(4.136)

where U may depend on wind speed in analogy to (4.118). The collector gain is now given by $E_c^{gain}=E_c^{sw}+E_c^{lw}+E_c^{sens},$ where the first term is proportional to A, while the two last terms are proportional to A_a. Assuming now a fluid flow $J_m^c$ through the absorber, the expression for determination of the amount of energy extracted becomes [cf. (4.128)]

$\begin{array}{ll}{J}_m^c{C}_p^c(T_{c,out}-T_{c,in})\hfill & =E_c^{gain}(\overline{T}c)-A_{a}C\prime \mathrm{d}\overline{T}c/\mathrm{d}t\hfill \\ \hfill & =E_c^{sw}+E_c^{lw}(\overline{T}c)+E_c^{sens}(\overline{T}c)-A_{a}C\prime \mathrm{d}\overline{T}c/\mathrm{d}t),\hfill \end{array}$ (4.137)

(4.137)

where A_aC′ is the heat capacity of the absorber and the relation between $\overline{T_c}$ and T_c,out may be of the form (4.129) or (4.37). Inserting (4.114), with use of (4.134), (4.135), and (4.136), into (4.137), it is apparent that the main additional parameters specifying the focusing system are the area concentration ratio,

$X=A/A_a,$ (4.138)

(4.138)

and the energy flux concentration ratio,

$C^{flux}=A{t}^{SW}(\mathrm{\Omega })/A_i,$ (4.139)

(4.139)

where t^SW is given in (4.134) and A_i is the area of the actual image (which may be different from A_a). Knowledge of A_i is important for determining the proper size of the absorber, but it does not appear in the temperature equation (4.137), except indirectly through the calculation of the transmission function t^SW.

For a stalled collector, $J_m^c=0$ and $\mathrm{d}{\overline{\mathit{T}}}_c/\mathrm{d}t=0$ give the maximum absorber temperature T_c,max. If this temperature is high, and the wind speed is low, the convective heat loss (4.138) is negligible in comparison with (4.135), yielding

$\sigma ({T_{c,max}}^4-{T_c}^4)=XD{\alpha }^{sw}{\epsilon }^{lw},$

where D is the incident short-wavelength flux in the direction of the Sun [cf. (4.114)], assuming the device accepts only the flux from this direction and assuming perfect transmission to the absorber (t^SW=1). With the same conditions, the best performance of an operating collector is obtained from (4.139), assuming a steady-state situation. The left-hand side of (4.139) is the amount of energy extracted from the collector per unit of time, E^extr:

$E^{extr}=A{\alpha }^{SW}D-A_a{\epsilon }^{lw}\sigma ({{\overline{T}}^4}_c-T_c^4),$ (4.140)

(4.140)

and the efficiency corresponding to the idealized assumptions

${\eta }_c^{ideal}=\frac{E^{extr}}{AD}={\alpha }^{SW}\left(1-\frac{{\epsilon }^{lw}\sigma }{X{\alpha }^{SW}D}({\overline{T}}_c^4-T_e^4)\right).$ (4.141)

(4.141)

This relation is illustrated in Fig. 4.90, assuming D=800 W m⁻², T_e=300 K, and α^SW=0.9, independent of absorber temperature. It is clear that the use of a selective absorber surface (large ratio α^SW/ε^lw) allows a higher efficiency to be obtained at the same temperature and that the efficiency drops rapidly with decreasing light intensity D. If the total short-wavelength flux on the collector is substantially different from the direct component, the actual efficiency relative to the incident flux $E_{s,\gamma }^{SW}$ of (3.15),

${\eta }_c=E^{extr}/E_{s,\gamma }^{SW},$

may be far less than the “ideal” efficiency (4.141).

4.4.4.2 Solar-thermal electricity generation

Instead of using photovoltaic or photo-electrochemical devices, one can convert solar radiation into electric energy in two steps: first, by converting radiation to heat, as described in section 4.4.3, or more likely by use of concentrating devices, as described above; and second, converting heat into electricity, using one of the methods described in sections 4.1.2 and 4.2.2. (Two-step conversion using chemical energy rather than heat as the intermediate energy form is also possible, for instance, by means of the photogalvanic conversion scheme briefly mentioned in section 4.5.2.) Examples of the layout of systems capable of performing solar heat to electric power conversion using thermodynamic cycles are given below, with discussion of efficiencies believed attainable.

4.4.4.3 Photo-thermoelectric converter design

A two-step conversion device may be of the general form shown in Fig. 4.91. The solar collector may be the flat-plate type or it may perform a certain measure of concentration, requiring partial or full tracking of the Sun. The second step is performed by a thermodynamic engine cycle, for example, a Rankine cycle with expansion through a turbine, as indicated in the figure. Owing to the vaporization of the engine-cycle working fluid in the heat exchanger linking the two conversion steps, the heat exchange performance does not follow the simple description leading to (4.37). The temperature behavior in the heat exchanger is more like the type shown in Fig. 4.92. The fluid in the collector circuit experiences a uniform temperature decrease as a function of the distance traveled through the heat exchanger, from x₁ to x₂. The working fluid entering the heat exchanger at x₂ is heated to the boiling point, after which further heat exchange is used to evaporate the working fluid (and perhaps superheat the gas), so that the temperature curve becomes nearly flat after a certain point.

The thermodynamic engine cycle (the right-hand side of Fig. 4.91) can be described by the method outlined in sections 4.1.1 and 4.1.2, with the efficiency of heat to electricity conversion given by (4.19) [limited by the ideal Carnot process efficiency (4.4)],

${\eta }_w=E^{electric}/{J_{Q,in}}^w\le (T_{w,in}-T_{w,out})/T_{w,in},$

where $E^{electric}=-J_q{F}_q$ is the electric power output [cf. (4.14)] and

$J_{Q,in}^w=J_m^c{C}_p^c(T_{c,out}-T_{c,in})$

is the rate of heat transfer from the collector circuit to the working-fluid circuit in the heat exchanger. The right-hand side is given by (4.137) for a concentrating solar collector and by (4.128) and (4.124) for a flat-plate collector. The overall conversion efficiency is the product of the efficiency η_c of the collector system and η_w,

$\eta ={\eta }_c{\eta }_w.$

Determination of the four temperatures T_c,in, T_c,out, T_w,in, and T_w,out (cf. Fig. 4.91) requires an equation for collector performance, for heat transfer to the collector fluid, for heat transfer in the heat exchanger, and for processes involving the working fluid of the thermodynamic cycle. If collector performance is assumed to depend only on an average collector temperature, ${\overline{T}}_c$ [e.g., given by (4.127)], and if T_w,in is considered to be roughly equal to ${\overline{T}}_c$ (Fig. 4.92 indicates that this may be a fair first-order approximation), then the upper limit for η_w is approximately

${\eta }_w\lesssim (Tc-{\overline{T}}_{w,out})/\overline{T}c.$

Taking T_w,out as 300 K (determined by the coolant flow through the condenser in Fig. 4.91), and replacing η_c by the idealized value (4.141), the overall efficiency may be estimated as shown in Fig. 4.93, as a function of ${\overline{T}}_c.$ The assumptions for ${\eta }_c^{ideal}$ are as in Fig. 4.90: no convective losses (this may be important for flat-plate collectors or collectors with a modest concentration factor) and an incident radiation flux of 800 W m⁻² reaching the absorber (for strongly focusing collectors this also has to be direct radiation). This is multiplied by ${\eta }_w^{ideal}$ (the Carnot limit) to yield ${\eta }_{total}^{ideal}$ depicted in Fig. 4.93, with the further assumptions regarding the temperature averaging and temperature drops in the heat exchanger that allow the introduction of ${\overline{T}}_c$ as the principal variable both in ${\eta }_w^{ideal}$ and in ${\eta }_c^{ideal}.$

In realistic cases, the radiation reaching the absorber of a focusing collector is perhaps half of the total incident flux $E_{s,\gamma }^{SW}$ and ${\eta }_w$ is maybe 60% of the Carnot value, i.e.,

${\eta }_{total}\approx 0.3{\eta }_{total}^{ideal}.$

This estimate may also be valid for small concentration values (or, rather, small values of the parameter Xα^sw/ε^lw characterizing the curves in Figs. 4.93 and 4.90), since the increased fraction of $E_{s,\gamma }^{SW}$ being absorbed is compensated for by high convective heat losses. Thus, the values of ${\eta }_{total}^{ideal}$ between 6% and 48%, obtained in Fig. 4.93 for suitable choices of ${\overline{T}}_c$ (which can be adjusted by altering the fluid flow rate $J_m^c$), may correspond to realistic efficiencies of 2–15%, for values of Xα^sw/ε^lw increasing from 1 to 200.

The shape of the curves in Fig. 4.93 is brought about by the increase in ${\eta }_w^{ideal}$ as a function of ${\overline{T}}_c,$ counteracted by the accelerated decrease in ${\eta }_c^{ideal}$ as a function of ${\overline{T}}_c$ which is seen in Fig. 4.90.

Photo-thermoelectric conversion systems based on flat-plate collectors have been discussed, for example, by Athey (1976); systems based on solar collectors of the type shown in Fig. 4.88d have been considered by Meinel and Meinel (1972). They estimate that a high temperature on the absorber can be obtained by use of selective surfaces, evacuated tubes (e.g., made of glass with a reflecting inner surface except for the window shown in Fig. 4.88d), and a crude Fresnel lens (see Fig. 4.89) concentrating the incoming radiation on the tube window. Molten sodium is suggested as the collector fluid. Finally, fully tracking systems based on the concept shown on the right-hand side of Fig. 4.87 (“power towers”) have been proposed by Teplyakov and Aparisi (1976) and by Hildebrandt and Vant-Hull (1977). This concept is used in several installations, the best known of which is the Odeillo plant in southern France (Trombe and Vinh, 1973), which originally aimed at achieving temperatures for metalwork that exceeded those possible at fossil-fuel furnaces.

4.4.4.4 Optical subsystem and concentrators

As indicated by the devices shown in Figs. 4.57 and 4.60, optical manipulation of incoming solar radiation is used in non-concentrating solar cells. These measures serve the purpose of reducing the reflection on the surface to below a few percent, over the wavelength interval of interest for direct and scattered solar radiation, as seen from Fig. 4.94. Their collection efficiency is far better than that shown on Fig. 4.54, exceeding 90% for wavelengths between 0.4 and 1.05 μm, and exceeding 95% in the interval 0.65–1.03 μm.

For large-factor concentration of light onto a photovoltaic cell much smaller than the aperture of the concentrator, the principles mentioned for thermal systems (discussion of Fig. 4.90) apply unchanged. Most of these concentrators are designed to focus the light onto a very small area, and thus tracking the Sun (in two directions) is essential, with the implications that scattered radiation largely cannot be used and that the expense and maintenance of non-stationary components have to be accepted.

One may think that abandoning very high concentration would allow the construction of concentrators capable of accepting light from most directions (including scattered light). However, this is not easy. Devices that accept all angles have a concentration factor of unity (no concentration), and even if the acceptance angular interval is diminished to, say, 0°–60°, which would be suitable because light from directions with a larger angle is reduced by the incident cosine factor, only a very small concentration can be obtained (examples like the design by Trombe are discussed in Meinel and Meinel, 1976).

The difficulty may be illustrated by a simple two-dimensional ray-tracing model of an (arbitrarily shaped) absorber, i.e., the PV cell, sitting inside an arbitrarily shaped concentrator (trough) with reflecting inner sides, and possibly with a cover glass at the top that could accommodate some texturing or lens structure. The one component that unfortunately is not available is a semitransparent cover that fully transmits solar radiation from one side and fully reflects the same wavelengths of radiation hitting the other side. With such a cover, nearly 100% of the radiation coming from any direction could reach the absorber, the only exception being rays being cyclically reflected into paths never reaching the absorber.

Figure 4.96 shows some ray-tracing experiments for a catenary-shaped concentrator holding a flat absorber at the bottom, with a concentration factor X=7 (ratio of absorber and aperture areas, here lines). The cover has an inverted pyramid shape. This is routinely used for flat-plate devices (see Fig. 4.57), with the purpose of extending the path of the light rays inside the semiconductor material so that absorption is more likely. For the concentrator, the purpose is only to make it more likely that some of the rays reach the absorber, by changing their direction from one that would not have led to impact onto the absorber. However, at the same time, some rays that otherwise would have reached the absorber may now become diverted. Thus, not all incident angles benefit from the concentrator, and it has to be decided whether the desired acceptance interval is broad (say, 0°–60°) or can be narrowed, e.g., to 0°–30°. As illustrated, the concentration of rays toward the center absorber is paid for by up to half of the incident rays’ being reflected upward after suffering total reflection on the lower inside of the cover glass.

The result of varying some of the parameters is shown in Fig. 4.95. An inverted pyramid angle of 0° corresponds to a flat-plate glass cover, and the figure shows that in this situation the concentration is limited to incident angles below 20°, going from a factor of four to unity. With increasing pyramid angle, the maximum concentration is moved toward higher incident angles, but its absolute value declines. The average concentration factor over the ranges of incident angles occurring in practice for a fixed collector is not even above unity, meaning that a PV cell without the concentrating device would have performed better.

Another idea is to direct the cover pyramids upward, but to avoid “stretching out” the rays again at the lower glass boundary by having the entire trough filled with the refractive substance (which might be a gel or plastic material, textured at the upper surface). This is illustrated in Figs. 4.97a–c for an X=5 concentrator. For a deep catenary-shaped collector, all rays up to 10° may be led to the absorber, but in this case (Figs 4.97a and 4.98), incident angles above 25° are not accepted. To get reasonable acceptance, pyramid angles around 30° would again have to be chosen, and Figs. 4.97a–c show that the concentration penalty in trying to extend the acceptance interval to angles above 35° (by decreasing the depth of the concentrator, see Fig. 4.99) is severe. The best choice is still the deep catenary shape, with maximum concentration between 2 and 3 and average concentration over the interesting interval of incident angles (say, to 60°) no more than about 1.5. This is better than the inverted pyramid structure in Fig. 4.96, but it is still doubtful whether the expense of the concentrator is warranted. The collector efficiency, as measured relative to total aperture area, is the ratio of absorber intensity and X, i.e., considerably less than the cell efficiency.

Generation 24 h a day at constant radiation intensity could be achieved by placing the photovoltaic device perpendicular to the direction of the Sun at the top of the Earth’s atmosphere. Use of photovoltaic energy systems is, as mentioned, common on space vehicles and also on satellites in orbit around the Earth. It has been suggested that, if a large photovoltaic installation were placed in a geosynchronous orbit, the power generated might be transmitted to a fixed location on the surface of the Earth, after having been converted to microwave frequency (see, for example, Glaser, 1977).

4.5.1.1 Reverse operation of fuel cells

A driven-cell conversion based on the dissociation of water may be accomplished by electrolysis, using components similar to those of the hydrogen–oxygen fuel cell to perform the inverse reactions. Thus, the efficiency of an ideal conversion may reach 1.20, according to (4.150), implying that the electrolysis process draws heat from the surroundings.

If a fuel cell is combined with an electrolysis unit, a regenerative system is obtained, and if hydrogen and oxygen can be stored, an energy storage system, a battery, results. The electric energy required for the electrolysis need not come from the fuel cell, but may be the result of an intermittent energy conversion process (e.g., wind turbine, solar photovoltaic cell, etc.).

Direct application of radiant energy to the electrodes of an electrochemical cell has been suggested, aiming at the achievement of the driven-cell process (e.g., dissociation of water) without having to supply electric energy. The electrodes could be made of suitable p- and n-type semiconductors, and the presence of photo-induced electron–hole excitations might modify the electrode potentials ϕ_a and ϕ_c in such a way that the driven-cell reactions become thermodynamically possible. Owing to the electrochemical losses discussed above, additional energy would have to be provided by solar radiation. Even if the radiation-induced electrode processes have a low efficiency, the overall efficiency may still be higher for photovoltaic conversion of solar energy into electricity followed by conventional electrolysis (Manassen et al., 1976). In recent years, several such photo-electrochemical devices have been constructed. Some of them are described in section 4.4.2. Direct hydrogen production by photo-electrochemical devices is discussed in Sørensen (2012).

4.5.1.2 Fuel-cell design

The idea of converting fuel into electricity by an electrode–electrolyte system originated in the 19th century (Grove, 1839). The basic principle behind a hydrogen–oxygen fuel cell is described above. The first practical applications were in powering space vehicles, starting during the 1970 s. During the late 1990 s, fuel-cell road vehicles were predicted to be ready within a decade, but it did not happen, due to problems with durability and cost. Hybrid vehicles allow a smaller (and thus less costly) fuel cell rating, but if combined with traction batteries the lowering of cost may turn out as difficult as for pure fuel cell vehicles. However, hybrids between either battery electric vehicles or fuel cell vehicles and conventional internal combustion engine vehicles may smooth the transition.

First, hybrids between conventional internal combustion engines and batteries reduce pollution in cities (they use batteries that are charged by the engine at convenient times or that are charged by off-road plug-in power supplies), while using the fuel-based engine for long highway trips will relieve demand on battery size. Second, if and when (presumably) both battery and fuel-cell technologies have come down in cost, the hybrid car could replace the internal combustion engine with a fuel cell plus electric motor, thus allowing total reliance on primary energy from hydrogen or grid electricity, which may both be renewable (see Sørensen, 2006; 2010; 2012).

Fuel technology is described in detail in Sørensen (2012). Here, only some general remarks are made about the development from early fuel-cell concepts to the present-day situation.

Originally developed for stationary power applications, phosphoric acid cells use porous carbon electrodes with a platinum catalyst and phosphoric acid as electrolyte and feed hydrogen to the negative electrode, with electrode reactions given by (4.144) and (4.145). Their operating temperature is 175–200°C, and water is continuously removed.

Alkaline cells use KOH as electrolyte and have electrode reactions of the form

$\begin{array}{l}{\mathrm{H}}_2+2{\mathrm{OH}}^{-}\to 2{\mathrm{H}}_2\mathrm{O}+2e^{-},\hfill \\ 1/2{\mathrm{O}}_2+{\mathrm{H}}_2\mathrm{O}+2e^{-}\to 2{\mathrm{OH}}^{-}.\hfill \end{array}$

Their operating temperature is 70–100°C, but specific catalysts require maintenance of fairly narrow temperature regimes. Also, the hydrogen fuel must be very pure and, notably, not contain any CO₂. Alkaline fuel cells have been used extensively on spacecraft and recently in an isolated attempt for road vehicles (Hoffmann, 1998). Their relative complexity and use of corrosive compounds requiring special care in handling make it unlikely that the cost can be reduced to levels acceptable for general-purpose use.

The third fuel cell in commercial use is the proton exchange membrane (PEM) cell. It has been developed over a fairly short period of time and is considered to hold the greatest promise for economic application in the transportation sector. It contains a solid polymer membrane sandwiched between two gas diffusion layers and electrodes. The membrane material may be polyperfluorosulfonic acid. A platinum or Pt–Ru alloy catalyst is used to break hydrogen molecules into atoms at the negative electrode, and the hydrogen atoms are then capable of penetrating the membrane and reaching the positive electrode, where they combine with oxygen to form water, again with the help of a platinum catalyst. The electrode reactions are again (4.144) and (4.145), and the operating temperature is 50–100°C (Wurster, 1997).

Figure 4.106 shows a typical layout of an individual cell. Several of them are then stacked on top of each other. This modularity implies that PEM cells can be used for applications requiring little power (1 kW or less). PEM cell stacks dominate the current wealth of demonstration projects in road transportation, portable power, and special applications. The efficiency of conversion for small systems is between 30% and 50%, but a 50 kW system has recently shown an efficiency in the laboratory of near 60%. As indicated in Fig. 4.107, an advantage of particular importance for automotive applications is the high efficiency at partial loads, which alone gives a factor of 2 improvement over current internal combustion engines.

For use in automobiles, compressed or liquefied hydrogen is limited by low energy density and safety precautions for containers, but most of all by the requirement for a new infrastructure for fueling. The hydrogen storage problem, which until recently limited fuel-cell projects to large vehicles like buses, may be solved by use of metal hydride or carbon nanofiber stores (see Chapter 5). In order to avoid having to make large changes to gasoline and diesel filling stations, methanol has been proposed as the fuel to be distributed to the vehicle fuel tank. The energy density of methanol is 4.4 kWh liter⁻¹, which is half that of gasoline. This is quite acceptable owing to methanol’s higher efficiency of conversion. Hydrogen is then produced onboard, by a methanol reformer, before being fed to the fuel cell to produce the electric power for an electric motor. The set-up is illustrated in Fig. 4.108. Prototype vehicles with this set-up have been tested (cf. Takahashi, 1998; Brown, 1998), but the reformer technology turned out to have many problems and most current vehicle fuel-cell concepts plan to use hydrogen directly, thereby also avoiding methanol emissions and the health issues connected with the methanol infrastructure.

Methanol, CH₃OH, may, however, still find use in fuel-cell development, in a direct methanol PEM fuel cell, without the extra step of reforming to H₂. PEM fuel cells that accept a mixture of methanol and water as feedstock have been developed (Bell, 1998) but still have lower energy conversion efficiency than hydrogen-fueled PEM cells. Because methanol can be produced from biomass, hydrogen may be eliminated from the energy system. On the other hand, handling of surplus production of wind or photovoltaic power might still conveniently involve hydrogen as an intermediate energy carrier, since it may have direct uses and thus may improve the system efficiency by avoiding the losses incurred in methanol production. The electric power to hydrogen conversion efficiency is about 65% in high-pressure alkaline electrolysis plants (Wagner et al., 1998; Sørensen, 2012), and the efficiency of the further hydrogen (plus CO or CO₂ and a catalyst) to methanol conversion is around 70%. Also, the efficiency of producing methanol from biomass is about 45%, whereas higher efficiencies are obtained if the feedstock is methane (natural gas) (Jensen and Sørensen, 1984; Nielsen and Sørensen, 1998).

For stationary applications, fuel cells of higher efficiency may be achieved by processes operating at higher temperatures. One line of research has been molten carbonate fuel cells, with electrode reactions

$\begin{array}{l}{\mathrm{H}}_2+{\mathrm{CO}}_3^{2-}\to {\mathrm{CO}}_2+{\mathrm{H}}_2\mathrm{O}+2e^{-},\hfill \\ 1/2{\mathrm{O}}_2+{\mathrm{CO}}_2+2e^{-}\to {\mathrm{CO}}_3^{2-}.\hfill \end{array}$

The electrolyte is a molten alkaline carbonate mixture retained within a porous aluminate matrix. The carbonate ions formed at the positive electrode travel through the matrix and combine with hydrogen at the negative electrode at an operating temperature of about 650°C. The process was originally aimed at hydrogen supplied from coal gasification or natural gas conversion. Test of a new 250-kW system is taking place at a power utility company in Bielefeld, Germany (Hoffmann, 1998). The expected conversion efficiency is about 55%, and some of the additional high-temperature heat may be utilized. Earlier experiments with molten carbon fuel cells encountered severe problems with material corrosion.

Considerable efforts are being dedicated to solid electrolyte cells. Solid oxide fuel cells (SOFC) use zirconia as the electrolyte layer to conduct oxygen ions formed at the positive electrode. Electrode reactions are

${\mathrm{H}}_2+{\mathrm{O}}^{2-}\to {\mathrm{H}}_2\mathrm{O}+2e^{-},$

$1/2{\mathrm{O}}_2+2e^{-}\to {\mathrm{O}}^{2-}.$

The reaction temperature is 700–1000°C. The lower temperatures are desirable, due to lower corrosion problems, and may be achieved by using a thin electrolyte layer (about 10 μm) of yttrium-stabilized zirconia sprayed onto the negative electrode as a ceramic powder (Kahn, 1996). A number of prototype plants (in the 100-kW size range) are in operation. Current conversion efficiency is about 55%, but it could reach 70–80% in the future (Hoffmann, 1998).

Particularly for vehicle applications of hydrogen-based technologies, efforts are needed to ensure a high level of safety in collisions and other accidents. Because of the difference between the physical properties of hydrogen fuel and the hydrocarbon fuels currently in use (higher diffusion speed, flammability, and explosivity over wider ranges of mixtures with air), there is a need for new safety-related studies, particularly where hydrogen is stored onboard in containers at pressures typically of 20–50 MPa. A few such studies have already been done (Brehm and Mayinger, 1989; see also Sørensen, 2012).

4.5.2.1 Conversion of ocean salinity gradients

As discussed in section 3.4.3 (Fig. 3.69), a salinity difference like the one existing between fresh (river) and saline (ocean) water, may be used to drive an osmotic pump, and the elevated water may in turn be used to generate electricity by an ordinary turbine (cf. section 4.3.4).

An alternative method, aiming directly at electricity production, takes advantage of the fact that the salts of saline water are ionized to a certain degree and thus may be used to derive an electrochemical cell of the type discussed in section 4.5.1 (Pattie, 1954; Weinstein and Leitz, 1976). Small amounts of power may even be generated by driving an electrolyte through small pores of, e.g., MoS₂ by a salinity or pressure gradient (Feng et al., 2016).

The electrochemical cell shown in Fig. 4.109 may be called a dialytic battery, since it is formally the same as the dialysis apparatus producing saline water with electric power input. The membrane allows one kind of ion to pass and thereby reach the corresponding electrode (in contrast to the osmotic pump in Fig. 3.69, where water could penetrate the membrane but ions could not). The free energy difference between the state with free Na⁺ and Cl⁻ ions (assuming complete ionization) and the state with no such ions is precisely the difference between saline and fresh water, calculated in section 3.4.3 and approximately given by (3.38). Assuming further that each Na⁺ ion reaching electrode A neutralizes one electron, and that correspondingly each Cl⁻ ion at electrode B gives rise to one electron (which may travel in the external circuit), then the electromotive force, i.e., the electric potential ϕ of the cell, is given in analogy to (4.146), with the number of positive ions per mole equal to $n_{{\mathrm{Na}}^{+}}$, and ϕ is related to the change in free energy (3.38), just as (4.146) is related to (4.143),

$n_e\mathcal{F}\varphi =n_{\mathrm{Na}}+\mathcal{F}\varphi =\delta G\approx 2RT{n}_{{\mathrm{Na}}^{+}},$

or

$\mathit{max}(\mathrm{\Delta }{\varphi }_{ext})=\varphi \approx 2RT/\mathcal{F}.$ (4.151)

(4.151)