable heat flux at the 1-side using the constant energy flux through the membrane under steady-state conditions. The choice of variables is not unique, but a practical one. Four flux equations follow. The equation for the

emf is obtained setting

$j=0$ in the equation for the electric current density. The result is well established, see, e.g., [

26,

53,

56].

Concentration differences contribute to the

emf, not only through the last two terms in Equation (

9) but also through the concentration dependence of the entropies. We identify the coefficient ratios by means of Onsager relations:

With the present choice of electrodes, the transfer of electrolytes follows the cation flux. The transport number

${t}_{{\mathrm{M}}^{+}}^{\mathrm{m}}$ is the fraction of the total charge transfer across the membrane carried by the cation. The water transference coefficient

${t}_{\mathrm{w}}^{\mathrm{m}}$ is the average number of moles of water transferred (reversibly) with electric current through the membrane per Faraday of charge. The reversible contribution to the heat flux is given by the entropy flux at isothermal conditions. There are two contributions: the entropies related to the nature of charge transporters and the entropy carried along with the components.

We assume that the reversible contribution to the total entropy flux can be attributed solely to the transport of the ions

The relation that can be taken to define the transported entropies

${S}_{i}^{*}$ of the ions. The last coefficient ratio in the expression for the

emf can now be identified through its Onsager relation

The

emf-contribution from the membrane becomes

where we have defined

${\eta}_{\mathrm{S}}^{\mathrm{m}}$ as

The entropy of the electrolyte is the standard entropy minus a term that depends on the logarithm of the activity

The expression for the membrane potential depends thus on the electrolyte activity via the electrolyte entropy, but also via the last terms in Equation (

14). We distinguish now between two well defined experimental situations.

In the first situation, the electrolyte solutions are stirred, and the last terms are zero. In the second case, the temperature gradient across the membrane leads to separation of components across the membrane. At the time this has occurred (

$t=\infty $,

${J}_{\mathrm{w}}=0,{J}_{\mathrm{MCl}}=0,j=0$), there is a balance of forces. This balance is called the Soret equilibrium. We may neglect coupling between the mass fluxes and find the balance of forces expressed by

The electrolyte contribution (the two last terms of Equation (

14)) can be determined in this case. The electrolyte contribution can also be contracted [

60] using the Gibbs–Duhem relation for the electrolyte solution on the integrated form, provided the concentration difference is small:

with

m as the molality and

${M}_{\mathrm{w}}$ as the molar mass of water. By using Equation (

18) to eliminate the chemical potential of water, we have

The bracketed combination is referred to in the literature as the apparent transport number of the membrane. The chemical potential differences (taken at constant temperature) between the two sides are most often zero in thermocell measurements. They are not zero, in the Soret equilibrium state, but this state is difficult to realize in the experiment with membranes, because of the time taken to obtain the state.

#### 3.4. The Seebeck Coefficient of the Complete Cell

We consider here the experimental situation with identical electrolytes on the two sides of the membrane. The total cell emf is obtained by adding the contributions derived above.

At initial time (

$t=0$, no Soret equilibrium), the Seebeck coefficient

${\eta}_{\mathrm{S}}$ of the cell is

where

and

We combined Equations (

15) and (

16) in the last step. These relations are suited to interpret experiments. We obtain a clear separation between electrode contribution, solution dependent contributions, and membrane-dependent contributions.

## 4. The emf of the Cell with Calomel Electrodes

Most experiments reported in the literature have been done with calomel electrodes, rather than with Ag|AgCl electrodes. We need the expression for the

emf to interpret these experiments. The cell has Hg|Hg

${}_{2}$Cl

${}_{2}$ electrodes which are kept at the temperature of the potentiometer,

${T}_{0}$. A salt bridge with saturated KCl is linking the electrode chambers to the electrolyte solution. The salt bridge on the left-hand side of the cell is exposed to a temperature difference

$T-{T}_{0}$, while the salt bridge on the right-hand side is exposed to a temperature difference

$T+\Delta T-{T}_{0}$. The membrane compartments are kept at temperatures

T and

$T+\Delta T$ as before. Therefore, there is no net contribution from the two electrode reactions. There is also no net contribution to the chemical potential differences at

$t=0$. As the two electrodes are symmetrically positioned, these contributions cancel. Furthermore, the expression for the membrane contribution has the same form as for the Ag|AgCl system. When the electrolyte concentrations in membrane compartments are the same, we obtain the membrane contribution

${\eta}_{\mathrm{S}}^{\mathrm{m}}$ as defined in Equation (

15). The membrane contributions to the

emf can thus be taken from

Section 3.

In addition to the contributions discussed, there is a thermocell contribution from the two liquid junctions. This point has so far been overlooked in the experimental literature.

The entropy production of the salt bridge is

with the measurable heat flux

With arguments similar to those in

Section 3, we find the contribution to the

emf from the liquid junctions (salt bridges)

The concentration contributions of the two salt bridges cancel, so that we are left with a thermoelectric contribution only. In the interface region between the salt bridges and the electrolyte solutions of MCl, another

emf-contribution may arise due to variations in the transport numbers and chemical potentials of the salts MCl and KCl in this region. In general, these variations give rise to a bi-ionic potential that will depend on the cation M

${}^{+}$ and the concentrations. For a detailed exposition, we refer to [

55]. In the present case, we simply consider M = K and assume that the transport number

${t}_{{\mathrm{K}}^{+}}=0.5$ throughout the liquid junctions. In this case, the total thermoelectric contribution of the salt bridges to the cell

emf can be quantified as

We expect that this expression provides a good lowest order approximation to the contribution [

55]. The observed Seebeck coefficient of a cell with calomel electrodes and KCl electrolyte is

Whenever the electrolyte is different from KCl, we expect non-trivial, but not necessarily large, deviations from this expression. This complication is avoided when Ag|AgCl electrodes are used, and we therefore recommend these electrodes for measuring thermoelectric potentials.

#### Theory of Tasaka and Coworkers

Ikeda [

24] made one of the first attempts to derive the thermoelectric potential, but Tasaka et al. [

27] were first to use irreversible thermodynamics. They found the following expression for the thermoelectric potential in 1965:

with

Lakshminarayanaiah [

28] derived a similar expression. Not all details were reported, but

${a}_{\pm}$ is the mean salt activity. The

$\eta $ appearing in Equation (

29) was called the differential or pure thermo electric coefficient. In the terminology used here, (

$F\eta ={t}_{{\mathrm{M}}^{+}}^{\mathrm{m}}{S}_{{\mathrm{M}}^{+}}^{*\mathrm{m}}-{t}_{{\mathrm{Cl}}^{-}}^{\mathrm{m}}{S}_{{\mathrm{Cl}}^{-}}^{*\mathrm{m}}$). The entropy

${S}_{i}^{o}$ is the partial molar entropy of an ion, and

${S}_{0}^{o}$ is the partial molar entropy of the solvent. The superscript

o refers to the reference-state, infinitely dilute system for all components. The quantity

${\tau}_{0}$ is the reduced transport number of water (

${\tau}_{0}={t}_{\mathrm{w}}^{\mathrm{m}}/F$). The transport numbers

${t}_{\pm}$ are, in our notation, the membrane transport numbers

${t}_{{\mathrm{M}}^{+}}^{\mathrm{m}}$ and

${t}_{{\mathrm{Cl}}^{-}}^{\mathrm{m}}$. In order to see the correspondence, we switch to the notation from the previous section. With this notation, Equation (

28) becomes

We identified the electrolyte entropy

${S}_{\mathrm{MCl}}^{o}={S}_{{\mathrm{M}}^{+}}^{o}+{S}_{{\mathrm{Cl}}^{-}}^{o}$. In their derivation, Tasaka and coworkers tacitly made the assumption

${\gamma}_{+}={\gamma}_{-}$, with

${\gamma}_{i}$ being the ion activity coefficients. This allowed their identification

${S}_{-}={S}_{-}^{o}-Rln{a}_{\pm}$. Making also the identification

${S}_{\mathrm{MCl}}={S}_{\mathrm{MCl}}^{o}-2Rln{a}_{\mathrm{MCl}}$ and setting the water entropy

${S}_{\mathrm{w}}^{o}\to {S}_{\mathrm{w}}$, we find

which gives the electrostatic potential difference

$\Delta \psi $ across the membrane for a given temperature difference. From invariance of the entropy production, the relation between

$\Delta \psi $ and the

emf-contribution

$\Delta \varphi $ between chloride-reversible electrodes is

We identify here

$\Delta {\mu}_{{\mathrm{Cl}}^{-}}=-{S}_{{\mathrm{Cl}}^{-}}\Delta T$, such that the corresponding

emf-contribution is given by

This is exactly our expression in Equation (

15) for the membrane contribution to the cell

emf, when the electrolyte concentration is uniform. For the KCl electrolyte, the addition of the salt bridge contributions derived in the previous section gives Equation (

27). By writing out the activity dependence of the electrolyte entropy, we have

where the observed activity dependence of the Seebeck coefficient is explained principally by the last term. The term in

$2{t}_{{\mathrm{K}}^{+}}$ is attributed to the membrane. The

$-1$-part, however, is here correctly attributed to the KCl salt bridges. This crucial point has not been pointed out earlier, and is important for interpretation of experimental results.

The electrolyte entropy terms provide here a linear dependence of the Seebeck coefficient on the logarithm of the electrolyte activity. As the electrolyte concentration is varied, the partial molar entropy of water will also change in general. Neglecting its variation with temperature, it can be related to the electrolyte activity by means of the Gibbs–Duhem relation

with

m as the electrolyte molality and

${M}_{\mathrm{w}}$ as the molar mass of water. Since the activity and molality are related by

${a}_{\mathrm{MCl}}={\gamma}_{\mathrm{MCl}}m$, the change in

${S}_{\mathrm{w}}$ will generally not be linear in

$ln{a}_{\mathrm{MCl}}$. For small variations at low concentrations, however, we may approximate

such that

where we have again identified what is commonly known as the apparent transport number

${t}_{a}^{\mathrm{m}}$ of the membrane. We see from this approximation that water transport gives a slight change in the slope of the Seebeck coefficient vs.

$ln{a}_{\mathrm{KCl}}$. While this analysis may be useful, it may also be good to keep the two entropy terms separate, because the activity of water in the electrolyte solutions is generally well known. We will, however, use Equation (

37) to interpret the dependence of the observed Seebeck coefficient on the electrolyte activity.

## 7. The Potential for Thermoelectric Energy Conversion

For thermoelectric energy conversion, not only the Seebeck coefficient is essential. Other properties need also be considered. To bring out the potential of ion-exchange membranes for thermoelectric energy conversion, we also need to know the thermal and electric conductivity. At a mean temperature

$\overline{T}$, the three transport properties can be combined in the lumped variable

N.

where

are the electrical conductivity at constant temperature and the thermal conductivity of the short-circuited membrane, respectively, and

${d}_{m}$ is the membrane thickness. The quantity

N has a lower and an upper bound, stemming from the requirement that the determinant of transport coefficients is positive [

55].

The open circuit thermal conductivity,

${\lambda}_{j=0}$, is related to the conductivity at short-circuit conditions, as follows:

The factor

$ZT$ is the so-called figure of merit, defined first by Ioffe [

74]. It is related to

N by

Furthermore,

$ZT$ is a non-negative quantity, but now there is no upper bound. When the figure of merit is equal to zero, there is no coupling between heat and charge transfer, and the process is completely irreversible. The other extreme is a completely reversible process, where

$N\to 1$ and thus

$ZT\to \infty $. In this situation, the entropy production is (close to) zero; there is no loss of work. In practice, we may seek to maximize

N or

$ZT$, which means that we want a small thermal conductivity, in order to inhibit irreversible heat transfer, and a large ionic conductivity, in order to reduce ohmic losses. In a dynamic approach, one may choose as objective function for minimization, the total entropy production of the system. It is then possible to also include geometric variables and constraints. This has been done successfully for several unit operations [

75].

Ion-exchange membranes are widely used in many applications involving electrochemical energy conversion, such as reverse electrodialysis or fuel cells. The electrochemical process efficiency (voltage efficiency) depends mainly on the overall electric resistance of the device, which is significantly affected by the electrical conductivity of the membrane. The ionic conductivity of ion-exchange membranes has been widely studied in the literature. Temperature, type and concentration of the electrolyte, and membrane water content are parameters that strongly affect the ionic conductivity of the membrane [

76,

77,

78,

79,

80]. A typical value for ionic conductivity of Nafion membranes at room temperature is 2.3 Sm

${}^{-1}$ [

77].

Data about thermal conductivity of ion-exchange membranes are, however, scarce. For Nafion membranes values between 0.18 and 0.25 WK

${}^{-1}$m

${}^{-1}$ have been measured at 20

${}^{\circ}$C, depending on its water content [

81]. Values in the range 0.10 and 0.20 WK

${}^{-1}$m

${}^{-1}$ were also reported [

82] for Nafion membranes, depending on the temperature. The thermal conductivity is found to decrease with increasing temperature [

82,

83].

An interest in thermoelectric studies has been recently increasing because its also gives direct access to a quantity more difficult to measure, namely the Peltier heat. It is increasingly recognized that local temperature effects and profiles can affect the performance of a fuel cell [

84,

85]. The temperature profile of such a cell cannot be accurately modeled without knowledge of the (large) Peltier effects [

14]. It was probed that thermoelectric contributions to the cell potential can improve the cell performance in the otherwise isothermal reverse electrodialysis cell [

20,

63].

The importance of the thermoelectric phenomena, their inherent symmetry and impact, lead us to conclude that there are many reasons to increase this area of research and obtain a more complete picture.

## 8. Conclusions and Perspectives

We have seen above that the following can be said about the Seebeck coefficient of thermoelectric cells with ion-exchange membranes:

It typically varies between absolute values 0.4 and 1 mV/K. Major contributions to this value come from the electrode compartment (including the electrodes) and from water and ion transport in the membrane. These values are relatively large, compared to values for semiconductors. In the calomel electrode system, the sign of the coefficient is typically positive for cation-exchange membranes and negative for anion-exchange membranes. This sign change is attributed mainly to the thermoelectric potential across the salt bridges in the system, but also to water transport when the electrolyte concentration is high.

It predicts a reduction for ideal membranes with the logarithm of the electrolyte activity. This variation is experimentally validated.

It has a possible optimal value with respect to the mean temperature across the membrane.

It does not depend significantly on water content in spite of its dependence on the water transference number.

It does not depend significantly on the membrane ion exchange capacity. One explanation is that the transport number is the ratio of the ionic and the total conductivity.

It depends on membrane heterogeneity, probably because this has an impact on the transported entropies of the ions in the membrane.

It increases in absolute value with the radius of the un-hydrated counter-ion.

It may have positive contributions from a membrane pressure difference.

In order to have access to the membrane dependent terms in the Seebeck coefficients, it is advantageous to use electrodes without salt bridges, such as the Ag|AgCl-electrode.

While most of these conclusions are as expected from theory, the ones related to membrane structure and water transport cannot be predicted. They also cannot be given a simple explanation, as they are still not fully understood (see, e.g., [

60]). Renewable energy technologies [

8,

15,

18] could benefit from more systematic studies of the impact of the properties on the transported entropies and on the water transference coefficient. The trends in the true Seebeck coefficient that depends on these properties are summarized in

Table 1.

Electrode choices and electrolyte conditions, including pressure gradients [

15], can help increase the value of the Seebeck coefficient. A temperature difference can lead to a thermo-osmotic pressure. The magnitude as well as the direction of the thermo-osmotic water transfer depends on the membrane [

86]. This additional term is little investigated. The understanding of the interplay of thermoelectric and thermo-osmotic phenomena in membrane systems is still lacking and should be pursued.

Not only is the Seebeck coefficient important when it comes to the application of these cells, but the thermal and electric conductivities are as well, as they enter in the figure of merit or the entropy production. Thermal conductivities should be small to help maintain a large temperature difference across the membrane, while electric conductivities should be large to reduce ohmic losses [

75,

87]. It is a challenge to find ion-exchange membranes that answer to all demands. Finding these is, however, of importance for a large number of interesting waste heat utilization processes.